Archive for Rational Mechanics and Analysis最新文献

In this paper, we rigorously prove the existence of self-similar converging shock wave solutions for the non-isentropic Euler equations for (gamma in (1,3]). These solutions are analytic away from the shock interface before collapse, and the shock wave reaches the origin at the time of collapse. The region behind the shock undergoes a sonic degeneracy, which causes numerous difficulties for smoothness of the flow and the analytic construction of the solution. The proof is based on continuity arguments, nonlinear invariances, and barrier functions.

We study the multiscale viscoelastic Doi model for suspensions of Brownian rigid rod-like particles, as well as its generalization by Saintillan and Shelley for self-propelled particles. We consider the regime of a small Weissenberg number, which corresponds to a fast rotational diffusion compared to the fluid velocity gradient, and we analyze the resulting hydrodynamic approximation. More precisely, we show the asymptotic validity of macroscopic nonlinear viscoelastic models, in form of so-called ordered fluid models, as an expansion in the Weissenberg number. The result holds for zero Reynolds number in 3D and for arbitrary Reynolds number in 2D. Along the way, we establish several new well-posedness and regularity results for nonlinear fluid models, which may be of independent interest.

This paper provides a new admissibility criterion for choosing physically relevant weak solutions of the equations of Lagrangian and continuum mechanics when non-uniqueness of solutions to the initial value problem occurs. The criterion is motivated by the classical least action principle but is now applied to initial value problems which exhibit non-unique solutions. Examples are provided for Lagrangian mechanics and the Euler equations of barotropic fluid mechanics. In particular, we show that the least action admissibility principle prefers the classical two shock solution to the Riemann initial value problem to certain solutions generated by convex integration. On the other hand, Dafermos’s entropy criterion prefers convex integration solutions to the two shock solutions. Furthermore, when the pressure is given by (p(rho )=rho ^2), we show that the two shock solution is always preferred whenever the convex integration solutions are defined for the same initial data.

We prove that p-harmonic systems with antisymmetric potentials of the form

((Omega ) is antisymmetric) can be written in divergence form as a conservation law

This extends to the p-harmonic framework the original work of the second author for (p=2) (see Rivière in Invent Math 168(1):1–22, 2007). We give applications of the existence of this divergence structure in the analysis (prightarrow 2).

We study the BCS critical temperature on half-spaces in dimensions (d=1,2,3) with Dirichlet or Neumann boundary conditions. We prove that the critical temperature on a half-space is strictly higher than on (mathbb {R}^d), at least at weak coupling in (d=1,2) and weak coupling and small chemical potential in (d=3). Furthermore, we show that the relative shift in critical temperature vanishes in the weak coupling limit.

This paper is focused on the local well-posedness of initial-boundary value and Cauchy problems to a one-dimensional quasilinear hyperbolic-parabolic coupled system with boundary or far field degenerate initial data. The governing system is derived from the theory of nematic liquid crystals, which couples a hyperbolic equation describing the crystal property and a parabolic equation describing the liquid property of the material. The hyperbolic equation is degenerate at the boundaries or spatial infinity, which results in the classical methods for the strictly hyperbolic-parabolic coupled systems being invalid. We introduce admissible weighted function spaces and apply the parametrix method to construct iteration mappings for these two degenerate problems separately. The local existence and uniqueness of classical solutions of the degenerate initial-boundary value and Cauchy problems are established by the contraction mapping principle in their selected function spaces. Moreover, the solutions have no loss of regularity and their existence times are independent of the spatial variable.

We study the periodic homogenization problem of state-constraint Hamilton–Jacobi equations on perforated domains in the convex setting and obtain the optimal convergence rate. We then consider a dilute situation in which the diameter of the holes is much smaller than the microscopic scale. Finally, a homogenization problem with domain defects where some holes are missing is analyzed.

We give a proof of local decay estimates for Schrödinger-type equations, which is based on the knowledge of Asymptotic Completeness. This approach extends to time dependent potential perturbations, as it does not rely on Resolvent Estimates or related methods. Global in time Strichartz estimates follow for quasi-periodic time-dependent potentials from our results.

Motivated by a model for lipid bilayer cell membranes, we study the minimization of the Willmore functional in the class of oriented closed surfaces with prescribed total mean curvature, prescribed area, and prescribed genus. Adapting methods previously developed by Keller–Mondino–Rivière, Bauer–Kuwert, and Ndiaye–Schätzle, we prove the existence of smooth minimizers for a large class of constraints. Moreover, we analyze the asymptotic behaviour of the energy profile close to the unit sphere and consider the total mean curvature of axisymmetric surfaces.

We solve explicitly a certain minimization problem for probability measures involving an interaction energy that is repulsive at short distances and attractive at large distances. We complement earlier works by showing that in an optimal part of the remaining parameter regime all minimizers are uniform distributions on a surface of a sphere, thus showing concentration on a lower dimensional set. Our method of proof uses convexity estimates on hypergeometric functions.