Archive for Rational Mechanics and Analysis最新文献

We construct examples of oscillating solutions with persistent oscillations for various hyperbolic-parabolic systems with singular diffusion matrices that appear in mechanics. These include an example for the equations of nonlinear viscoelasticity of Kelvin–Voigt type with stored energy that violates rank-one convexity, which amounts to a time-dependent variant of twinning solutions. We also present an example pertaining to the system of gas dynamics with thermal effects for a viscous, adiabatic gas. Finally, we show an example for the compressible Navier–Stokes system in one-space dimension with nonmonotone pressure function. We also study the existence of oscillating solutions for linear hyperbolic-parabolic systems with singular diffusion matrices.

This paper concerns smooth transonic flows with nonzero vorticity in De Laval nozzles for a quasi two dimensional steady Euler flow model which is a generalization of the classical quasi one dimensional model. First we examine the existence and uniqueness of smooth transonic flows to the quasi one-dimensional model, which start from a subsonic state at the entrance and accelerate to reach a sonic state at the throat and then become supersonic are proved by a reduction of degeneracy of the velocity near the sonic point and the implicit function theorem. These flows can have positive or zero acceleration at their sonic points and the degeneracy types near the sonic point are classified precisely. We then establish the structural stability of the smooth one dimensional transonic flow with positive acceleration at the sonic point for the quasi two dimensional steady Euler flow model under small perturbations of suitable boundary conditions, which yields the existence and uniqueness of a class of smooth transonic flows with nonzero vorticity and positive acceleration to the quasi two dimensional model. The positive acceleration of the one dimensional transonic solutions plays an important role in searching for an appropriate multiplier for the linearized second order mixed type equations. A deformation-curl decomposition for the quasi two dimensional model is utilized to deal with the transonic flows with nonzero vorticity.

We study the homogenization of the Dirichlet problem for the Stokes equations in (mathbb {R}^3) perforated by m spherical particles. We assume the positions and velocities of the particles to be identically and independently distributed random variables. In the critical regime, when the radii of the particles are of order (m^{-1}), the homogenization limit u is given as the solution to the Brinkman equations. We provide optimal rates for the convergence (u_m rightarrow u) in (L^2), namely (m^{-beta }) for all (beta < 1/2). Moreover, we consider the fluctuations. In the central limit scaling, we show that these converge to a Gaussian field, locally in (L^2(mathbb {R}^3)), with an explicit covariance. Our analysis is based on explicit approximations for the solutions (u_m) in terms of u as well as the particle positions and their velocities. These are shown to be accurate in (dot{H}^1(mathbb {R}^3)) to order (m^{-beta }) for all (beta < 1). Our results also apply to the analogous problem regarding the homogenization of the Poisson equations.

We introduce an evolution model à la Firey for a convex stone which tumbles on a beach and undertakes an erosion process depending on some variational energy, such as torsional rigidity, a principal Dirichlet Laplacian eigenvalue, or Newtonian capacity. Relying on the assumption of the existence of a solution to the corresponding parabolic flow, we prove that the stone tends to become asymptotically spherical. Indeed, we identify an ultimate shape of these flows with a smooth convex body whose ground state satisfies an additional boundary condition, and we prove symmetry results for the corresponding overdetermined elliptic problems. Moreover, we extend the analysis to arbitrary convex bodies: we introduce new notions of cone variational measures and we prove that, if such a measure is absolutely continuous with constant density, the underlying body is a ball.

In this paper, we prove the asymptotic stability of Couette flow in a strong uniform magnetic field for the Euler-MHD system, when the perturbations are in Gevrey-(frac{1}{s}), ((frac{1}{2}<sleqq 1)) and of size smaller than the resistivity coefficient (mu ). More precisely, we prove that

1. (1)

   the (mu ^{-frac{1}{3}})-amplification of the perturbed vorticity, namely, the size of the vorticity grows from (Vert omega _{textrm{in}}Vert _{mathcal {G}^{lambda _{0}}}lesssim mu ) to (Vert omega _{infty }Vert _{mathcal {G}^{lambda '}}lesssim mu ^{frac{2}{3}});

2. (2)

   the polynomial decay of the perturbed current density, namely, (left| j_{ne }right| _{L^2}lesssim frac{c_0 }{langle trangle ^2 }min left{ mu ^{-frac{1}{3}},langle t rangle right} );

3. (3)

   and the damping for the perturbed velocity and magnetic field, namely,

   $$begin{aligned} left| (u^1_{ne },b^1_{ne })right| _{L^2}lesssim frac{c_0mu }{langle trangle }min left{ mu ^{-frac{1}{3}},langle t rangle right} , quad left| (u^2,b^2)right| _{L^2}lesssim frac{c_0mu }{langle trangle ^2 }min left{ mu ^{-frac{1}{3}},langle t rangle right} . end{aligned}$$

We also confirm that the strong uniform magnetic field stabilizes the Euler-MHD system near Couette flow.

The main result of this work is a homogenization theorem via variational convergence for elastic materials with stiff checkerboard-type heterogeneities under the assumptions of physical growth and non-self-interpenetration. While the obtained energy estimates are rather standard, determining the effective deformation behavior, or in other words, characterizing the weak Sobolev limits of deformation maps whose gradients are locally close to rotations on the stiff components, is the challenging part. To this end, we establish an asymptotic rigidity result, showing that, under suitable scaling assumptions, the attainable macroscopic deformations are affine conformal contractions. This identifies the composite as a mechanical metamaterial with a negative Poisson’s ratio. Our proof strategy is to tackle first an idealized model with full rigidity on the stiff tiles to acquire insight into the mechanics of the model and then transfer the findings and methodology to the model with diverging elastic constants. The latter requires, in particular, a new quantitative geometric rigidity estimate for non-connected squares touching each other at their vertices and a tailored Poincaré type inequality for checkerboard structures.

We prove the first regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain (Omega ) is obtained as the integral of a cost function j(u, x) depending on the solution u of a certain PDE problem on (Omega ). The main feature of these functionals is that the minimality of a domain (Omega ) cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions (j(u,x)=-g(x)u+Q(x)), where u is the solution of the PDE (-Delta u=f) with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal u from the inwards/outwards optimality of (Omega ) and then we use the stability of (Omega ) with respect to variations with smooth vector fields in order to study the blow-up limits of the state function u. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose (partial Omega ) into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of (partial Omega ) we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove (C^infty ) regularity of the regular part of the free boundary when the data are smooth.

The Kelvin–Planck statement of the second law of thermodynamics is a stricture on the nature of heat receipt by any body suffering a cyclic process. It makes no mention of temperature or of entropy. Beginning with a Kelvin–Planck statement of the Second Law, we show that entropy and temperature—in particular, existence of functions that relate the local specific entropy and thermodynamic temperature to the local state in a material body—emerge immediately and simultaneously as consequences of the Hahn–Banach theorem. The existence of such functions of state requires no stipulation that their domains be restricted to equilibrium states. Further properties, including uniqueness, are addressed in a companion paper.

In a companion article it was shown in a certain precise sense that, for any thermodynamical theory that respects the Kelvin–Planck second law, the Hahn–Banach theorem immediately ensures the existence of a pair of continuous functions of the local material state—a specific entropy (entropy per mass) and a thermodynamic temperature—that together satisfy the Clausius–Duhem inequality for every process. There was no requirement that the local states considered be states of equilibrium. This article addresses questions about properties of the entropy and thermodynamic temperature functions so obtained: To what extent do such temperature functions provide a faithful reflection of “hotness”? In precisely which Kelvin–Planck theories is such a temperature function essentially unique, and, among those theories, for which is the entropy function also essentially unique? What is a thermometer for a Kelvin–Planck theory, and, for the theory, what properties does the existence of a thermometer confer? In all of these questions, the Hahn–Banach Theorem again plays a crucial role.

In this paper, we introduce Prodi–Serrin like criteria which enable us to provide a priori estimates for the solutions to the spatially homogeneous Landau equation for all classical soft potentials and dimensions (d geqq 3). The physical case of Coulomb interaction in dimension (d=3) is included in our analysis; this generalizes the work of Silvestre (J Differ Equ 262:3034–3055, 2017). Our approach is quantitative and does not require a preliminary knowledge of elaborate tools for nonlinear parabolic equations.