Authors:Jeffrey Humpherys; Gregory Lyng; Kevin Zumbrun Pages: 923 - 973 Abstract: 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: 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: 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: 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: 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: 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: 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: 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: 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:Zdzisław Brzeźniak; Ben Goldys; Terence Jegaraj Pages: 497 - 558 Abstract: Abstract We study a stochastic Landau–Lifshitz equation on a bounded interval and with finite dimensional noise. We first show that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Next, we prove the large deviations principle for the small noise asymptotic of solutions using the weak convergence method. An essential ingredient of the proof is the compactness, or weak to strong continuity, of the solution map for a deterministic Landau–Lifschitz equation when considered as a transformation of external fields. We then apply this large deviations principle to show that small noise can cause magnetisation reversal. We also show the importance of the shape anisotropy parameter for reducing the disturbance of the solution caused by small noise. The problem is motivated by applications from ferromagnetic nanowires to the fabrication of magnetic memories. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1117-0 Issue No:Vol. 226, No. 2 (2017)

Authors:Vieri Benci; Donato Fortunato; Filippo Gazzola Pages: 559 - 585 Abstract: Abstract This paper studies the existence of solitons, namely stable solitary waves, in an idealized suspension bridge. The bridge is modeled as an unbounded degenerate plate, that is, a central beam with cross sections, and displays two degrees of freedom: the vertical displacement of the beam and the torsional angles of the cross sections. Under fairly general assumptions, we prove the existence of solitons. Under the additional assumption of large tension in the sustaining cables, we prove that these solitons have a nontrivial torsional component. This appears relevant for security since several suspension bridges collapsed due to torsional oscillations. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1138-8 Issue No:Vol. 226, No. 2 (2017)

Authors:Zhigang Wu; Weike Wang Pages: 587 - 638 Abstract: Abstract The Cauchy problem of the bipolar Navier–Stokes–Poisson system (1.1) in dimension three is considered. We obtain the pointwise estimates of the time-asymptotic shape of the solution, which exhibit a generalized Huygens’ principle as the Navier–Stokes system. This phenomenon is the most important difference from the unipolar Navier–Stokes–Poisson system. Due to the non-conservative structure of the system (1.1) and the interplay of two carriers which counteract the influence of the electric field (a nonlocal term), some new observations are essential for the proof. We fully use the conservative structure of the system for the total density and total momentum, and the mechanism of the linearized unipolar Navier–Stokes–Poisson system together with the special form of the nonlinear terms in the system for the difference of densities and the difference of momentums. Lastly, as a byproduct, we extend the usual \({L^2({\mathbb{R}}^3)}\) -decay rate to the \({L^p({\mathbb{R}}^3)}\) -decay rate with \({p > 1}\) and also improve former decay rates. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1140-1 Issue No:Vol. 226, No. 2 (2017)

Authors:Blanche Buet; Gian Paolo Leonardi; Simon Masnou Pages: 639 - 694 Abstract: Abstract We show that the theory of varifolds can be suitably enriched to open the way to applications in the field of discrete and computational geometry. Using appropriate regularizations of the mass and of the first variation of a varifold we introduce the notion of approximate mean curvature and show various convergence results that hold, in particular, for sequences of discrete varifolds associated with point clouds or pixel/voxel-type discretizations of d-surfaces in the Euclidean n-space, without restrictions on dimension and codimension. The variational nature of the approach also allows us to consider surfaces with singularities, and in that case the approximate mean curvature is consistent with the generalized mean curvature of the limit surface. A series of numerical tests are provided in order to illustrate the effectiveness and generality of the method. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1141-0 Issue No:Vol. 226, No. 2 (2017)

Authors:Scott Armstrong; Tuomo Kuusi; Jean-Christophe Mourrat; Christophe Prange Pages: 695 - 741 Abstract: 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:Davit Harutyunyan Pages: 743 - 766 Abstract: Abstract In the paper we deal with shells with non-zero Gaussian curvature. We derive sharp Korn’s first (linear geometric rigidity estimate) and second inequalities on that kind of shell for zero or periodic Dirichlet, Neumann, and Robin type boundary conditions. We prove that if the Gaussian curvature is positive, then the optimal constant in the first Korn inequality scales like h, and if the Gaussian curvature is negative, then the Korn constant scales like h 4/3, where h is the thickness of the shell. These results have a classical flavour in continuum mechanics, in particular shell theory. The Korn first inequalities are the linear version of the famous geometric rigidity estimate by Friesecke et al. for plates in Arch Ration Mech Anal 180(2):183–236, 2006 (where they show that the Korn constant in the nonlinear Korn’s first inequality scales like h 2), extended to shells with nonzero curvature. We also recover the uniform Korn–Poincaré inequality proven for “boundary-less” shells by Lewicka and Müller in Annales de l’Institute Henri Poincare (C) Non Linear Anal 28(3):443–469, 2011 in the setting of our problem. The new estimates can also be applied to find the scaling law for the critical buckling load of the shell under in-plane loads as well as to derive energy scaling laws in the pre-buckled regime. The exponents 1 and 4/3 in the present work appear for the first time in any sharp geometric rigidity estimate. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1143-y Issue No:Vol. 226, No. 2 (2017)

Authors:Yuan Chen; Yong Yu Pages: 767 - 808 Abstract: Abstract In this article we construct a global solution of the simplified Ericksen-Leslie system. We show that the velocity of the solution can be decomposed into the sum of three parts. The main flow is governed by the Oseen vortex with the same circulation Reynolds number as the initial fluid. The secondary flow has finite kinetic energy and decay in the speed (1 + t)−2 as \({t \rightarrow \infty}\) . The third part is a minor flow whose kinetic energy decays faster than the secondary flow. As for the orientation variable, our solution has a phase function which diverges logarithmically to \({\infty}\) as \({t \rightarrow \infty}\) . This indicates that the orientation variable will keep rotating around the z-axis while \({t \rightarrow \infty}\) . This phenomenon results from a non-trivial coupling between the orientation variable and a fluid with a non-zero circulation Reynolds number. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1144-x Issue No:Vol. 226, No. 2 (2017)

Authors:U. S. Fjordholm; S. Lanthaler; S. Mishra Pages: 809 - 849 Abstract: Abstract We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on p-integrable functions. To do so, we prove the equivalence between probability measures on L p spaces and infinite families of correlation measures. Each member of this family, termed a correlation marginal, is a Young measure on a finite-dimensional tensor product domain and provides information about multi-point correlations of the underlying integrable functions. We also prove that any probability measure on a L p space is uniquely determined by certain moments (correlation functions) of the equivalent correlation measure. We utilize this equivalence to define statistical solutions of multi-dimensional conservation laws in terms of an infinite set of equations, each evolving a moment of the correlation marginal. These evolution equations can be interpreted as augmenting entropy measure-valued solutions, with additional information about the evolution of all possible multi-point correlation functions. Our concept of statistical solutions can accommodate uncertain initial data as well as possibly non-atomic solutions, even for atomic initial data. For multi-dimensional scalar conservation laws we impose additional entropy conditions and prove that the resulting entropy statistical solutions exist, are unique and are stable with respect to the 1-Wasserstein metric on probability measures on L 1. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1145-9 Issue No:Vol. 226, No. 2 (2017)

Authors:Michaela A. C. Vollmer Pages: 851 - 922 Abstract: Abstract We study the bifurcation diagram of the Onsager free-energy functional for liquid crystals with orientation parameter on the sphere. In particular, we concentrate on the bifurcations from the isotropic solution for a general class of two-body interaction potentials including the Onsager kernel. Reformulating the problem as a non-linear eigenvalue problem for the kernel operator, we prove that spherical harmonics are the corresponding eigenfunctions and we present a direct relationship between the coefficients of the Taylor expansion of this class of interaction potentials and their eigenvalues. We find explicit expressions for all bifurcation points corresponding to bifurcations from the isotropic state of the Onsager free-energy functional equipped with the Onsager interaction potential. A substantial amount of our analysis is based on the use of spherical harmonics and a special algorithm for computing expansions of products of spherical harmonics in terms of spherical harmonics is presented. Using a Lyapunov–Schmidt reduction, we derive a bifurcation equation depending on five state variables. The dimension of this state space is further reduced to two dimensions by using the rotational symmetry of the problem and the invariant theory of groups. On the basis of these results, we show that the first bifurcation from the isotropic state of the Onsager interaction potential is a transcritical bifurcation and that the corresponding solution is uniaxial. In addition, we prove some global properties of the bifurcation diagram such as the fact that the trivial solution is the unique local minimiser if the bifurcation parameter is high, that it is not a local minimiser if the bifurcation parameter is small, the boundedness of all equilibria of the functional and that the bifurcation branches are either unbounded or that they meet another bifurcation branch. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1146-8 Issue No:Vol. 226, No. 2 (2017)

Authors:Richard M. Höfer; Juan J. L. Velázquez Abstract: Abstract We study the convergence of the method of reflections for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains which exist in the exterior of balls. We prove that the method converges if the balls are contained in a bounded region and the density of the electrostatic capacity of the balls is sufficiently small. If the capacity density is too large or the balls extend to the whole space, the method diverges, but we provide a suitable modification of the method that converges to the solution of the Dirichlet problem also in this case. We give new proofs of classical homogenization results for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains using the (modified) method of reflections. PubDate: 2017-10-14 DOI: 10.1007/s00205-017-1182-4

Authors:Alberto Bressan; Marta Lewicka Abstract: Abstract We consider a free boundary problem for a system of PDEs, modeling the growth of a biological tissue. A morphogen, controlling volume growth, is produced by specific cells and then diffused and absorbed throughout the domain. The geometric shape of the growing tissue is determined by the instantaneous minimization of an elastic deformation energy, subject to a constraint on the volumetric growth. For an initial domain with \({\mathcal{C}^{2,\alpha}}\) boundary, our main result establishes the local existence and uniqueness of a classical solution, up to a rigid motion. PubDate: 2017-10-12 DOI: 10.1007/s00205-017-1183-3