Archive for Rational Mechanics and Analysis最新文献

We consider a nonlocal reaction-diffusion equation that physically arises from the classical Peierls–Nabarro model for dislocations in crystalline structures. Our initial configuration corresponds to multiple slip loop dislocations in (mathbb {R}^n), (n ge 2). After suitably rescaling the equation with a small phase parameter (varepsilon >0), the rescaled solution solves a fractional Allen–Cahn equation. We show that, as (varepsilon rightarrow 0), the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature.

In this paper, we establish the transverse linear asymptotic stability of one-dimensional small-amplitude solitary waves of the gravity water-waves system. More precisely, we show that the semigroup of the linearized operator about the solitary wave decays exponentially within a spectral subspace supplementary to the space generated by the spectral projection on continuous resonant modes. The key element of the proof is to establish suitable uniform resolvent estimates. To achieve this, we use different arguments depending on the size of the transverse frequencies. For high transverse frequencies, we use reductions based on pseudodifferential calculus, for intermediate ones, we use an energy-based approach relying on the design of various appropriate energy functionals for different regimes of longitudinal frequencies and for low frequencies, we use the KP-II approximation. As a corollary of our main result, we also get the spectral stability in the unweighted energy space.

One of the unexplored benefits of studying layer potentials on smooth, closed hypersurfaces of Euclidean space is the factorization of the Neumann-Poincaré operator into a product of two self-adjoint transforms. Resurrecting some pertinent indications of Carleman and M. G. Krein, we exploit this grossly overlooked structure by confining the spectral analysis of the Neumann-Poincaré operator to the amenable (L^2)-space setting, rather than bouncing back and forth the computations between Sobolev spaces of negative or positive fractional order. An enhanced, fresh new look at symmetrizable linear transforms enters into the picture in the company of geometric/microlocal analysis techniques. The outcome is manyfold, complementing recent advances on the theory of layer potentials, in the smooth boundary setting.

We consider the homogeneous Landau equation in ({mathbb {R}}^3) with Coulomb potential and initial data in polynomially weighted (L^{3/2}). We show that there exists a smooth solution that is bounded for all positive times. The proof is based on short-time regularization estimates for the Fisher information, which, combined with the recent result of Guillen and Silvestre, yields the existence of a global-in-time smooth solution. Additionally, if the initial data belongs to (L^p) with (p>3/2), there is a unique solution. At the crux of the result is a new (varepsilon )-regularity criterion in the spirit of the Caffarelli–Kohn–Nirenberg theorem: a solution which is small in weighted (L^{3/2}) is regular. Although the (L^{3/2}) norm is a critical quantity for the Landau–Coulomb equation, using this norm to measure the regularity of solutions presents significant complications. For instance, the (L^{3/2}) norm alone is not enough to control the (L^infty ) norm of the competing reaction and diffusion coefficients. These analytical challenges caused prior methods relying on the parabolic structure of the Landau–Coulomb to break down. Our new framework is general enough to handle slowly decaying and singular initial data, and provides the first proof of global well-posedness for the Landau–Coulomb equation with rough initial data.

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.