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Abstract: A relativistic version of the Kinetic Theory for polyatomic gas is considered and a new hierarchy of moments that takes into account the total energy composed by the rest energy and the energy of the molecular internal modes is presented. In the first part, we prove via classical limit that the truncated system of moments dictates a precise hierarchy of moments in the classical framework. In the second part, we consider the particular physical case of fifteen moments closed via maximum entropy principle in a neighborhood of equilibrium state. We prove that this symmetric hyperbolic system satisfies all the general assumptions of some theorems that guarantee the global existence of smooth solutions for initial data sufficiently small. PubDate: 2022-05-16

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Abstract: In this work, we consider the model of \({{\,\mathrm{{\mathbb {S}}^2\times {\mathbb {R}}}\,}}\) isometric to \({\mathbb {R}}^3{\setminus } \{0\}\) , endowed with a metric conformally equivalent to the Euclidean metric of \({\mathbb {R}}^3\) , and we define a Gauss map for surfaces in this model likewise in the Euclidean 3-space. We show as a main result that any two minimal conformal immersions in \({{\,\mathrm{{\mathbb {S}}^2\times {\mathbb {R}}}\,}}\) with the same non-constant Gauss map differ by only two types of ambient isometries: either \(f=({{\,\mathrm{\mathrm {Id}}\,}},T)\) , where T is a translation on \({\mathbb {R}}\) , or \(f=({\mathcal {A}},T)\) , where \({\mathcal {A}}\) denotes the antipodal map on \({\mathbb {S}}^2\) . This means that any minimal immersion is determined by its conformal structure and its Gauss map, up to those isometries. PubDate: 2022-05-16

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Abstract: In this paper, we establish the existence of weak solutions to the ellipsoidal BGK model (ES-BGK model) of the Boltzmann equation with the correct Prandtl number, which corresponds to the case when the Knudsen parameter is \(-1/2\) . PubDate: 2022-05-16

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Abstract: We investigate the numerical implementation of the limiting equation for the phonon transport equation in the small Knudsen number regime. The main contribution is that we derive the limiting equation that achieves the second order convergence, and provide a numerical recipe for computing the Robin coefficients. These coefficients are obtained by solving an auxiliary half-space equation. Numerically the half-space equation is solved by a spectral method that relies on the even-odd decomposition to eliminate corner-point singularity. Numerical evidences will be presented to justify the second order asymptotic convergence rate. PubDate: 2022-05-13

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Abstract: A method of fundamental solutions (MFS) is proposed and analyzed for the ill-posed problem of finding the wave motion from given lateral Cauchy data in annular domains. A finite difference scheme, known as the Houbolt method, is applied for the time-discretisation rendering a sequence of elliptic systems corresponding to the number of time steps. The solution of the elliptic problems is sought as a linear combination of elements in what is known as a fundamental sequence with source points placed outside of the solution domain. Collocating on the boundary part where Cauchy data is given, a sequence of linear equations is obtained for finding the coefficients in the MFS approximation. Tikhonov regularization is employed to generate a stable solution to the obtained systems of linear equations. It is outlined that the elements in the fundamental sequence constitute a linearly independent and dense set on the boundary of the solution domain in the \(L_2\) -sense. Numerical results both in two and three-dimensional domains confirm the applicability of the proposed strategy for the considered lateral Cauchy problem for the wave equation both for exact and noisy data. PubDate: 2022-05-09

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Abstract: We consider the initial-boundary value problem for the heat equation in the half space with an exponential nonlinear boundary condition. We prove the existence of global-in-time solutions under the smallness condition on the initial data in the Orlicz space \(\mathrm {exp}L^2({\mathbb {R}}^N_+)\) . Furthermore, we derive decay estimates and the asymptotic behavior for small global-in-time solutions. PubDate: 2022-05-06

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Abstract: Abstract We provide bounds in a Sobolev-space framework for transport equations with nontrivial inflow and outflow. We give, for the first time, bounds on the gradient of the solution with the type of inflow boundary conditions that occur in Poiseuille flow. Following ground-breaking work of the late Charles Amick, we name a generalization of this type of flow domain in his honor. We prove gradient bounds in Lebesgue spaces for general Amick domains which are crucial for proving well posedness of the grade-two fluid model. We include a complete review of transport equations with inflow boundary conditions, providing novel proofs in most cases. To illustrate the theory, we review and extend an example of Bernard that clarifies the singularities of solutions of transport equations with nonzero inflow boundary conditions. PubDate: 2022-04-30

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Abstract: Abstract In this paper, we analyze and implement the Dirichlet spectral-Galerkin method for approximating simply supported vibrating plate eigenvalues with variable coefficients. This is a Galerkin approximation that uses the approximation space that is the span of finitely many Dirichlet eigenfunctions for the Laplacian. Convergence and error analysis for this method is presented for two and three dimensions. Here we will assume that the domain has either a smooth or Lipschitz boundary with no reentrant corners. An important component of the error analysis is Weyl’s law for the Dirichlet eigenvalues. Numerical examples for computing the simply supported vibrating plate eigenvalues for the unit disk and square are presented. In order to test the accuracy of the approximation, we compare the spectral-Galerkin method to the separation of variables for the unit disk. Whereas for the unit square we will numerically test the convergence rate for a variable coefficient problem. PubDate: 2022-04-21

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Abstract: Abstract We consider the classical Yang–Mills system coupled with a Dirac equation in 3+1 dimensions in temporal gauge. Using that most of the nonlinear terms fulfill a null condition we prove local well-posedness for small data with minimal regularity assumptions. This problem for smooth data was solved forty years ago by Y. Choquet-Bruhat and D. Christodoulou. The corresponding problem in Lorenz gauge was considered recently by the author in [14]. PubDate: 2022-04-10

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Abstract: Abstract In this paper, we prove several inequalities such as Sobolev, Poincaré, logarithmic Sobolev, which involve a general norm with accurate information of extremals, and are valid for some symmetric functions. We use Ioku’s transformation, which is a special case of p-harmonic transplantation, between symmetric functions. PubDate: 2022-04-07

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Abstract: Abstract The aim of this paper is to extend the modeling of a hyperelastic rod undergoing large displacements with tangential self-friction to their modeling with rotational self-friction. The discontinuity of contact force into a contact region not known in advance with taking into account the effects of friction in this problem type leads to serious difficulties in modelization, mathematical and numerical analysis. In this paper, we present an accurate modeling of rotational and tangential self-friction with Coulomb’s law and describe also an augmented Lagrangian method to present a weak variational formulation approach of this problem. We then use the minimization method of the total energy to present an existence result of solution for the nonlinear penalized formulation. Finally, we give the linearization and the finite-element discretization of the weak variational formulation that can be useful for a numerical implementation. PubDate: 2022-04-06

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Abstract: Abstract In this paper, we establish a sharp weighted Sobolev inequality on the upper half-space. We also discourse existence and nonexistence of minimizer . As an application, we study a quasilinear problem on the upper half-space. PubDate: 2022-04-05

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Abstract: Abstract We consider the semilinear heat equation \(\partial _t u -\Delta u=u^{N/(N-2)}\) in \(\Omega \) with \(u=0\) on \(\partial \Omega \) , where \(N\ge 3\) and \(\Omega \) is a smooth bounded domain in \(\mathbf {R}^N\) . Let \(\xi :\mathbf {R}\rightarrow \Omega \) satisfy \(\overline{\{\xi (t);t\in \mathbf {R}\}}\subset \Omega \) . Under some assumption on the uniform Hölder continuity of \(\xi \) , we construct a nonnegative solution u defined for all \(t\in \mathbf {R}\) satisfying \(u(x,t)\rightarrow \infty \) for each \(t\in \mathbf {R}\) as \(x\rightarrow \xi (t)\) . PubDate: 2022-03-29

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Abstract: Abstract We establish an invariance principle corresponding to the universality of random matrices. More precisely, we prove the dynamical universality of random matrices in the sense that, if the random point fields \( \mu ^N \) of N-particle systems describing the eigenvalues of random matrices or log-gases with general self-interaction potentials V converge to some random point field \( \mu \) , then the associated natural \( \mu ^N \) -reversible diffusions represented by solutions of stochastic differential equations (SDEs) converge to some \( \mu \) -reversible diffusion given by the solution of an infinite-dimensional SDE (ISDE). Our results are general theorems that can be applied to various random point fields related to random matrices such as sine, Airy, Bessel, and Ginibre random point fields. In general, the representations of finite-dimensional SDEs describing N-particle systems are very complicated. Nevertheless, the limit ISDE has a simple and universal representation that depends on a class of random matrices appearing in the bulk, and at the soft- and at hard-edge positions. Thus, we prove that ISDEs such as the infinite-dimensional Dyson model and the Airy, Bessel, and Ginibre interacting Brownian motions are universal dynamical objects. PubDate: 2022-03-23

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Abstract: Abstract In this paper, we show the existence of a minimizer for the \(L^2\) -constrained minimization problem associated with a nonlinear Schrödinger system with three wave interaction without assuming symmetry for potentials. PubDate: 2022-03-23

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Abstract: Abstract Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess nontrivial entire solutions) guarantee optimal universal estimates of solutions of related initial and initial-boundary value problems. Assume that \(p>1\) is subcritical in the Sobolev sense. In the case of nonnegative solutions and the system $$\begin{aligned} U_t-\varDelta U=F(U)\quad \hbox {in}\quad \mathbb {R}^n\times \mathbb {R}\end{aligned}$$ where \(U=(u_1,\dots ,u_N)\) , \(F=\nabla G\) is p-homogeneous and satisfies the positivity assumptions \(G(U)>0\) for \(U\ne 0\) and \(\xi \cdot F(U)>0\) for some \(\xi \in \mathbb {R}^N\) and all \(U\ge 0\) , \(U\ne 0\) , it has recently been shown in [P. Quittner, Duke Math. J. 170 (2021), 1113–1136] that the parabolic Liouville theorem is true whenever the corresponding elliptic Liouville theorem for the system \(-\varDelta U=F(U)\) is true. By modifying the arguments in that proof we show that the same result remains true without the positivity assumptions on G and F, and that the class of solutions can also be enlarged to contain (some or all) sign-changing solutions. In particular, in the scalar case \(N=1\) and \(F(u)= u ^{p-1}u\) , our results cover the main result in [T. Bartsch, P. Poláčik and P. Quittner, J. European Math. Soc. 13 (2011), 219–247]. We also prove a parabolic Liouville theorem for solutions in \(\mathbb {R}^n_+\times \mathbb {R}\) satisfying homogeneous Dirichlet boundary conditions on \(\partial \mathbb {R}^n_+\times \mathbb {R}\) since such theorem is also needed if one wants to prove universal estimates of solutions of related systems in \(\varOmega \times (0,T)\) , where \(\varOmega \subset \mathbb {R}^n\) is a smooth domain. Finally, we use our Liouville theorems to prove universal estimates for particular parabolic systems. PubDate: 2022-03-22

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Abstract: Abstract We study the nonlinear problem \(-\Delta u + V(x)u = f(u), x \in \mathbb {R}^{N}, \lim _{ x \rightarrow \infty } u(x) = 0\) , where the Schrödinger operator \(-\Delta + V\) is positive and f is asymptotically linear. Moreover, \(\lim _{ x \rightarrow \infty } V(x) = \sigma _{0}\) . We allow the interference of essential spectrum, i.e. \(\sup _{t \ne 0}f(t)/t \ge \sigma _{0}\) . If \(\sup _{t \ne 0}2F(t)/t^{2} < \sigma _{0}\) , the existence of four solutions will be proved by Morse theory. If \(\sup _{t \ne 0}2F(t)/t^{2} \ge \sigma _{0}\) , we can find a positive solution when \(mes(\{x \in \mathbb {R}^{N}: V(x)> \sigma _{0}\}) > 0\) . PubDate: 2022-03-21

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Abstract: Abstract Given \(2\le p<\infty \) , \(s\in (0, 1)\) and \(t\in (1, 2s)\) , we establish interior \(W^{t,p}\) Calderón-Zygmund estimates for solutions of nonlocal equations of the form $$\begin{aligned} \int _{\Omega } \int _{\Omega } K\left( x, x-y ,\frac{x-y}{ x-y }\right) \frac{(u(x)-u(y))(\varphi (x)-\varphi (y))}{ x-y ^{n+2s}} dx dy = g[\varphi ], \quad \forall \phi \in C_c^{\infty }(\Omega ) \end{aligned}$$ where \(\Omega \subset \mathbb {R}^{n}\) is an open set. Here we assume K is bounded, nonnegative and continuous in the first entry – and ellipticity is ensured by assuming that K is strictly positive in a cone. The setup is chosen so that it is applicable for nonlocal equations on manifolds, but the structure of the equation is general enough that it also applies to the certain fractional p-Laplace equations around points where \(u \in C^1\) and \( \nabla u \ne 0\) . PubDate: 2022-03-21

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Abstract: Abstract This paper is devoted to the global in time solvability for a superlinear parabolic equation $$\begin{aligned} \partial _t u = \Delta u + f(u), \quad x\in {\mathbb {R}}^N, \quad t>0, \quad u(x,0) = u_0(x), \quad x\in {\mathbb {R}}^N,\quad \hbox {(P)} \end{aligned}$$ where f(u) denotes superlinear nonlinearity of problem (P) and \(u_0\) is a nonnegative initial function. As a continuation of the paper in 2018 by the authors of this paper, we consider the global in time existence and nonexistence of nonnegative solutions for problem (P). We prove the existence of global in time solutions based on a quasi-scaling property for (P). We also discuss the nonexistence of nontrivial nonnegative global in time solutions by focusing on the behavior of f(u) as \(u\rightarrow +0\) . These results enable us to generalize the Fujita exponent, which is known as the critical exponent classifying the global in time solvability for a power-type semilinear heat equation. PubDate: 2022-03-11

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Abstract: Abstract In this paper we investigate the existence of ground states and dual ground states for Maxwell’s Equations in \({\mathbb {R}}^3\) in nonlocal nonlinear metamaterials. We prove that several nonlocal models admit ground states in contrast to their local analogues. PubDate: 2022-03-09