Archive for Rational Mechanics and Analysis最新文献

Surface tension at cavity walls can play havoc with the mechanical properties of perforated soft solids when the cavities are filled with a fluid. This study is an investigation of the macroscopic elastic properties of elastomers embedding spherical cavities filled with a pressurized liquid in the presence of surface tension, starting with the linearization of the fully nonlinear model and ending with the enhancement properties of the linearized model when many such liquid filled cavities are present.

We investigate the existence of minimizers of variational models featuring Eulerian–Lagrangian formulations. We consider energy functionals depending on the deformation of a body, defined on its reference configuration, and an Eulerian map defined on the unknown deformed configuration in the actual space. Our existence theory moves beyond the purely elastic setting and accounts for material failure by addressing free-discontinuity problems where both deformations and Eulerian fields are allowed to jump. To do this, we build upon the work of Henao and Mora-Corral regarding the variational modeling of cavitation and fracture in nonlinear elasticity. Two main settings are considered by modeling deformations as Sobolev and SBV-maps, respectively. The regularity of Eulerian maps is specified in each of these two settings according to the geometric and topological properties of the deformed configuration. We present some applications to specific models of liquid crystals, phase transitions, and ferromagnetic elastomers. Effectiveness and limitations of the theory are illustrated by means of explicit examples.

For any (analytic) axisymmetric toroidal domain (Omega subset mathbb {R}^3) we prove that there is a locally generic set of divergence-free vector fields that are not topologically equivalent to any magnetohydrostatic (MHS) state in (Omega ). Each vector field in this set is Morse–Smale on the boundary, does not admit a nonconstant first integral, and exhibits fast growth of periodic orbits; in particular this set is residual in the Newhouse domain. The key dynamical idea behind this result is that a vector field with a dense set of nondegenerate periodic orbits cannot be topologically equivalent to a generic MHS state. On the analytic side, this geometric obstruction is implemented by means of a novel rigidity theorem for the relaxation of generic magnetic fields with a suitably complex orbit structure.

We provide a suitable generalisation of Pansu’s differentiability theorem to general Radon measures on Carnot groups and we show that if Lipschitz maps between Carnot groups are Pansu-differentiable almost everywhere for some Radon measures (mu ), then (mu ) must be absolutely continuous with respect to the Haar measure of the group.

We consider steady states of the two-dimensional incompressible Euler equations on ({mathbb {T}}^2) and construct smooth and singular steady states around a particular singular steady state. More precisely, we construct families of smooth and singular steady solutions that converge to the Bahouri–Chemin patch.

Ideal magnetohydrodynamic (MHD) equilibria on a Riemannian 3-manifold satisfy the stationary Euler equations for ideal fluids. A stationary solution X admits a large set of “adapted” metrics on M for which X solves the corresponding MHD equilibrium equations with the same pressure function. We prove different versions of the following statement: an MHD equilibrium with non-constant pressure on a compact three-manifold with or without boundary admits no continuous Killing symmetries for an open and dense set of adapted metrics. This contrasts with the classical conjecture of Grad which loosely states that an MHD equilibrium on a toroidal Euclidean domain in ({mathbb {R}}^3) with pressure function foliating the domain with nested toroidal surfaces must admit Euclidean symmetries.

We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. On the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed.

We prove the well-posed character of a regularity structure formulation of the quasilinear generalized (KPZ) equation and give an explicit form for a renormalized equation in the full subcritical regime. Under the assumption that the BPHZ models associated with a non-translation invariant operator converge, we obtain a convergence result for the solutions of the regularized renormalized equations. This conditional result covers the spacetime white noise case.

The purpose of this paper is to prove new fine regularity results for nonlocal drift-diffusion equations via pointwise potential estimates. Our analysis requires only minimal assumptions on the divergence free drift term, enabling us to include drifts of critical order belonging merely to BMO. In particular, our results allow us to derive new estimates for the dissipative surface quasi-geostrophic equation.

We report a proof that under natural assumptions shock profiles viewed as heteroclinic travelling wave solutions to a hyperbolically regularized system of conservation laws are spectrally stable if the shock amplitude is sufficiently small. This means that an associated Evans function (mathcal {E}:Lambda rightarrow mathbb {C}) with (Lambda subset mathbb {C}) an open superset of the closed right half plane (mathbb {H}^+equiv {kappa in mathbb {C}:text {Re},kappa geqq 0}) has only one zero, namely, a simple zero at 0. The result is analogous to the one obtained in Freistühler and Szmolyan (Arch Ration Mech Anal 164:287–309, 2002) and Plaza and Zumbrun (Discrete Contin Dyn Syst 10(4):885–924, 2004) for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in Mascia and Zumbrun (Partial Differ Equ 34(1–3):119–136, 2009), Plaza and Zumbrun (2004) and Ueda (Math Methods Appl Sci 32(4):419–434, 2009).