Archive for Rational Mechanics and Analysis最新文献

We establish the existence of infinitely many statistically stationary solutions, as well as ergodic statistically stationary solutions, to the three dimensional Navier–Stokes and Euler equations in both deterministic and stochastic settings, driven by additive noise. These solutions belong to the regularity class (C({{mathbb {R}}};H^{vartheta })cap C^{vartheta }({{mathbb {R}}};L^{2})) for some (vartheta >0) and satisfy the equations in an analytically weak sense. The solutions to the Euler equations are obtained as vanishing viscosity limits of statistically stationary solutions to the Navier–Stokes equations. Furthermore, regardless of their construction, every statistically stationary solution to the Euler equations within this regularity class, which satisfies a suitable moment bound, is a limit in law of statistically stationary analytically weak solutions to Navier–Stokes equations with vanishing viscosities. Our results are based on a novel stochastic version of the convex integration method, which provides uniform moment bounds in the aforementioned function spaces.

For axisymmetric flows without swirl and compactly supported initial vorticity, we prove the upper bound of (t^{4/3}) for the growth of the vorticity maximum, which was conjectured by Childress (Phys. D 237(14-17):1921-1925, 2008) and supported by numerical computations from Childress–Gilbert–Valiant (J. Fluid Mech. 805:1-30, 2016). The key is to estimate the velocity maximum by the kinetic energy together with conserved quantities involving the vorticity.

We consider a nonlinear model equation, known as the Localized Induction Equation, describing the motion of a vortex filament immersed in an incompressible and inviscid fluid. We show stability estimates for an arc-shaped vortex filament, which is an exact solution to an initial-boundary value problem for the Localized Induction Equation. An arc-shaped filament travels along an axis at a constant speed without changing its shape, and is oriented in such a way that the arc stays in a plane that is perpendicular to the axis. We prove that an arc-shaped filament is stable in the Lyapunov sense for general perturbations except in the axis-direction, for which the perturbation can grow linearly in time. We also show that this estimate is optimal. We then apply the obtained stability estimates to study the stability of a circular vortex filament under some symmetry assumptions on the initial perturbation. We do this by dividing the circular filament into arcs, apply the stability estimate to each arc-shaped filament, and combine the estimates to obtain estimates for the whole circle. The optimality of the stability estimates for an arc-shaped filament also shows that a circular filament is not stable in the Lyapunov sense, namely, certain perturbations can grow linearly in time.

We analyze the ground state energy of N fermions in a two-dimensional box interacting with an impurity particle via two-body point interactions. We show that for weak coupling, the ground state energy is asymptotically described by the polaron energy, as proposed by F. Chevy in the physics literature. The polaron energy is the solution of a nonlinear equation involving the Green’s function of the free Fermi gas and the binding energy of the two-body point interaction. We provide quantitative error estimates that are uniform in the thermodynamic limit.

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.