Authors:Jeffrey Humpherys; Gregory Lyng; Kevin Zumbrun Pages: 923 - 973 Abstract: Extending results of Humpherys–Lyng–Zumbrun in the one-dimensional case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the multidimensional stability of planar Navier–Stokes shocks across the full range of shock amplitudes, including the infinite-amplitude limit, for monatomic or diatomic ideal gas equations of state and viscosity and heat conduction coefficients \({\mu}\) , \({\mu +\eta}\) , and \({\nu=\kappa/c_v}\) constant and in the physical ratios predicted by statistical mechanics, and Mach number \({M > 1.035}\) . Our results indicate unconditional stability within the parameter range considered; this agrees with the results of Erpenbeck and Majda for the corresponding inviscid case of Euler shocks. Notably, this study includes the first successful numerical computation of an Evans function associated with the multidimensional stability of a viscous shock wave. The methods introduced can be used in principle to decide stability for shocks in any polytropic gas, or indeed for shocks of other models, including in, particular, viscoelasticity, combustion, and magnetohydrodynamics (MHD). PubDate: 2017-12-01 DOI: 10.1007/s00205-017-1147-7 Issue No:Vol. 226, No. 3 (2017)

Authors:Hermano Frid; Yachun Li Pages: 975 - 1008 Abstract: We consider a mixed type boundary value problem for a class of degenerate parabolic–hyperbolic equations. Namely, we consider a Cartesian product domain and split its boundary into two parts. In one of them we impose a Dirichlet boundary condition; in the other, we impose a Neumann condition. We apply a normal trace formula for L 2-divergence-measure fields to prove a new strong trace property in the part of the boundary where the Neumann condition is imposed. We prove the existence and uniqueness of the entropy solution. PubDate: 2017-12-01 DOI: 10.1007/s00205-017-1148-6 Issue No:Vol. 226, No. 3 (2017)

Authors:Amit Acharya; Gui-Qiang G. Chen; Siran Li; Marshall Slemrod; Dehua Wang Pages: 1009 - 1060 Abstract: We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections for the case of two spatial dimensions, and demonstrate that the continuum mechanical equations can be mapped into a corresponding geometric framework and the inherent direct application of the theory of isometric embeddings and the Gauss–Codazzi equations through examples for the Euler equations for fluids and the Euler–Lagrange equations for elastic solids. These results show that the geometric theory provides an avenue for addressing the admissibility criteria for nonlinear conservation laws in continuum mechanics. PubDate: 2017-12-01 DOI: 10.1007/s00205-017-1149-5 Issue No:Vol. 226, No. 3 (2017)

Authors:Christophe Gomez; Olivier Pinaud Pages: 1061 - 1138 Abstract: This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the white-noise limit, where random fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity field, we derive the fractional Itô–Schrödinger equation, that is, a Schrödinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence. PubDate: 2017-12-01 DOI: 10.1007/s00205-017-1150-z Issue No:Vol. 226, No. 3 (2017)

Authors:Jan Cristina; Lassi Päivärinta Pages: 1139 - 1160 Abstract: We study the evolution equation \({\partial_{t}u=-\Lambda_{t}u}\) where \({\Lambda_{t}}\) is the Dirichlet–Neumann operator of a decreasing family of Riemannian manifolds with boundary \({\Sigma_{t}}\) . We derive a lower bound for the solution of such an equation, and apply it to a quantitative density estimate for the restriction of harmonic functions on \({\mathcal{M}=\Sigma_{0}}\) to the boundaries of \({\partial\Sigma_{t}}\) . Consequently we are able to derive a lower bound for the difference of the Dirichlet–Neumann maps in terms of the difference of a background metrics g and an inclusion metric \({g+\chi_{\Sigma}(h-g)}\) on a manifold \({\mathcal{M}}\) . PubDate: 2017-12-01 DOI: 10.1007/s00205-017-1151-y Issue No:Vol. 226, No. 3 (2017)

Authors:Michael Ruzhansky; Niyaz Tokmagambetov Pages: 1161 - 1207 Abstract: Given a Hilbert space \({\mathcal{H}}\) , we investigate the well-posedness of the Cauchy problem for the wave equation for operators with a discrete non-negative spectrum acting on \({\mathcal{H}}\) . We consider the cases when the time-dependent propagation speed is regular, Hölder, and distributional. We also consider cases when it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of “very weak solutions” to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the harmonic and anharmonic oscillators, the Landau Hamiltonian on \({\mathbb{R}^n}\) , uniformly elliptic operators of different orders on domains, Hörmander’s sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others. PubDate: 2017-12-01 DOI: 10.1007/s00205-017-1152-x Issue No:Vol. 226, No. 3 (2017)

Authors:Alexey B. Stepanov; Stuart S. Antman Pages: 1209 - 1247 Abstract: This paper treats radially symmetric motions of nonlinearly viscoelastic circular-cylindrical and spherical shells subjected to the live loads of centrifugal force and (time-dependent) hydrostatic pressures. The governing equations are exact versions of those for 3-dimensional continuum mechanics (so shell does not connote an approximate via some shell theory). These motions are governed by quasilinear third-order parabolic-hyperbolic equations having but one independent spatial variable. The principal part of such a partial differential equation is determined by a general family of nonlinear constitutive equations. The presence of strains in two orthogonal directions requires a careful treatment of constitutive restrictions that are physically natural and support the analysis. The interaction of geometrically exact formulations, the compatible use of general constitutive equations for material response, and the presence of live loads show how these factors play crucial roles in the behavior of solutions. In particular, for different kinds of live loads there are thresholds separating materials that produce qualitatively different dynamical behavior. The analysis (using classical methods) covers infinite-time blowup for cylindrical shells subject to centrifugal forces, infinite-time blowup for cylindrical shells subject to steady and time-dependent hydrostatic pressures, finite-time blowup for spherical shells subject to steady and time-dependent hydrostatic pressures, and the preclusion of total compression. This paper concludes with a sketch (using some modern methods) of the existence of regular solutions until the time of blowup. PubDate: 2017-12-01 DOI: 10.1007/s00205-017-1153-9 Issue No:Vol. 226, No. 3 (2017)

Authors:Qinglong Zhou; Yiming Long Pages: 1249 - 1301 Abstract: In this paper, we use the central configuration coordinate decomposition to study the linearized Hamiltonian system near the 3-body elliptic Euler solutions. Then using the Maslov-type \({\omega}\) -index theory of symplectic paths and the theory of linear operators we compute the \({\omega}\) -indices and obtain certain properties of linear stability of the Euler elliptic solutions of the classical three-body problem. PubDate: 2017-12-01 DOI: 10.1007/s00205-017-1154-8 Issue No:Vol. 226, No. 3 (2017)

Authors:Alberto Bressan; Geng Chen Pages: 1303 - 1343 Abstract: The nonlinear wave equation \({u_{tt}-c(u)(c(u)u_x)_x=0}\) determines a flow of conservative solutions taking values in the space \({H^1(\mathbb{R})}\) . However, this flow is not continuous with respect to the natural H 1 distance. The aim of this paper is to construct a new metric which renders the flow uniformly Lipschitz continuous on bounded subsets of \({H^1(\mathbb{R})}\) . For this purpose, H 1 is given the structure of a Finsler manifold, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, one can carefully estimate how the weighted length grows in time. By the generic regularity result proved in [7], these piecewise regular paths are dense and can be used to construct a geodesic distance with the desired Lipschitz property. PubDate: 2017-12-01 DOI: 10.1007/s00205-017-1155-7 Issue No:Vol. 226, No. 3 (2017)

Authors:Scott Armstrong; Tuomo Kuusi; Jean-Christophe Mourrat; Christophe Prange Pages: 695 - 741 Abstract: We prove quantitative estimates on the rate of convergence for the oscillating Dirichlet problem in periodic homogenization of divergence-form uniformly elliptic systems. The estimates are optimal in dimensions larger than three and new in every dimension. We also prove a regularity estimate on the homogenized boundary condition. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1142-z Issue No:Vol. 226, No. 2 (2017)

Authors:Mitsuo Higaki; Yasunori Maekawa; Yuu Nakahara Abstract: We study the two-dimensional stationary Navier–Stokes equations describing the flows around a rotating obstacle. The unique existence of solutions and their asymptotic behavior at spatial infinity are established when the rotation speed of the obstacle and the given exterior force are sufficiently small. PubDate: 2017-11-22 DOI: 10.1007/s00205-017-1201-5

Authors:Aifang Qu; Wei Xiang Abstract: In this paper, we study the stability of the three-dimensional jet created by a supersonic flow past a concave cornered wedge with the lower pressure at the downstream. The gas beyond the jet boundary is assumed to be static. It can be formulated as a nonlinear hyperbolic free boundary problem in a cornered domain with two characteristic free boundaries of different types: one is the rarefaction wave, while the other one is the contact discontinuity, which can be either a vortex sheet or an entropy wave. A more delicate argument is developed to establish the existence and stability of the square jet structure under the perturbation of the supersonic incoming flow and the pressure at the downstream. The methods and techniques developed here are also helpful for other problems involving similar difficulties. PubDate: 2017-11-21 DOI: 10.1007/s00205-017-1197-x

Authors:Cristian E. Gutiérrez; Ahmad Sabra Abstract: We show the existence of a lens, when its lower face is given, such that it refracts radiation emanating from a planar source, with a given field of directions, into the far field that preserves a given distribution of energies. Conditions are shown under which the lens obtained is physically realizable. It is shown that the upper face of the lens satisfies a pde of Monge-Ampère type. PubDate: 2017-11-17 DOI: 10.1007/s00205-017-1196-y

Authors:Diego Chamorro; Pierre-Gilles Lemarié-Rieusset; Kawther Mayoufi Abstract: We study the role of the pressure in the partial regularity theory for weak solutions of the Navier–Stokes equations. By introducing the notion of dissipative solutions, due to Duchon and Robert (Nonlinearity 13:249–255, 2000), we will provide a generalization of the Caffarelli, Kohn and Nirenberg theory. Our approach sheels new light on the role of the pressure in this theory in connection to Serrin’s local regularity criterion. PubDate: 2017-11-15 DOI: 10.1007/s00205-017-1191-3

Authors:Alessio Figalli; Connor Mooney Abstract: A developable cone (“d-cone”) is the shape made by an elastic sheet when it is pressed at its center into a hollow cylinder by a distance \({\varepsilon}\) . Starting from a nonlinear model depending on the thickness h > 0 of the sheet, we prove a \({\Gamma}\) -convergence result as \({h \rightarrow 0}\) to a fourth-order obstacle problem for curves in \({\mathbb{S}^2}\) . We then describe the exact shape of minimizers of the limit problem when \({\varepsilon}\) is small. In particular, we rigorously justify previous results in the physics literature. PubDate: 2017-11-11 DOI: 10.1007/s00205-017-1195-z

Authors:Yuning Liu; Wei Wang Abstract: We study the relationship between Onsager’s molecular theory, which involves the effects of nonlocal molecular interactions and the Oseen–Frank theory for nematic liquid crystals. Under the molecular setting, we prove the existence of global minimizers for the generalized Onsager’s free energy, subject to a nonlocal boundary condition which prescribes the second moment of the number density function near the boundary. Moreover, when the re-scaled interaction distance tends to zero, the global minimizers will converge to a uniaxial distribution predicted by a minimizing harmonic map. This is achieved through the investigations of the compactness property and the boundary behaviors of the corresponding second moments. A similar result is established for critical points of the free energy that fulfill a natural energy bound. PubDate: 2017-11-08 DOI: 10.1007/s00205-017-1180-6

Authors:Alessio Brancolini; Carolin Rossmanith; Benedikt Wirth Abstract: We consider two different variational models of transport networks: the so-called branched transport problem and the urban planning problem. Based on a novel relation to Mumford–Shah image inpainting and techniques developed in that field, we show for a two-dimensional situation that both highly non-convex network optimization tasks can be transformed into a convex variational problem, which may be very useful from analytical and numerical perspectives. As applications of the convex formulation, we use it to perform numerical simulations (to our knowledge this is the first numerical treatment of urban planning), and we prove a lower bound for the network cost that matches a known upper bound (in terms of how the cost scales in the model parameters) which helps better understand optimal networks and their minimal costs. PubDate: 2017-11-04 DOI: 10.1007/s00205-017-1192-2

Authors:Onur Alper Abstract: In [2], Hardt, Lin and the author proved that the defect set of minimizers of the modified Ericksen energy for nematic liquid crystals consists locally of a finite union of isolated points and Hölder continuous curves with finitely many crossings. In this article, we show that each Hölder continuous curve in the defect set is of finite length. Hence, locally, the defect set is rectifiable. For the most part, the proof closely follows the work of De Lellis et al. (Rectifiability and upper minkowski bounds for singularities of harmonic q-valued maps, arXiv:1612.01813, 2016) on harmonic \({\mathcal{Q}}\) -valued maps. The blow-up analysis in Alper et al. (Calc Var Partial Differ Equ 56(5):128, 2017) allows us to simplify the covering arguments in [11] and locally estimate the length of line defects in a geometric fashion. PubDate: 2017-11-03 DOI: 10.1007/s00205-017-1193-1

Authors:Claude Bardos; Edriss S. Titi Abstract: The goal of this note is to show that, in a bounded domain \({\Omega \subset \mathbb{R}^n}\) , with \({\partial \Omega\in C^2}\) , any weak solution \({(u(x,t),p(x,t))}\) , of the Euler equations of ideal incompressible fluid in \({\Omega\times (0,T) \subset \mathbb{R}^n\times\mathbb{R}_t}\) , with the impermeability boundary condition \({u\cdot \vec n =0}\) on \({\partial\Omega\times(0,T)}\) , is of constant energy on the interval (0,T), provided the velocity field \({u \in L^3((0,T); C^{0,\alpha}(\overline{\Omega}))}\) , with \({\alpha > \frac13.}\) PubDate: 2017-11-02 DOI: 10.1007/s00205-017-1189-x

Authors:Xiangdi Huang; Jing Li Abstract: For the three-dimensional full compressible Navier–Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid, we establish the global existence and uniqueness of classical solutions with smooth initial data which are of small energy but possibly large oscillations where the initial density is allowed to vanish. Moreover, for the initial data, which may be discontinuous and contain vacuum states, we also obtain the global existence of weak solutions. These results generalize previous ones on classical and weak solutions for initial density being strictly away from a vacuum, and are the first for global classical and weak solutions which may have large oscillations and can contain vacuum states. PubDate: 2017-11-02 DOI: 10.1007/s00205-017-1188-y