Archive for Rational Mechanics and Analysis最新文献

We study the limiting behavior of minimizing p-harmonic maps from a bounded Lipschitz domain (Omega subset mathbb {R}^{3}) to a compact connected Riemannian manifold without boundary and with finite fundamental group as (p nearrow 2). We prove that there exists a closed set (S_{*}) of finite length such that minimizing p-harmonic maps converge to a locally minimizing harmonic map in (Omega setminus S_{*}). We prove that locally inside (Omega ) the singular set (S_{*}) is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains. Furthermore, we establish local and global estimates for the limiting singular harmonic map. Under additional assumptions, we prove that globally in (overline{Omega }) the set (S_{*}) is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains, which is defined by a given boundary datum and (Omega ).

We consider the three-dimensional incompressible Euler equations on a bounded domain (Omega ) with (C^4) boundary. We prove that if the velocity field (u in C^{0,alpha } (Omega )) with (alpha > 0) (where we are omitting the time dependence), it follows that the corresponding pressure p of a weak solution to the Euler equations belongs to the Hölder space (C^{0, alpha } (Omega )). We also prove that away from the boundary p has (C^{0,2alpha }) regularity. In order to prove these results we use a local parametrisation of the boundary and a very weak formulation of the boundary condition for the pressure of the weak solution, as was introduced in Bardos and Titi (Philos Trans R Soc A 380, 20210073, 2022), which is different than the commonly used boundary condition for classical solutions of the Euler equations. Moreover, we provide an explicit example illustrating the necessity of this new very weak formulation of the boundary condition for the pressure. Furthermore, we also provide a rigorous derivation of this new formulation of the boundary condition for weak solutions of the Euler equations. This result is of importance for the proof of the first half of the Onsager Conjecture, the sufficient conditions for energy conservation of weak solutions to the three-dimensional incompressible Euler equations in bounded domains. In particular, the results in this paper remove the need for separate regularity assumptions on the pressure in the proof of the Onsager conjecture.

We consider the incompressible Euler equations on an analytic domain (Omega ) with a nonhomogeneous boundary condition (ucdot {textsf{n}} = {overline{u}}cdot {textsf{n}}) on (partial Omega ), where ({overline{u}}) is a given divergence-free analytic vector field. We establish the local well-posedness for u in analytic spaces without any compatibility conditions in all space dimensions. We also prove the global well-posedness in the 2D case if ({overline{u}}) decays in time sufficiently fast.

We study Maxwell’s equations with periodic coefficients in a closed waveguide. A functional analytic approach is used to formulate and to solve the radiation problem. Furthermore, we characterize the set of all bounded solutions to the homogeneous problem. The case of a compact perturbation of the medium is included, and the scattering problem and the limiting absorption principle are discussed.

Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and dampens electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in (mathbb R^3). The velocity equation in this system is the 3D Navier–Stokes equation with dissipation only in the (x_1)-direction, while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field (0, 1, 0) is globally stable in the Sobolev setting (H^3({mathbb {R}}^3)). In addition, explicit decay rates in (H^2({mathbb {R}}^3)) are also obtained. For when there is no presence of a magnetic field, the 3D anisotropic Navier–Stokes equation is not well understood and the small data global well-posedness in (mathbb R^3) remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps to stabilize the fluid.

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.