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 Showing 1 - 82 of 82 Journals sorted alphabetically Advances in Applied Mathematics       (Followers: 15) Advances in Applied Mathematics and Mechanics       (Followers: 6) Advances in Applied Mechanics       (Followers: 15) AKCE International Journal of Graphs and Combinatorics American Journal of Applied Mathematics and Statistics       (Followers: 11) American Journal of Applied Sciences       (Followers: 22) American Journal of Modeling and Optimization       (Followers: 3) Annals of Actuarial Science       (Followers: 2) Applied Mathematical Modelling       (Followers: 22) Applied Mathematics and Computation       (Followers: 31) Applied Mathematics and Mechanics       (Followers: 4) Applied Mathematics and Nonlinear Sciences Applied Mathematics and Physics       (Followers: 2) Biometrical Letters British Actuarial Journal       (Followers: 2) Bulletin of Mathematical Sciences and Applications Communication in Biomathematical Sciences       (Followers: 2) Communications in Applied and Industrial Mathematics       (Followers: 1) Communications on Applied Mathematics and Computation       (Followers: 1) Differential Geometry and its Applications       (Followers: 4) Discrete and Continuous Models and Applied Computational Science Discrete Applied Mathematics       (Followers: 10) Doğuş Üniversitesi Dergisi e-Journal of Analysis and Applied Mathematics Engineering Mathematics Letters       (Followers: 1) European Actuarial Journal Foundations and Trends® in Optimization       (Followers: 3) Frontiers in Applied Mathematics and Statistics       (Followers: 1) Fundamental Journal of Mathematics and Applications International Journal of Advances in Applied Mathematics and Modeling       (Followers: 1) International Journal of Applied Mathematics and Statistics       (Followers: 3) International Journal of Computer Mathematics : Computer Systems Theory International Journal of Data Mining, Modelling and Management       (Followers: 10) International Journal of Engineering Mathematics       (Followers: 7) International Journal of Fuzzy Systems International Journal of Swarm Intelligence       (Followers: 2) International Journal of Theoretical and Mathematical Physics       (Followers: 13) International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems       (Followers: 3) Journal of Advanced Mathematics and Applications       (Followers: 1) Journal of Advances in Mathematics and Computer Science Journal of Applied & Computational Mathematics Journal of Applied Intelligent System Journal of Applied Mathematics & Bioinformatics       (Followers: 6) Journal of Applied Mathematics and Physics       (Followers: 9) Journal of Computational Geometry       (Followers: 3) Journal of Innovative Applied Mathematics and Computational Sciences       (Followers: 6) Journal of Mathematical Sciences and Applications       (Followers: 2) Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance       (Followers: 12) Journal of Mathematics and Statistics Studies Journal of Physical Mathematics       (Followers: 2) Journal of Symbolic Logic       (Followers: 2) Letters in Biomathematics       (Followers: 1) Mathematical and Computational Applications       (Followers: 3) Mathematical Models and Computer Simulations       (Followers: 3) Mathematics and Computers in Simulation       (Followers: 3) Modeling Earth Systems and Environment       (Followers: 1) Moscow University Computational Mathematics and Cybernetics Multiscale Modeling and Simulation       (Followers: 2) Pacific Journal of Mathematics for Industry Partial Differential Equations in Applied Mathematics       (Followers: 1) Ratio Mathematica Results in Applied Mathematics       (Followers: 1) Scandinavian Actuarial Journal       (Followers: 2) SIAM Journal on Applied Dynamical Systems       (Followers: 3) SIAM Journal on Applied Mathematics       (Followers: 11) SIAM Journal on Computing       (Followers: 11) SIAM Journal on Control and Optimization       (Followers: 18) SIAM Journal on Discrete Mathematics       (Followers: 8) SIAM Journal on Financial Mathematics       (Followers: 3) SIAM Journal on Imaging Sciences       (Followers: 7) SIAM Journal on Mathematical Analysis       (Followers: 4) SIAM Journal on Matrix Analysis and Applications       (Followers: 3) SIAM Journal on Numerical Analysis       (Followers: 7) SIAM Journal on Optimization       (Followers: 12) SIAM Journal on Scientific Computing       (Followers: 16) SIAM Review       (Followers: 9) SIAM/ASA Journal on Uncertainty Quantification       (Followers: 2) Swarm Intelligence       (Followers: 3) Theory of Probability and its Applications       (Followers: 2) Uniform Distribution Theory Universal Journal of Applied Mathematics       (Followers: 2) Universal Journal of Computational Mathematics       (Followers: 3)
Similar Journals
 SIAM Journal on Mathematical AnalysisJournal Prestige (SJR): 2.431 Citation Impact (citeScore): 2Number of Followers: 4      Hybrid journal (It can contain Open Access articles) ISSN (Print) 0036-1410 - ISSN (Online) 1095-7154 Published by Society for Industrial and Applied Mathematics  [17 journals]
• Global Solutions of Wave-Klein--Gordon Systems in 2+1 Dimensional
Space-Time with Strong Couplings in Divergence Form

Authors: Senhao Duan, Yue Ma
Pages: 2691 - 2726
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 2691-2726, June 2022.
In this paper we establish the global stability on a type of totally geodesic wave maps and the Klein--Gordon--Zakharov system in $\mathbb{R}^{2+1}$. Both systems belong to a special type of wave-Klein--Gordon system which has its strong coupling terms in divergence form. We cope with the problems by constructing auxiliary systems with the shifted primitives of the original unknowns. This permits us to reveal some hidden structures about Hessian form and null form and then establish the global stability.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-05-02T07:00:00Z
DOI: 10.1137/20M1377229
Issue No: Vol. 54, No. 3 (2022)

• Exponential Convergence Towards Consensus for Non-Symmetric Linear
First-Order Systems in Finite and Infinite Dimensions

Authors: Laurent Boudin, Francesco Salvarani, Emmanuel Trélat
Pages: 2727 - 2752
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 2727-2752, June 2022.
We consider finite and infinite-dimensional first-order consensus systems with time-constant interaction coefficients. For symmetric coefficients, convergence to consensus is classically established by proving, for instance, that the usual variance is an exponentially decreasing Lyapunov function. We investigate here the convergence to consensus in the non-symmetric case: we identify a positive weight which allows us to define a weighted mean corresponding to the consensus and obtain exponential convergence towards consensus. Moreover, we compute the sharp exponential decay rate.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-05-02T07:00:00Z
DOI: 10.1137/21M1416102
Issue No: Vol. 54, No. 3 (2022)

• Inverse Boundary Problem for the Two Photon Absorption Transport Equation

Authors: Plamen Stefanov, Yimin Zhong
Pages: 2753 - 2767
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 2753-2767, June 2022.
We study the inverse boundary problem for the nonlinear two photon absorption radiative transport equation. We show that the absorption coefficients and the scattering coefficient can be uniquely determined from the albedo operator. If the scattering is absent, we do not require smallness of the incoming source, and the reconstruction of the absorption coefficients is explicit.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-05-02T07:00:00Z
DOI: 10.1137/21M1417387
Issue No: Vol. 54, No. 3 (2022)

• Leray's Backward Self-Similar Solutions to the 3D Navier--Stokes Equations
in Morrey Spaces

Authors: Yanqing Wang, Quansen Jiu, Wei Wei
Pages: 2768 - 2791
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 2768-2791, June 2022.
In this paper, it is shown that there does not exist a nontrivial Leray backward self-similar solution to the three-dimensional (3D) Navier--Stokes equations with profiles in Morrey spaces $\dot{\mathcal{M}}^{q,1}(\mathbb{R}^{3})$ provided $3/2 Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-03T07:00:00Z DOI: 10.1137/20M1346055 Issue No: Vol. 54, No. 3 (2022) • Long Time Behavior of Stochastic Nonlocal Partial Differential Equations and Wong--Zakai Approximations • Free pre-print version: Loading... Authors: Jiaohui Xu, Tomás Caraballo Pages: 2792 - 2844 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 2792-2844, June 2022. This paper is devoted to investigating the well-posedness and asymptotic behavior of a class of stochastic nonlocal partial differential equations driven by nonlinear noise. First, the existence of a weak martingale solution is established by using the Faedo--Galerkin approximation and an idea analogous to Da Prato and Zabczyk [Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992]. Second, we show the uniqueness and continuous dependence on initial values of solutions to the above stochastic nonlocal problem when there exist some variational solutions. Third, the asymptotic local stability of steady-state solutions is analyzed either when the steady-state solution of the deterministic problem is also a solution of the stochastic one or when this does not happen. Next, to study the global asymptotic behavior, namely, the existence of attracting sets of solutions, we consider an approximation of the noise given by Wong and Zakai's technique using the so called colored noise. For this model, we can use the power of the theory of random dynamical systems and prove the existence of random attractors. Eventually, particularizing in the cases of additive and multiplicative noise, it is proved that the Wong--Zakai approximation models possess random attractors which converge upper-semicontinuously to the respective random attractors of the stochastic equations driven by standard Brownian motions. This fact justifies the use of this colored noise technique to approximate the asymptotic behavior of the models with general nonlinear noises, although the convergence of attractors and solutions is still an open problem. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-03T07:00:00Z DOI: 10.1137/21M1412645 Issue No: Vol. 54, No. 3 (2022) • Local Well-Posedness for the Boltzmann Equation with Very Soft Potential and Polynomially Decaying Initial Data • Free pre-print version: Loading... Authors: Christopher Henderson, Weinan Wang Pages: 2845 - 2875 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 2845-2875, June 2022. In this paper, we address the local well-posedness of the spatially inhomogeneous noncutoff Boltzmann equation when the initial data decays polynomially in the velocity variable. We consider the case of very soft potentials$\gamma + 2s < 0$. Our main result completes the picture for local well-posedness in this decay class by removing the restriction$\gamma + 2s> -3/2$of previous works. Our approach is entirely based on the Carleman decomposition of the collision operator into a lower order term and an integro-differential operator similar to the fractional Laplacian. Interestingly, this yields a very short proof of local well-posedness when$\gamma \in (-3,0]$and$s \in (0,1/2)$in a weighted$C^1$space that we include as well. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-05T07:00:00Z DOI: 10.1137/21M1427504 Issue No: Vol. 54, No. 3 (2022) • Periodic Perturbations of a Composite Wave of Two Viscous Shocks for 1-d Full Compressible Navier--Stokes Equations • Free pre-print version: Loading... Authors: Qian Yuan, Yuan Yuan Pages: 2876 - 2905 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 2876-2905, June 2022. This paper is concerned with the asymptotic stability of a composite wave of two viscous shocks under spatially periodic perturbations for the 1-d full compressible Navier--Stokes equations. It is proved that as time increases, the solution approaches the background composite wave with a shift for each shock, where the shifts can be uniquely determined if both the periodic perturbations and strengths of two shocks are small. The key of the proof is to construct a suitable ansatz such that the antiderivative method works. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-10T07:00:00Z DOI: 10.1137/21M1421489 Issue No: Vol. 54, No. 3 (2022) • On a Relativistic BGK Model for Polyatomic Gases Near Equilibrium • Free pre-print version: Loading... Authors: Byung-Hoon Hwang, Tommaso Ruggeri, Seok-Bae Yun Pages: 2906 - 2947 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 2906-2947, June 2022. Recently, a novel relativistic polyatomic BGK model was suggested by Pennisi and Ruggeri [J. Phys. Conf. Ser., 1035 (2018), 012005] to overcome drawbacks of the Anderson--Witting model and the Marle model. In this paper, we prove the unique existence and asymptotic behavior of classical solutions to the relativistic polyatomic BGK model when the initial data is sufficiently close to a global equilibrium. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-12T07:00:00Z DOI: 10.1137/21M1404946 Issue No: Vol. 54, No. 3 (2022) • Incompressible Limit of Isentropic Navier--Stokes Equations with Ill-Prepared Data in Bounded Domains • Free pre-print version: Loading... Authors: Yaobin Ou, Lu Yang Pages: 2948 - 2989 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 2948-2989, June 2022. In this paper, we study the incompressible limit of strong solutions to the isentropic compressible Navier--Stokes equations with ill-prepared initial data and slip boundary condition in a three-dimensional bounded domain. Previous results only deal with the cases of the weak solutions or the cases without solid boundary, where the uniform estimates are much easier to be shown. We propose a new weighted energy functional to establish the uniform estimates, in particular for the time derivatives and the high-order spatial derivatives. The estimates of highest-order spatial derivatives of fast variables are subtle and crucial for the uniform bounds of solutions. The incompressible limit is shown by applying the Helmholtz decomposition, the weak convergence of the velocity, and the strong convergence of its divergence-free component. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-12T07:00:00Z DOI: 10.1137/20M1380491 Issue No: Vol. 54, No. 3 (2022) • Convergence of a Lagrangian Discretization for Barotropic Fluids and Porous Media Flow • Free pre-print version: Loading... Authors: Thomas O. Gallouët, Quentin Mérigot, Andrea Natale Pages: 2990 - 3018 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 2990-3018, June 2022. When expressed in Lagrangian variables, the equations of motion for compressible (barotropic) fluids have the structure of a classical Hamiltonian system in which the potential energy is given by the internal energy of the fluid. The dissipative counterpart of such a system coincides with the porous medium equation, which can be cast in the form of a gradient flow for the same internal energy. Motivated by these related variational structures, we propose a particle method for both problems in which the internal energy is replaced by its Moreau--Yosida regularization in the$L^2$sense, which can be efficiently computed as a semidiscrete optimal transport problem. Using a modulated energy argument which exploits the convexity of the problem in Eulerian variables, we prove quantitative convergence estimates towards smooth solutions. We verify such estimates by means of several numerical tests. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-16T07:00:00Z DOI: 10.1137/21M1422756 Issue No: Vol. 54, No. 3 (2022) • New Regularity Results for Scalar Conservation Laws, and Applications to a Source-Destination Model for Traffic Flows on Networks • Free pre-print version: Loading... Authors: Simone Dovetta, Elio Marconi, Laura V. Spinolo Pages: 3019 - 3053 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3019-3053, June 2022. We focus on entropy admissible solutions of scalar conservation laws in one space dimension and establish new regularity results with respect to time. First, we assume that the flux function$f$is strictly convex and show that, for every$ x \in \mathbb{R}$, the total variation of the composite function$f \circ u(\cdot, x)$is controlled by the total variation of the initial datum. Next, we assume that$f$is monotone and, under no convexity assumption, we show that, for every$x$, the total variation of the left and the right trace$u(\cdot, x^\pm)$is controlled by the total variation of the initial datum. We also exhibit a counterexample showing that in the first result the total variation bound does not extend to the function$u$, or equivalently that in the second result we cannot drop the monotonicity assumption. We then discuss applications to a source-destination model for traffic flows on road networks. We introduce a new approach, based on the analysis of transport equations with irregular coefficients, and, under the assumption that the network only contains so-called T-junctions, we establish existence and uniqueness results for merely bounded data in the class of solutions where the traffic is not congested. Our assumptions on the network and the traffic congestion are basically necessary to obtain well-posedness in view of a counterexample due to Bressan and Yu. We also establish stability and propagation of$BV$regularity, and this is again interesting in view of recent counterexamples. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-23T07:00:00Z DOI: 10.1137/21M1434283 Issue No: Vol. 54, No. 3 (2022) • Approximation of the Filter Equation for Multiple Timescale, Correlated, Nonlinear Systems • Free pre-print version: Loading... Authors: Ryne Beeson, Navaratnam S. Namachchivaya, Nicolas Perkowski Pages: 3054 - 3090 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3054-3090, June 2022. This paper considers the approximation of the continuous time filtering equation for the case of a multiple timescale signal (slow-intermediate and fast scales) that may have correlation between the slow-intermediate process and the observation process. The signal process is considered fully coupled, taking values in a multidimensional Euclidean space and without periodicity assumptions on coefficients. It is proved that in the weak topology, the solution of the filtering equation converges in probability to a solution of a lower dimensional averaged filtering equation in the limit of large timescale separation. The method of proof uses the perturbed test function approach (method of corrector) to handle the intermediate timescale in showing tightness and characterization of limits. The correctors are solutions of Poisson equations. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-24T07:00:00Z DOI: 10.1137/20M1379265 Issue No: Vol. 54, No. 3 (2022) • Zero-Inertia Limit: From Particle Swarm Optimization to Consensus-Based Optimization • Free pre-print version: Loading... Authors: Cristina Cipriani, Hui Huang, Jinniao Qiu Pages: 3091 - 3121 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3091-3121, June 2022. Recently a continuous description of particle swarm optimization (PSO) based on a system of stochastic differential equations was proposed by Grassi and Pareschi in [Math. Models Methods Appl. Sci., 31 (2021), pp. 1625--1657] where the authors formally showed the link between PSO and the consensus-based optimization (CBO) through the zero-inertia limit. This paper is devoted to solving this theoretical open problem proposed in [S. Grassi and L. Pareschi, Math. Methods Appl. Sci., 31 (2021), pp. 1625--1657] by providing a rigorous derivation of CBO from PSO through the limit of zero inertia, and a quantified convergence rate is obtained as well. The proofs are based on a probabilistic approach by investigating both the weak and strong convergence of the corresponding stochastic differential equations of Mckean type in the continuous path space and the results are illustrated with some numerical examples. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-24T07:00:00Z DOI: 10.1137/21M1412323 Issue No: Vol. 54, No. 3 (2022) • The Average Distance Problem with Perimeter-to-Area Ratio Penalization • Free pre-print version: Loading... Authors: Qiang Du, Xin Yang Lu, Chong Wang Pages: 3122 - 3138 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3122-3138, June 2022. In this paper we consider the functional$E_{p,\lambda}(\Omega):=\int_\Omega {\rm dist}^p(x,\partial \Omega ){\,{\operatorname{d}}} x+\lambda \frac{\mathcal{H}^1(\partial \Omega)}{\mathcal{H}^2(\Omega)}.$Here$p\geq 1$,$\lambda>0$are given parameters, the unknown$\Omega$varies among compact, convex, Hausdorff two-dimensional sets of$\mathbb{R}^2$,$\partial \Omega$denotes the boundary of$\Omega$, and${\rm dist}(x,\partial \Omega):=\inf_{y\in\partial \Omega} x-y $. The integral term$\int_\Omega {\rm dist}^p(x,\partial \Omega ){\,{\operatorname{d}}} x$quantifies the “easiness” for points in$\Omega$to reach the boundary, while$\frac{\mathcal{H}^1(\partial \Omega)}{\mathcal{H}^2(\Omega)}$is the perimeter-to-area ratio. The main aim is to prove existence and$C^{1,1}$-regularity of minimizers of${E_{p,\lambda}}$. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-24T07:00:00Z DOI: 10.1137/21M1422987 Issue No: Vol. 54, No. 3 (2022) • Continued Gravitational Collapse for Gaseous Star and Pressureless Euler--Poisson System • Free pre-print version: Loading... Authors: Feimin Huang, Yue Yao Pages: 3139 - 3160 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3139-3160, June 2022. The gravitational collapse of an isolated self-gravitating gaseous star for$\gamma$-law pressure$p(\rho)=\rho^\gamma$($10$. Moreover, it is proved that there exists a class of spherically symmetric solutions of a gaseous star, which formulates a continued gravitational collapse in finite time, based on the background of the pressureless solutions if$\chi_1(r)< v^*(r)$for all$r\in[0,1]$. It is noted that$\chi_1(r)$could be positive; that is, the star might expand initially but finally collapse. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-24T07:00:00Z DOI: 10.1137/21M1450902 Issue No: Vol. 54, No. 3 (2022) • Global Regularity for a Nonlocal PDE Describing Evolution of Polynomial Roots Under Differentiation • Free pre-print version: Loading... Authors: Alexander Kiselev, Changhui Tan Pages: 3161 - 3191 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3161-3191, June 2022. In this paper, we analyze a nonlocal nonlinear partial differential equation formally derived by Steinerberger [Proc. Amer. Math. Soc., 147 (2019), pp. 4733--4744] to model dynamics of roots of polynomials under differentiation. This partial differential equation is critical and bears striking resemblance to hydrodynamic models used to describe collective behavior of agents (such as birds, fish, or robots) in mathematical biology. We consider a periodic setting and show global regularity and exponential in time convergence to uniform density for solutions corresponding to strictly positive smooth initial data. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-24T07:00:00Z DOI: 10.1137/21M1422859 Issue No: Vol. 54, No. 3 (2022) • Global Well-Posedness of Classical Solutions to the Cauchy Problem of Two-Dimensional Barotropic Compressible Navier--Stokes System with Vacuum and Large Initial Data • Free pre-print version: Loading... Authors: Xiangdi Huang, Jing Li Pages: 3192 - 3214 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3192-3214, June 2022. The Cauchy problem for the barotropic compressible Navier--Stokes equations on the whole two-dimensional space with vacuum as far field density is considered. When the shear viscosity is a positive constant and the bulk one is a power function of density with the power bigger than four-thirds, the global existence and uniqueness of strong and classical solutions is established. It should be remarked that there are no restrictions on the size of the data. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-26T07:00:00Z DOI: 10.1137/21M1440943 Issue No: Vol. 54, No. 3 (2022) • Weak-Strong Uniqueness for Maxwell--Stefan Systems • Free pre-print version: Loading... Authors: Xiaokai Huo, Ansgar Jüngel, Athanasios E. Tzavaras Pages: 3215 - 3252 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3215-3252, June 2022. The weak-strong uniqueness for Maxwell--Stefan systems and some generalized systems is proved. The corresponding parabolic cross-diffusion equations are considered in a bounded domain with no-flux boundary conditions. The key points of the proofs are various inequalities for the relative entropy associated with the systems and the analysis of the spectrum of a quadratic form capturing the frictional dissipation. The latter task is complicated by the singular nature of the diffusion matrix. This difficulty is addressed by proving its positive definiteness on a subspace and using the Bott--Duffin matrix inverse. The generalized Maxwell--Stefan systems are shown to cover several known cross-diffusion systems for the description of tumor growth and physical vapor deposition processes. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-26T07:00:00Z DOI: 10.1137/21M145210X Issue No: Vol. 54, No. 3 (2022) • The Defocusing Energy-Supercritical Nonlinear Schrödinger Equation in High Dimensions • Free pre-print version: Loading... Authors: Jing Li, Kuijie Li Pages: 3253 - 3274 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3253-3274, June 2022. This work is a continuation of the study on the energy-supercritical nonlinear Schrödinger equation following the previous work of Killip and Visan [Comm. Partial Differential Equations, 35 (2010), pp. 945--987], where the equation in high dimensions ($d\ge 5$) with nonlinearities$ u ^pu$for a certain range of$p$was considered. In this paper, we cover almost all of the cases in which$p$belongs to the local existence range. This improvement is based on the delicate analysis on the regularity of the almost periodic solutions. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-05-26T07:00:00Z DOI: 10.1137/21M1456443 Issue No: Vol. 54, No. 3 (2022) • On the Placement of an Obstacle so as to Optimize the Dirichlet Heat Content • Free pre-print version: Loading... Authors: Liangpan Li Pages: 3275 - 3291 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3275-3291, June 2022. We prove that, among all doubly connected domains bounded by two spheres of given radii, the Dirichlet heat content at any fixed time achieves its minimum when the spheres are concentric. This is shown to be a special case of a more general theorem concerning the optimal placement of a convex obstacle inside some larger domain so as to maximize or minimize the Dirichlet heat content. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-06-02T07:00:00Z DOI: 10.1137/21M1433411 Issue No: Vol. 54, No. 3 (2022) • The Cahn--Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity • Free pre-print version: Loading... Authors: Pierluigi Colli, Takeshi Fukao, Luca Scarpa Pages: 3292 - 3315 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3292-3315, June 2022. An asymptotic analysis for a system with an equation and a dynamic boundary condition of Cahn--Hilliard type is carried out as the coefficient of the surface diffusion acting on the phase variable tends to 0, thus obtaining a forward-backward dynamic boundary condition at the limit. This is done in a very general setting, with nonlinear terms admitting maximal monotone graphs both in the bulk and on the boundary. The two graphs are related by a growth condition with the boundary graph that dominates the other one. It turns out that in the limiting procedure the solution of the problem loses some regularity and the limit equation has to be properly interpreted in the sense of a subdifferential inclusion. However, the limit problem is still well-posed since a continuous dependence estimate can be proved. Moreover, in the case when the two graphs exhibit the same growth, it is shown that the solution enjoys more regularity and the boundary condition holds almost everywhere. An error estimate can also be shown for a suitable order of the diffusion parameter. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-06-02T07:00:00Z DOI: 10.1137/21M142441X Issue No: Vol. 54, No. 3 (2022) • Regularity of Boltzmann Equation with Cercignani--Lampis Boundary in Convex Domain • Free pre-print version: Loading... Authors: Hongxu Chen Pages: 3316 - 3378 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3316-3378, June 2022. The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the regularity of both dynamical and steady Boltzmann equations in a strictly convex domain with the Cercignani--Lampis (C-L) boundary condition. The C-L boundary condition describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We construct a local weighted$C^1$dynamical solution using repeated interaction through the characteristic. When we assume small fluctuation to the wall temperature and accommodation coefficients, we construct weighted$C^1$steady solution. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-06-02T07:00:00Z DOI: 10.1137/21M1421635 Issue No: Vol. 54, No. 3 (2022) • The Calderón Problem for the Fractional Wave Equation: Uniqueness and Optimal Stability • Free pre-print version: Loading... Authors: Pu-Zhao Kow, Yi-Hsuan Lin, Jenn-Nan Wang Pages: 3379 - 3419 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3379-3419, June 2022. We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial dimension$n\in \mathbb{N}$. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-06-06T07:00:00Z DOI: 10.1137/21M1444941 Issue No: Vol. 54, No. 3 (2022) • Invisibility Cloaking and Transformation Optics for Three Dimensional Manifolds and Applications in Cosmology • Free pre-print version: Loading... Authors: Tracey Balehowsky, Matti Lassas, Pekka Pankka, Ville Sirviö Pages: 3420 - 3456 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3420-3456, June 2022. We consider how transformation optics and invisibility cloaking can be used to construct models in subsets$\mathbb R^3$with a varying metric, where the time-harmonic waves for a given angular wavenumber$k$are equivalent to the waves in some closed orientable manifold. Our model can be used in two ways. First, the model makes it possible to do mathematical analysis on a compact 3-manifold$M$by smoothly embedding the set$M\setminus L$, where$L$is a finite union of closed curves, to an open subset of$\mathbb R^3$. Second, the model could in principle be physically implemented using a device built from metamaterials. In particular the measurements in the metamaterial device given by the Helmholtz source-to-solution operator are equivalent to Helmholtz source-to-solution measurements in a universe given by$(\mathbb{R}_+\times M, -dt^2 +g)$, where$(M,g)$is a closed, orientable,$C^\infty$-smooth, 3-dimensional Riemannian manifold. Thus the obtained device could be used to simulate cosmological models using metamaterial devices. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-06-07T07:00:00Z DOI: 10.1137/21M1431096 Issue No: Vol. 54, No. 3 (2022) • On Bounded Two-Dimensional Globally Dissipative Euler Flows • Free pre-print version: Loading... Authors: Björn Gebhard, József J. Kolumbán Pages: 3457 - 3479 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3457-3479, June 2022. We examine the two-dimensional Euler equations including the local energy (in)equality as a differential inclusion and show that the associated relaxation essentially reduces to the known relaxation for the Euler equations considered without local energy (im)balance. Concerning bounded solutions we provide a sufficient criterion for a globally dissipative subsolution to induce infinitely many globally dissipative solutions having the same initial data, pressure, and dissipation measure as the subsolution. The criterion can easily be verified in the case of a flat vortex sheet giving rise to the Kelvin--Helmholtz instability. As another application we show that there exists initial data for which associated globally dissipative solutions realize every dissipation measure from an open set in${\mathcal C}^0(\mathbb{T}^2\times[0,T])$. In fact the set of such initial data is dense in the space of solenoidal$L^2(\mathbb{T}^2;\mathbb{R}^2)$vector fields. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-06-07T07:00:00Z DOI: 10.1137/21M1454675 Issue No: Vol. 54, No. 3 (2022) • Boundary Layer Solution of the Boltzmann Equation for Diffusive Reflection Boundary Conditions in Half-Space • Free pre-print version: Loading... Authors: Feimin Huang, Yong Wang Pages: 3480 - 3534 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3480-3534, June 2022. We study the steady Boltzmann equation in half-space, which arises in the Knudsen boundary layer problem, with diffusive reflection boundary conditions. Under certain admissible conditions, we establish the existence of a boundary layer solution for both linear and nonlinear Boltzmann equations in half-space with diffusive reflection boundary condition in$L^\infty_{x,v}$when the far-field Mach number of the Maxwellian is zero. The continuity and the spacial decay of the solution are obtained. The uniqueness is established under some constraint conditions. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-06-13T07:00:00Z DOI: 10.1137/21M142945X Issue No: Vol. 54, No. 3 (2022) • Convergence of Large Population Games to Mean Field Games with Interaction Through the Controls • Free pre-print version: Loading... Authors: Mathieu Laurière, Ludovic Tangpi Pages: 3535 - 3574 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3535-3574, June 2022. This work considers stochastic differential games with a large number of players, whose costs and dynamics interact through the empirical distribution of both their states and their controls. We develop a new framework to prove convergence of finite-player games to the asymptotic mean field game. Our approach is based on the concept of propagation of chaos for forward and backward weakly interacting particles which we investigate by stochastic analysis methods, and which appear to be of independent interest. These propagation of chaos arguments allow us to derive moment and concentration bounds for the convergence of Nash equilibria. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-06-13T07:00:00Z DOI: 10.1137/22M1469328 Issue No: Vol. 54, No. 3 (2022) • An Analytical Study of Flatness and Intermittency through Riemann's Nondifferentiable Functions • Free pre-print version: Loading... Authors: Daniel Eceizabarrena, Victor VilaDa Rocha Pages: 3575 - 3608 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3575-3608, June 2022. In the study of turbulence, intermittency is a measure of how much Kolmogorov's theory of 1941 deviates from experimental measurements. It is quantified with the flatness of the velocity of the fluid, usually based on structure functions in physical space. However, it can also be defined with Fourier high-pass filters. Experimental and numerical simulations suggest that the two approaches do not always give the same results. Our purpose is to compare them from the analytical point of view of functions. We do that by studying generalizations of Riemann's nondifferentiable function, yielding computations that are related to some classical problems in Fourier analysis. The conclusion is that the result strongly depends on regularity. To visualize this, we establish an analogy between these generalizations and the influence of viscosity in turbulent flows. This article is motivated by the mathematical works on the multifractal formalism and the discovery of Riemann's nondifferentiable function as a trajectory of polygonal vortex filaments. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-06-16T07:00:00Z DOI: 10.1137/21M1411512 Issue No: Vol. 54, No. 3 (2022) • Gradient Estimates for Stokes and Navier--Stokes Systems with Piecewise DMO Coefficients • Free pre-print version: Loading... Authors: Jongkeun Choi, Hongjie Dong, Longjuan Xu Pages: 3609 - 3635 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3609-3635, June 2022. We study stationary Stokes systems in divergence form with piecewise Dini mean oscillation (DMO) coefficients and data in a bounded domain containing a finite number of subdomains with$C^{1,{Dini}}$boundaries. We prove that if$(u, p)$is a weak solution of the system, then$(Du, p)$is bounded and piecewise continuous. The corresponding results for stationary Navier--Stokes systems are also established, from which the Lipschitz regularity of the stationary$H^1$-weak solution in dimensions$d=2,3,4$is obtained. Our results can be applied to stationary Stokes systems and Navier--Stokes systems with the second-order term$\operatorname{div} (\tau \mathcal{S}u)$, where$\mathcal{S}u=\frac{1}{2}(Du+(Du)^{\top})$is the strain tensor and$\tau$is a positive piecewise DMO scalar function. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-06-16T07:00:00Z DOI: 10.1137/21M1423518 Issue No: Vol. 54, No. 3 (2022) • Symmetry Properties of Minimizers of a Perturbed Dirichlet Energy with a Boundary Penalization • Free pre-print version: Loading... Authors: Giovanni Di Fratta, Antonin Monteil, Valeriy Slastikov Pages: 3636 - 3653 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3636-3653, June 2022. We consider$\mathbb{S}^2$-valued maps on a domain$\Omega\subset\mathbb{R}^N$minimizing a perturbation of the Dirichlet energy with vertical penalization in$\Omega$and horizontal penalization on$\partial\Omega$. We first show the global minimality of universal constant configurations in a specific range of the physical parameters using a Poincaré-type inequality. Then we prove that any energy minimizer takes its values into a fixed half-meridian of the sphere$\mathbb{S}^2$and deduce uniqueness of minimizers up to the action of the appropriate symmetry group. We also prove a comparison principle for minimizers with different penalizations. Finally, we apply these results to a problem on a ball and show radial symmetry and monotonicity of minimizers. In dimension$N=2$our results can be applied to the Oseen--Frank energy for nematic liquid crystals and the micromagnetic energy in a thin-film regime. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-06-21T07:00:00Z DOI: 10.1137/21M143011X Issue No: Vol. 54, No. 3 (2022) • ThreeFold Weyl Points for the Schrödinger Operator with Periodic Potentials • Free pre-print version: Loading... Authors: Haimo Guo, Meirong Zhang, Yi Zhu Pages: 3654 - 3695 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3654-3695, June 2022. Weyl points are degenerate points on the spectral bands at which energy bands intersect conically. They are the origins of many novel physical phenomena and have attracted much attention recently. In this paper, we investigate the existence of such points in the spectrum of the three-dimensional Schrödinger operator$H = - \Delta +V(x)$with$V(x)$being in a large class of periodic potentials. Specifically, we give very general conditions on the potentials which ensure the existence of threefold Weyl points on the associated energy bands. Different from two-dimensional honeycomb structures which possess Dirac points where two adjacent band surfaces touch each other conically, the threefold Weyl points are conically intersection points of two energy bands with an extra band sandwiched in between. To ensure the threefold and three-dimensional conical structures, more delicate, new symmetries are required. As a consequence, new techniques combining more symmetries are used to justify the existence of such conical points under the conditions proposed. This paper provides a comprehensive proof of such threefold Weyl points. In particular, the role of each symmetry endowed to the potential is carefully analyzed. Our proof extends the analysis on the conical spectral points to a higher dimension and higher multiplicities. We also provide some numerical simulations on typical potentials to demonstrate our analysis. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-06-21T07:00:00Z DOI: 10.1137/21M1410464 Issue No: Vol. 54, No. 3 (2022) • Normalized Solutions for Lower Critical Choquard Equations with Critical Sobolev Perturbation • Free pre-print version: Loading... Authors: Shuai Yao, Haibo Chen, D. Rădulescu, Juntao Sun Pages: 3696 - 3723 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3696-3723, June 2022. We study normalized solutions for the following Choquard equations with lower critical exponent and a local perturbation$ -\Delta u+\lambda u=\gamma (I_{\alpha }\ast u ^{\frac{\alpha }{N}+1}) u ^{\frac{\alpha }{N}-1}u+\mu u ^{q-2}u \quad \text{in}\quad \mathbb{R}^{N}, \int_{\mathbb{R}^{N}} u ^{2}dx=c^{2},$where$\gamma ,\,\mu,\,c$are given positive numbers and$2
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-06-23T07:00:00Z
DOI: 10.1137/21M1463136
Issue No: Vol. 54, No. 3 (2022)

• Global Existence of Weak Solutions to the Compressible Navier--Stokes
Equations with Temperature-Depending Viscosity Coefficients

Authors: Guodong Wang, Bijun Zuo
Pages: 3724 - 3756
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, Page 3724-3756, June 2022.
This paper is devoted to the global existence of weak solutions to the three-dimensional compressible Navier--Stokes equations with heat-conducting effects in a bounded domain. The viscosity and the heat conductivity coefficients are assumed to be functions of the temperature, and the shear viscosity coefficient may vanish as the temperature goes to zero. The proof is to apply the Galerkin method to a suitable approximate system with several parameters and obtain uniform estimates for the approximate solutions. The key ingredient in obtaining the required estimates is to apply De Giorgi's iteration to the modified temperature equation, from which we can get a lower bound for the temperature not depending on the artificial viscosity coefficient introduced in the modified momentum equation, which makes the compactness argument available as the artificial viscous term vanishes.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-06-23T07:00:00Z
DOI: 10.1137/21M1405915
Issue No: Vol. 54, No. 3 (2022)

• Recovery of a Time-Dependent Hermitian Connection and Potential Appearing
in the Dynamic Schrödinger Equation

Authors: Alexander Tetlow
Pages: 1347 - 1369
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1347-1369, April 2022.
We consider, on a trivial vector bundle over a Riemannian manifold with boundary, the inverse problem of uniquely recovering time- and space-dependent coefficients of the dynamic, vector-valued Schrödinger equation from the knowledge of the Dirichlet-to-Neumann (D-to-N) map. We show that the D-to-N map uniquely determines both the connection form and the potential appearing in the Schrödinger equation, under the assumption that the manifold is either (a) two-dimensional and simple or (b) of higher dimension with strictly convex boundary and admits a smooth, strictly convex function.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-01T08:00:00Z
DOI: 10.1137/20M1353265
Issue No: Vol. 54, No. 2 (2022)

• A New Approach to Space-Time Boundary Integral Equations for the Wave
Equation

Authors: Olaf Steinbach, Carolina Urzúa-Torres
Pages: 1370 - 1392
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1370-1392, April 2022.
We present a new approach for boundary integral equations for the wave equation with zero initial conditions. Unlike previous attempts, our mathematical formulation allows us to prove that the associated boundary integral operators are continuous and satisfy inf-sup conditions in trace spaces of the same regularity, which are closely related to standard energy spaces with the expected regularity in space and time. This feature is crucial from a numerical perspective, as it provides the foundations to derive sharper error estimates and paves the way to devise efficient adaptive space-time boundary element methods, which will be tackled in future work. On the other hand, the proposed approach is compatible with the current time dependent boundary element method's implementations, and we predict that it explains many of the behaviors observed in practice but that were not understood with the existing theory.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-01T08:00:00Z
DOI: 10.1137/21M1420034
Issue No: Vol. 54, No. 2 (2022)

• Optimal Regularity of Mixed Dirichlet-Conormal Boundary Value Problems for
Parabolic Operators

Authors: Jongkeun Choi, Hongjie Dong, Zongyuan Li
Pages: 1393 - 1427
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1393-1427, April 2022.
We obtain the regularity of solutions in Sobolev spaces for the mixed Dirichlet-conormal problem for parabolic operators in cylindrical domains with time-dependent separations, which is the first of its kind. Assuming the boundary of the domain to be Reifenberg-flat and the separation to be locally sufficiently close to a Lipschitz function of $m$ variables, where $m=0,\ldots,d-2$, with respect to the Hausdorff distance, we prove the unique solvability for $p\in (2(m+2/(m+3),2(m+2)/(m+1)))$. In the case when $m=0$, the range $p\in(4/3,4)$ is optimal in view of the known results for Laplace equations.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-01T08:00:00Z
DOI: 10.1137/21M1461344
Issue No: Vol. 54, No. 2 (2022)

• Vanishing Capillarity Limit of the Navier--Stokes--Korteweg System in One
Dimension with Degenerate Viscosity Coefficient and Discontinuous Initial
Density

Authors: Cosmin Burtea, Boris Haspot
Pages: 1428 - 1469
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1428-1469, April 2022.
In the first main result of this paper, we prove that one can approximate discontinuous solutions of the 1d Navier--Stokes system with solutions of the 1d Navier--Stokes--Korteweg system as the capillarity parameter tends to 0. Moreover, we allow the viscosity coefficients $\mu=\mu\left( \rho\right)$ to degenerate near vacuum. In order to obtain this result, we propose two main technical novelties. First, we provide an upper bound for the density verifying NSK that does not degenerate when the capillarity coefficient tends to 0. Second, we are able to show that the positive part of the effective velocity is bounded uniformly w.r.t. the capillary coefficient. This turns out to be crucial in providing a lower bound for the density. The second main result states the existence of a unique finite-energy global strong solution for the 1d Navier--Stokes system assuming only that $\rho_{0},1/\rho_{0}\in L^{\infty}$. This last result finds itself a natural application in the context of the mathematical modeling of multiphase flows.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-03T08:00:00Z
DOI: 10.1137/21M1428686
Issue No: Vol. 54, No. 2 (2022)

• Equilibria of Charged Hyperelastic Solids

Authors: Elisa Davoli, Anastasia Molchanova, Ulisse Stefanelli
Pages: 1470 - 1487
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1470-1487, April 2022.
We investigate equilibria of charged deformable materials via the minimization of an electroelastic energy. This involves the coupling of elastic response and electrostatics by means of a capacitary term, which is naturally defined in Eulerian coordinates. The ensuing electroelastic energy is then of mixed Lagrangian--Eulerian type. We prove that minimizers exist by investigating the continuity properties of the capacitary terms under convergence of the deformations.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-03T08:00:00Z
DOI: 10.1137/21M1413286
Issue No: Vol. 54, No. 2 (2022)

• Multiscale Steady Vortex Patches for 2D Incompressible Euler Equations

Authors: Daomin Cao, Jie Wan
Pages: 1488 - 1514
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1488-1514, April 2022.
In this paper, we study the existence and qualitative properties of multiscale steady vortex patches for Euler equations in a 2D bounded domain. By considering certain maximization problems for the vorticity, we obtain the existence of double vortex patches which are trapped in a neighborhood of two points. Limiting localizations of these two points are determined by the Robin function and the boundary of the domain, rather than critical points of the Kirchhoff--Routh function $H_2$, which is quite different from all the known results. Moreover, the strengths of two components of vorticity are of different order. Multiscale vortex patches concentrating near $k$ points are also constructed for any integer $k\geq 2.$
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-07T08:00:00Z
DOI: 10.1137/21M1390529
Issue No: Vol. 54, No. 2 (2022)

• On a Supersonic-Sonic Patch in the Three-Dimensional Steady Axisymmetric
Transonic Flows

Authors: Yanbo Hu
Pages: 1515 - 1542
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1515-1542, April 2022.
This paper focuses on the structure of solutions in a supersonic bubble arising from the three-dimensional transonic flows. Given a velocity distribution on a streamline, we construct a small smooth supersonic-sonic patch for the three-dimensional axisymmetric steady isentropic irrotational Euler equations. This patch can be regarded as the region near the upstream vertex of a supersonic bubble. The main difficulty is the coupling of nonhomogeneous terms and sonic degeneracy. To overcome it, we adopt the idea of characteristic decompositions to solve the problem in a partial hodograph coordinate system composed by the Mach angle and the potential function. By converting to the physical plane, a smooth solution for the original problem is established and the uniform regularity of solution up to the sonic curve is also discussed.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-07T08:00:00Z
DOI: 10.1137/21M1393108
Issue No: Vol. 54, No. 2 (2022)

• Localized Sensitivity Analysis at High-Curvature Boundary Points of
Reconstructing Inclusions in Transmission Problems

Authors: Habib Ammari, Yat Tin Chow, Hongyu Liu
Pages: 1543 - 1592
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1543-1592, April 2022.
In this paper, we are concerned with the recovery of the geometric shapes of inhomogeneous inclusions from the associated far field data in electrostatics and acoustic scattering. We present a local resolution analysis and show that the local shape around a boundary point with a mean curvature of high magnitude can be reconstructed more easily and stably. In proving this, we develop a novel mathematical scheme by analyzing the generalized polarization tensors (GPTs) and the scattering coefficients (SCs) coming from the associated scattered fields, which in turn boils down to the analysis of the layer potential operators that sit inside the GPTs and SCs via microlocal analysis. In a delicate and subtle manner, we decompose the reconstruction process into several steps, where all but one step depends on the global geometry, and one particular step depends on the mean curvature at the given boundary point. Then by a sensitivity analysis with respect to local perturbations of the curvature of the boundary surface, we establish the local resolution effects. Our study opens up a new field of mathematical analysis on wave superresolution imaging.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-09T08:00:00Z
DOI: 10.1137/20M1323576
Issue No: Vol. 54, No. 2 (2022)

• Parabolic Approximation of Quasilinear Wave Equations with Applications in
Nonlinear Acoustics

Authors: Barbara Kaltenbacher, Vanja Nikolić
Pages: 1593 - 1622
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1593-1622, April 2022.
This work deals with the convergence analysis of parabolic perturbations to quasilinear wave equations on smooth bounded domains. In particular, we consider wave equations with nonlinearities of quadratic type, which cover the two classical models of nonlinear acoustics, the Westervelt and Kuznetsov equations. By employing a high-order energy analysis, we obtain convergence of their solutions to the corresponding inviscid equations' solutions in the standard energy norm with a linear rate, assuming small data and a sufficiently short time. The smallness of initial data can, however, be imposed in a lower-order norm than the one needed in the energy analysis. It arises only from ensuring the nondegeneracy of the studied wave equations. In addition, we address the open questions of local well-posedness of the two classical models on bounded domains with a nonnegative, possibly vanishing sound diffusivity.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-09T08:00:00Z
DOI: 10.1137/20M1380430
Issue No: Vol. 54, No. 2 (2022)

• Generalized Maslov Indices for Non-Hamiltonian Systems

Authors: Thomas J. Baird, Paul Cornwell, Graham Cox, Christopher Jones, Robert Marangell
Pages: 1623 - 1668
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1623-1668, April 2022.
We extend the definition of the Maslov index to a broad class of non-Hamiltonian dynamical systems. To do this, we introduce a family of topological spaces---which we call Maslov--Arnold spaces---that share key topological features with the Lagrangian Grassmannian and hence admit a similar index theory. This family contains the Lagrangian Grassmannian and much more. We construct a family of examples, called hyperplane Maslov--Arnold spaces, that are dense in the Grassmannian and hence are much larger than the Lagrangian Grassmannian (which is a submanifold of positive codimension). The resulting index is then used to study eigenvalue problems for nonsymmetric reaction-diffusion systems. A highlight of our analysis is a topological interpretation of the Turing instability: the bifurcation that occurs as one increases the ratio of diffusion coefficients corresponds to a change in the generalized Maslov index.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-09T08:00:00Z
DOI: 10.1137/20M1381319
Issue No: Vol. 54, No. 2 (2022)

• NonMonotonicity of Traveling Wave Profiles for a Unimodal Recursive System

Authors: Jian Fang, Yingli Pan
Pages: 1669 - 1694
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1669-1694, April 2022.
Recursive systems of unimodal type may exhibit rich propagation dynamics. The shape of traveling wave profiles is one of the unsolved questions in this challenging topic. In this paper, we obtain criteria for the nonmonotonicity of wave profiles in terms of an eigenvalue problem. The proofs rely on establishing several nonlocal Harnack type inequalities, which may be of interest on their own. The existence and uniqueness of traveling waves are also obtained based on these inequalities.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-14T07:00:00Z
DOI: 10.1137/21M139236X
Issue No: Vol. 54, No. 2 (2022)

• Defects in Liquid Crystal Flows

Authors: Zaihui Gan, Xianpeng Hu, Fanghua Lin
Pages: 1695 - 1717
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1695-1717, April 2022.
This paper concerns the dynamical properties of topological defects in two dimensional flows of liquid crystals modeled by the Ginzburg--Landau approximations. The fluid is transported by a nonlocal (an averaged) velocity and is coupled with effects of the elastic stress. The defects move along the trajectories of the flow associated with this averaged velocity, that is, $\frac{d}{dt}a_j(t)={u}(a_j(t), t).$
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-14T07:00:00Z
DOI: 10.1137/21M1396010
Issue No: Vol. 54, No. 2 (2022)

• Asymptotics for Semidiscrete Entropic Optimal Transport

Authors: Jason M. Altschuler, Jonathan Niles-Weed, Austin J. Stromme
Pages: 1718 - 1741
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1718-1741, April 2022.
We compute exact second-order asymptotics for the cost of an optimal solution to the entropic optimal transport problem in the continuous-to-discrete, or semidiscrete, setting. In contrast to the discrete-discrete or continuous-continuous case, we show that the first-order term in this expansion vanishes but the second-order term does not, so that in the semidiscrete setting the difference in cost between the unregularized and the regularized solutions is quadratic in the inverse regularization parameter, with a leading constant that depends explicitly on the value of the density at the points of discontinuity of the optimal unregularized map between the measures. We develop these results by proving new pointwise convergence rates of the solutions to the dual problem, which may be of independent interest.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-14T07:00:00Z
DOI: 10.1137/21M1440165
Issue No: Vol. 54, No. 2 (2022)

• Boundary-Layer Analysis of Repelling Particles Pushed to an Impenetrable
Barrier

Authors: Patrick van Meurs
Pages: 1742 - 1774
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1742-1774, April 2022.
This paper considers the equilibrium positions of $n$ particles in one dimension. Two forces act on the particles; a nonlocal repulsive particle-interaction force and an external force which pushes them to an impenetrable barrier. While the continuum limit as $n \to \infty$ is known for a certain class of potentials, numerical simulations show that a discrete boundary layer appears at the impenetrable barrier, i.e., the positions of $o(n)$ particles do not fit to the particle density predicted by the continuum limit. In this paper we establish a first-order $\Gamma$-convergence result which guarantees that these $o(n)$ particles converge to a specific continuum boundary-layer profile.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-15T07:00:00Z
DOI: 10.1137/21M1420198
Issue No: Vol. 54, No. 2 (2022)

• Strong Traces to Degenerate Parabolic Equations

Authors: Marko Erceg, Darko Mitrović
Pages: 1775 - 1796
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1775-1796, April 2022.
We prove existence of strong traces at $t=0$ for quasi-solutions to (multidimensional) degenerate parabolic equations with no nondegeneracy conditions. In order to solve the problem, we combine the blow-up method and a strong precompactness result for quasi-solutions to degenerate parabolic equations with the induction argument with respect to the space dimension.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-17T07:00:00Z
DOI: 10.1137/21M1425530
Issue No: Vol. 54, No. 2 (2022)

• Stability and Bifurcation of Mixing in the Kuramoto Model with Inertia

Authors: Hayato Chiba, Georgi S. Medvedev
Pages: 1797 - 1819
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1797-1819, April 2022.
The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on Erdös--Rényi and small-world graphs. Applications of the second order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-22T07:00:00Z
DOI: 10.1137/21M1427000
Issue No: Vol. 54, No. 2 (2022)

• Global Smooth Sonic-Supersonic Flows in a Class of Critical Nozzles

Authors: Chunpeng Wang
Pages: 1820 - 1859
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1820-1859, April 2022.
This paper concerns the global existence of smooth sonic-supersonic potential flows in a two-dimensional expanding nozzle with the critical geometry at the inlet. The flow is sonic and its velocity is along the normal direction at the inlet, which is a segment vertical to the walls of the nozzle, and is supersonic in the nozzle. Such a sonic-supersonic model is governed by a quasilinear nonstrictly hyperbolic equation with degeneracy at the inlet, and the degeneracy is strong in the sense that all characteristics from the inlet coincide with the inlet and never approach the domain. Furthermore, the asymptotic behaviors of the average speed on the cross section and the normal acceleration on the walls are of the same order with respect to the distance to the inlet, and of different order with respect to the height of the inlet. We seek the flows whose asymptotic behavior near the inlet is the same as the average speed on the cross section. An interesting phenomenon is that the existence of such sonic-supersonic flows depends on the height of the inlet. More precisely, it is shown by means of a method of characteristics with many elaborate estimates that there exists uniquely such a flow if it is suitably small, while it does not if it is suitably large. The results in the paper, together with other works, describe completely the geometry of the de Laval nozzles where there are smooth transonic flows of Meyer type whose sonic points are all exceptional.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-24T07:00:00Z
DOI: 10.1137/20M1373906
Issue No: Vol. 54, No. 2 (2022)

• Propagation Dynamics for Time-Periodic and Partially Degenerate
Reaction-Diffusion Systems

Authors: Mingdi Huang, Shi-Liang Wu, Xiao-Qiang Zhao
Pages: 1860 - 1897
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1860-1897, April 2022.
This paper is concerned with the propagation dynamics for partially degenerate reaction-diffusion systems with monostable and time-periodic nonlinearity. In the cooperative case, we prove the existence of periodic traveling fronts and the exponential stability of continuous periodic traveling fronts. In the noncooperative case, we establish the existence of the minimal wave speed of periodic traveling waves and its coincidence with the spreading speed. More specifically, when the system is nondegenerate, the existence of periodic traveling waves is proved by using Schauder's fixed point theorem and the regularity of analytic semigroup, while in the partially degenerate case, due to the lack of compactness and standard parabolic estimates, the existence result is obtained by appealing to the asymptotic fixed point theorem with the help of some properties of the Kuratowski measure of noncompactness. It may be the first work to study periodic traveling waves of partially degenerate reaction-diffusion systems with noncooperative and time-periodic nonlinearity.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-29T07:00:00Z
DOI: 10.1137/21M1397234
Issue No: Vol. 54, No. 2 (2022)

• Double Phase Implicit Obstacle Problems with Convection and Multivalued
Mixed Boundary Value Conditions

Authors: Shengda Zeng, Vicenţiu D. Rădulescu, Patrick Winkert
Pages: 1898 - 1926
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1898-1926, April 2022.
In this paper we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called a double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms, and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such an implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani-Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-31T07:00:00Z
DOI: 10.1137/21M1441195
Issue No: Vol. 54, No. 2 (2022)

• Bounds on the Heat Transfer Rate via Passive Advection

Authors: Gautam Iyer, Truong-Son Van
Pages: 1927 - 1965
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1927-1965, April 2022.
In heat exchangers, an incompressible fluid is heated initially and cooled at the boundary. The goal is to transfer the heat to the boundary as efficiently as possible. In this paper we study a related steady version of this problem where a steadily stirred fluid is uniformly heated in the interior and cooled on the boundary. For a given large Péclet number, how should one stir to minimize some norm of the temperature' This version of the problem was previously studied by Marcotte et al. [SIAM J. Appl. Math., 78 (2018), pp. 591--608] in a disk, where the authors used matched asymptotics to show that when the Péclet number, Pe, is sufficiently large one can stir the fluid in a manner that ensures the total heat is $O(1/{Pe})$. In this paper we Pconfirm their results with rigorous proofs and also provide an almost matching lower bound. For simplicity, we work on the infinite strip instead of the unit disk and the proof uses probabilistic techniques.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-31T07:00:00Z
DOI: 10.1137/21M1394497
Issue No: Vol. 54, No. 2 (2022)

• Homoclinic Solutions of Periodic Discrete Schrödinger Equations with

Authors: Genghong Lin, Jianshe Yu
Pages: 1966 - 2005
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 1966-2005, April 2022.
We deal with the existence and multiplicity of homoclinic solutions for a class of periodic discrete Schrödinger equations with local superquadratic conditions. The arising problem intensely involves major difficulties including the indefiniteness of the associated variational problem, the restriction of local superquadratic conditions, and the boundedness of Cerami sequences. New methods including weak*-compactness and an approximation scheme are developed in this work to conquer these difficulties. This allows us to obtain a ground state solution and infinitely many geometrically distinct solutions. To the best of our knowledge, this is the first time in the existing literature to obtain the existence and multiplicity results of such a discrete problem under local superquadratic conditions. Even for the uniform superquadratic case, our results also significantly improve the well-known ones as special cases. Moreover, our weaker conditions may be suitable to other types of variational problems.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-03-31T07:00:00Z
DOI: 10.1137/21M1413201
Issue No: Vol. 54, No. 2 (2022)

• The Initial-Value Problem to the Modified Two-Component
Euler--Poincaré Equations

Authors: Kan Yan, Yue Liu
Pages: 2006 - 2039
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2006-2039, April 2022.
We are concerned with the initial-value problem for the modified two-component Euler--Poincaré equations including the classical Euler--Poincaré equations, the integrable two-component Camassa--Holm system, and its two-component modified version. We first construct the global and blow-up strong solutions by using the orthogonal and symmetric transform invariances. We then show rigorously that the equations will recover to a symmetric hyperbolic system of conservation laws as the dispersion parameters vanish. Finally, we prove the Liouville-type theorem for the stationary weak solutions to the equations.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-04-11T07:00:00Z
DOI: 10.1137/20M138274X
Issue No: Vol. 54, No. 2 (2022)

• Cloaking Property of a Plasmonic Structure in Doubly Complementary Media
and Three-Sphere Inequalities with Partial Data

Authors: Hoai-Minh Nguyen
Pages: 2040 - 2096
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2040-2096, April 2022.
We investigate the cloaking property of negative-index metamaterials in the time-harmonic electromagnetic setting for the so-called doubly complementary media. These are media consisting of negative-index metamaterials in a shell (plasmonic structure) and positive-index materials in its complement for which the shell is complementary to a part of the core and a part of the exterior of the core-shell structure. We show that an arbitrary object is invisible when it is placed close to a plasmonic structure of a doubly complementary medium as long as its cross section is smaller than a threshold given by the property of the plasmonic structure. To handle the loss of the compactness and of the ellipticity of the modeling Maxwell equations with sign-changing coefficients, we first obtain Cauchy's problems associated with two Maxwell systems using reflections. We then derive information from them, and combine it with the removing localized singularity technique to deal with the localized resonance. A central part of the analysis on Cauchy's problems is to establish three-sphere inequalities with partial data for general elliptic systems, which are interesting in themselves. The proof of these inequalities first relies on an appropriate change of variables, inspired by conformal maps, and is then based on Carleman's estimates for a class of degenerate elliptic systems.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-04-11T07:00:00Z
DOI: 10.1137/21M1394722
Issue No: Vol. 54, No. 2 (2022)

• Global Limit Theorem for Parabolic Equations with a Potential

Authors: L. Koralov, B. Vainberg
Pages: 2097 - 2113
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2097-2113, April 2022.
We obtain the asymptotics, as $t + x \rightarrow \infty$, of the fundamental solution to the heat equation with a compactly supported potential. It is assumed that the corresponding stationary operator has at least one positive eigenvalue. Two regions with different types of behavior are distinguished: Inside a certain conical surface in the $(t,x)$ space, the asymptotics is determined by the principal eigenvalue and the corresponding eigenfunction; outside of the conical surface, the main term of the asymptotics is a product of a bounded function and the fundamental solution of the unperturbed operator, with the contribution from the potential becoming negligible if $x /t \rightarrow \infty$. A formula for the global asymptotics, as $t + x \rightarrow \infty$, of the solution in the entire half-space $t> 0$ is provided. In probabilistic terms, the result describes the asymptotics of the density of particles in a branching diffusion with compactly supported branching and killing potentials.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-04-11T07:00:00Z
DOI: 10.1137/21M1429370
Issue No: Vol. 54, No. 2 (2022)

• On the Two-Dimensional Quantum Confined Stark Effect in Strong Electric
Fields

Authors: Horia Cornean, David Krejčiřík, Thomas G. Pedersen, Nicolas Raymond, Edgardo Stockmeyer
Pages: 2114 - 2127
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2114-2127, April 2022.
We consider a Stark Hamiltonian on a two-dimensional bounded domain with Dirichlet boundary conditions. In the strong electric field limit we derive, under certain local convexity conditions, a three-term asymptotic expansion of the low-lying eigenvalues. This shows that the excitation frequencies are proportional to the square root of the boundary curvature at a certain point determined by the direction of the electric field.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-04-11T07:00:00Z
DOI: 10.1137/20M1386712
Issue No: Vol. 54, No. 2 (2022)

• On Solutions with Compact Spectrum to Nonlinear Klein--Gordon and
Schrödinger Equations

Authors: Andrew Comech
Pages: 2128 - 2141
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2128-2141, April 2022.
We consider finite energy solutions to the nonlinear Schrödinger equation and nonlinear Klein--Gordon equation and find conditions on the nonlinearity under which the standard, one-frequency solitary waves are the only solutions with compact spectrum. We also construct an example of a four-frequency solitary wave solution to the nonlinear Dirac equation in three dimensions.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-04-14T07:00:00Z
DOI: 10.1137/21M1411330
Issue No: Vol. 54, No. 2 (2022)

• Global Existence of a Solution for Isentropic Gas Flow in the Laval Nozzle
with a Friction Term

Authors: Naoki Tsuge
Pages: 2142 - 2162
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2142-2162, April 2022.
We are concerned with isentropic gas flow in the Laval nozzle with a friction term due to viscosity. It is well known that the flow attains the sonic state at the throat, where the cross section is minimum in the Laval nozzle. However, the present friction term changes the position of the sonic state into downstream, which is called chooking. Our goal in this paper is to investigate these phenomena mathematically. From the mathematical point of view, the friction term is different from normal friction terms such as $-\alpha m$ and difficult to treat with. In spite of its physical importance, the friction term has not received much attention until now. For the case without the friction term, the global existence of a solution was obtained by the author. However, for the case with the friction term, there are only restrictive results. The most difficult point is to obtain the bounded estimate of solutions. To solve this problem, we introduce an invariant region depending on the mass. Adjusting the invariant region, we invent a new difference scheme, which yields approximate solutions including the mass.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-04-14T07:00:00Z
DOI: 10.1137/21M1415029
Issue No: Vol. 54, No. 2 (2022)

• Schauder Estimates for Fractional Laplacians and NonLocal, One-Dimensional
Singular SPDEs

Authors: Leandro Chiarini, Claudio Landim
Pages: 2163 - 2215
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2163-2215, April 2022.
We prove a Schauder estimate for the semigroup associated to the fractional Laplacian $-\, (-\Delta)^{\rho/2}$, $1\le \rho Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-04-14T07:00:00Z DOI: 10.1137/20M1382829 Issue No: Vol. 54, No. 2 (2022) • A Mathematical Study of a Hyperbolic Metamaterial in Free Space • Free pre-print version: Loading... Authors: Patrick Ciarlet, Maryna Kachanovska Pages: 2216 - 2250 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2216-2250, April 2022. Wave propagation in hyperbolic metamaterials is described by the Maxwell equations with a frequency-dependent tensor of dielectric permittivity, whose eigenvalues are of different signs. In this case the problem becomes hyperbolic (Klein--Gordon equation) for a certain range of frequencies. The principal theoretical and numerical difficulty comes from the fact that this hyperbolic equation is posed in a free space, without initial conditions provided. The subject of the work is the theoretical justification of this problem. In particular, this includes the construction of a radiation condition, a well-posedness result, a limiting absorption principle, and regularity estimates on the solution. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-04-19T07:00:00Z DOI: 10.1137/21M1404223 Issue No: Vol. 54, No. 2 (2022) • A McKean--Vlasov SDE and Particle System with Interaction from Reflecting Boundaries • Free pre-print version: Loading... Authors: Michele Coghi, Wolfgang Dreyer, Peter K. Friz, Paul Gajewski, Clemens Guhlke, Mario Maurelli Pages: 2251 - 2294 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2251-2294, April 2022. We consider a one-dimensional McKean--Vlasov SDE on a domain and the associated mean-field interacting particle system. The peculiarity of this system is the combination of the interaction, which keeps the average position prescribed, and the reflection at the boundaries; these two factors make the effect of reflection nonlocal. We show pathwise well-posedness for the McKean--Vlasov SDE and convergence for the particle system in the limit of large particle number. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-04-19T07:00:00Z DOI: 10.1137/21M1409421 Issue No: Vol. 54, No. 2 (2022) • The Wave Breaking for Whitham-Type Equations Revisited • Free pre-print version: Loading... Authors: Jean-Claude Saut, Yuexun Wang Pages: 2295 - 2319 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2295-2319, April 2022. We prove wave breaking (shock formation) for some Whitham-type equations which include the Burgers--Hilbert equation, the fractional Korteweg--de Vries equation, and the classical Whitham equation. The result seems to be new for the Burgers--Hilbert equation and has been proven independently in (Yang, SIAM J. Math. Anal., 53 (2021), pp. 5756--5802). In all cases we provide simpler proofs than the known ones. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-04-19T07:00:00Z DOI: 10.1137/20M1345207 Issue No: Vol. 54, No. 2 (2022) • A Rigorous Derivation of a Boltzmann System for a Mixture of Hard-Sphere Gases • Free pre-print version: Loading... Authors: Ioakeim Ampatzoglou, Joseph K. Miller, Nataša Pavlović Pages: 2320 - 2372 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2320-2372, April 2022. In this paper, we rigorously derive a Boltzmann equation for mixtures from the many body dynamics of two types of hard-sphere gases. We prove that the microscopic dynamics of two gases with different masses and diameters is well defined, and we introduce the concept of a two parameter BBGKY hierarchy to handle the nonsymmetric interaction of these gases. As a corollary of the derivation, we prove Boltzmann's propagation of chaos assumption for the case of a mixture of gases. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-04-19T07:00:00Z DOI: 10.1137/21M1424779 Issue No: Vol. 54, No. 2 (2022) • Elliptic Equations with Degenerate Weights • Free pre-print version: Loading... Authors: Anna Kh. Balci, Lars Diening, Raffaella Giova, Antonia Passarelli di Napoli Pages: 2373 - 2412 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2373-2412, April 2022. We obtain new local Calderón--Zygmund estimates for elliptic equations with matrix-valued weights for linear as well as nonlinear equations. We introduce a novel log-BMO condition on the weight$\mathbb M$. In particular, we assume smallness of the logarithm of the matrix-valued weight in BMO. This allows us to include degenerate, discontinuous weights. The assumption on the smallness parameter is sharp and linear in terms of the integrability exponent of the gradient. This is a novelty even in the linear setting with nondegenerate weights compared to previously known results, where the dependency was exponential. We provide examples that show the sharpness of the estimates in terms of the log-BMO norm. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-04-21T07:00:00Z DOI: 10.1137/21M1412529 Issue No: Vol. 54, No. 2 (2022) • Weakly Nonlinear Multiphase Geometric Optics for Hyperbolic Quasilinear Boundary Value Problems: Construction of a Leading Profile • Free pre-print version: Loading... Authors: Corentin Kilque Pages: 2413 - 2507 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2413-2507, April 2022. We investigate in this paper the existence of the leading profile of a WKB expansion for quasilinear initial boundary value problems with a highly oscillating forcing boundary term. The framework is weakly nonlinear, as the boundary term is of order$ O(\epsilon) $where the frequencies are of order$ O(1/\epsilon) $. We consider here multiple phases on the boundary, generating a countable infinite number of phases inside the domain, and we therefore use an almost-periodic functional framework. The major difficulties of this work are the lack of symmetry in the leading profile equation and the occurrence of infinitely many resonances (opposite to the simple phase case studied earlier). The leading profile is constructed as the solution of a quasilinear problem, which is solved using a priori estimates without loss of derivatives. The assumptions of this work are illustrated with the example of isentropic Euler equations in space dimension two. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-04-21T07:00:00Z DOI: 10.1137/21M1413596 Issue No: Vol. 54, No. 2 (2022) • Hydrodynamic Limit of 3dimensional Evolutionary Boltzmann Equation in Convex Domains • Free pre-print version: Loading... Authors: Lei Wu, Zhimeng Ouyang Pages: 2508 - 2569 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2508-2569, April 2022. This is the second half of our work on the hydrodynamic limit (the first half [L. Wu and Z. Ouyang, manuscript] focuses on the stationary problem). We consider the 3D evolutionary Boltzmann equation in convex domains with diffusive-reflection boundary condition. We rigorously derive the unsteady incompressible Navier--Stokes--Fourier system and justify the asymptotic convergence as the Knudsen number$\varepsilon$shrinks to zero. The proof is based on an innovative remainder estimate and an intricate analysis of boundary layers with geometric correction. Citation: SIAM Journal on Mathematical Analysis PubDate: 2022-04-27T07:00:00Z DOI: 10.1137/20M1375735 Issue No: Vol. 54, No. 2 (2022) • Local Continuity of Weak Solutions to the Stefan Problem Involving the Singular$p$-Laplacian • Free pre-print version: Loading... Authors: Naian Liao Pages: 2570 - 2586 Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2570-2586, April 2022. We establish the local continuity of locally bounded weak solutions (temperatures) to the doubly singular parabolic equation modeling the phase transition of a material:$\partial_t \beta(u)-\Delta_p u\ni 0\,\text{for }\tfrac{2N}{N+1}
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-04-27T07:00:00Z
DOI: 10.1137/21M1402443
Issue No: Vol. 54, No. 2 (2022)

• Formation and Construction of a Multidimensional Shock Wave for the
First-Order Hyperbolic Conservation Law with Smooth Initial Data

Authors: Yin Huicheng, Zhu Lu
Pages: 2587 - 2610
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2587-2610, April 2022.
In the paper, the problem on formation and construction of a multidimensional shock wave is studied for the first-order conservation law $\partial_t u+\partial_x F(u)+\partial_y G(u)=0$ with smooth initial data $u_0(x,y)$. It is well known that the smooth solution $u$ will blow up on the time $T^*=-\frac{1}{\min{H(\xi,\eta)}}$ when $\min{H(\xi,\eta})T^*$. Additionally, in the neighborhood of $\Gamma$, some detailed and precise descriptions on the singularities of solution $u$ are given.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-04-28T07:00:00Z
DOI: 10.1137/21M1406581
Issue No: Vol. 54, No. 2 (2022)

• On the Ill-Posedness of the Triple Deck Model

Authors: Helge Dietert, David Gérard-Varet
Pages: 2611 - 2633
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2611-2633, April 2022.
We analyze the stability properties of the so-called triple deck model, a classical refinement of the Prandtl equation to describe boundary layer separation. Combining the methodology introduced in [A.-L. Dalibard et al., SIAM J. Math. Anal., 50 (2018), pp. 4203--4245], based on complex analysis tools, and stability estimates inspired from Dietert and Gérard-Varet [Anal. PDE, 5 (2019), 8], we exhibit unstable linearizations of the triple deck equation. The growth rates of the corresponding unstable eigenmodes scale linearly with the tangential frequency. This shows that the recent result of Iyer and Vicol [Comm. Pure Appl. Math., 74 (2021), pp. 1641--1684] of local well-posedness for analytic data is essentially optimal.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-04-28T07:00:00Z
DOI: 10.1137/21M1427401
Issue No: Vol. 54, No. 2 (2022)

• Benjamin--Ono Soliton Dynamics in a Slowly Varying Potential Revisited

Authors: Justin Holmer, Katherine Zhiyuan Zhang
Pages: 2634 - 2690
Abstract: SIAM Journal on Mathematical Analysis, Volume 54, Issue 2, Page 2634-2690, April 2022.
The Benjamin--Ono equation with a slowly varying potential is $\text{(pBO)} \ u_t + (Hu_x-Vu + \tfrac12 u^2)_x=0$ with $V(x)=W(hx)$, $0< h \ll 1$, and $W\in C_c^\infty(\mathbb{R})$, and $H$ denotes the Hilbert transform. The soliton profile is $Q_{a,c}(x) = cQ(c(x-a)) \,, \text{ where } Q(x) = \frac{4}{1+x^2}$ and $a\in \mathbb{R}$, $c>0$ are parameters. For initial condition $u_0(x)$ to (pBO) close to $Q_{0,1}(x)$, it was shown in [K. Z. Zhang, Nonlinearity, 33 (2020), pp. 1064--1093] that the solution $u(x,t)$ to (pBO) remains close to $Q_{a(t),c(t)}(x)$ and approximate parameter dynamics for $(a,c)$ were provided, on a dynamically relevant time scale. In this paper, we prove exact $(a,c)$ parameter dynamics. This is achieved using the basic framework of the paper [K. Z. Zhang, Nonlinearity, 33 (2020), pp. 1064--1093] but adding a local virial estimate for the linearization of (pBO) around the soliton. This is a local-in-space estimate averaged in time, often called a local smoothing estimate, showing that effectively the remainder function in the perturbation analysis is smaller near the soliton than globally in space. A weaker version of this estimate is proved in [C. E. Kenig and Y. Martel, Rev. Mat. Iberoam., 25 (2009), pp. 909--970] as part of a “linear Liouville” result, and we have adapted and extended their proof for our application.
Citation: SIAM Journal on Mathematical Analysis
PubDate: 2022-04-28T07:00:00Z
DOI: 10.1137/21M1425177
Issue No: Vol. 54, No. 2 (2022)

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