Archive for Rational Mechanics and Analysis最新文献

In this paper, we study the nonlinear asymptotic stability of Couette flow for the two-dimensional Navier-Stokes equation with small viscosity (nu >0) in (mathbb {T}times mathbb {R}). It is well known that the nonlinear asymptotic stability of the Couette flow depends closely on the size and regularity of the initial perturbation, which yields the stability threshold problem. This work studies the relationship between the regularity and the size of the initial perturbation that makes the nonlinear asymptotic stability hold. More precisely, we prove that if the initial perturbation is in some Gevrey-(frac{1}{s}) class with size (varepsilon nu ^{beta }) where (sin [0,frac{1}{2}]) and (beta ge frac{1-2s}{3-3s}), then the nonlinear asymptotic stability holds. We think this index is sharp.

We develop a Birman–Schwinger principle for the spherically symmetric, asymptotically flat Einstein–Vlasov system. The principle characterizes the stability properties of steady states such as the positive definiteness of an Antonov-type operator or the existence of exponentially growing modes in terms of a one-dimensional variational problem for a Hilbert–Schmidt operator. This requires a refined analysis of the operators arising from linearizing the system, which uses action-angle type variables. For the latter, a single-well structure of the effective potential for the particle flow of the steady state is required. This natural property can be verified for a broad class of singularity-free steady states. As a particular example for the application of our Birman–Schwinger principle we consider steady states where a Schwarzschild black hole is surrounded by a shell of Vlasov matter. We prove the existence of such steady states and derive linear stability if the mass of the Vlasov shell is small compared to the mass of the black hole.

We resolve a question of Carrapatoso et al. (Arch Ration Mech Anal 243(3):1565–1596, 2022) on Gaussian optimality for the sharp constant in Poincaré-Korn inequalities, under a moment constraint. We also prove stability, showing that measures with a near-optimal constant are quantitatively close to standard Gaussian.

In this paper, we establish the existence of global self-similar solutions to the 3D Muskat equation when the two fluids have the same viscosity but different densities. These self-similar solutions are globally defined in both space and time, with exact cones as their initial data. Furthermore, we estimate the difference between our self-similar solutions and solutions of the linearized equation around the flat interface in terms of critical spaces and some weighted (dot{W}^{k,infty }(mathbb {R}^2)) spaces for (k=1,2). The main ingredients of the proof are new estimates in the sense of (dot{H}^{s_1}(mathbb {R}^2) cap dot{H}^{s_2}(mathbb {R}^2)) with (3/2<s_1<2<s_2<3), which is continuously embedded in critical spaces for the 3D Muskat problem: (dot{H}^2(mathbb {R}^2)) and (dot{W}^{1,infty }(mathbb {R}^2)).

We revisit the interface fluctuation problem for the 1D Allen-Cahn equation perturbed by a small space-time white noise. We show that if the initial data is a standing wave solution to the deterministic equation, then under proper long time scale, the solution is still close to the family of traveling wave solutions. Furthermore, the motion of the interface converges to an explicit stochastic differential equation. This extends the classical result in Funaki (Probab Theory Relat Fields 102(2):221–288, 1995) to a full small noise regime, and recovers the result in Brassesco et al. (J Theor Probab 11:25–80, 1998). The proof builds on the analytic framework in Funaki (Probab Theory Relat Fields 102(2):221–288, 1995). Our main novelty is the construction of a series of functional correctors that are designed to recursively cancel potential divergences. Moreover, to show that these correctors are well-behaved, we develop a systematic decomposition of Fréchet derivatives of the deterministic Allen-Cahn flow of all orders. This decomposition is of its own interest, and may be useful in other situations as well.

Given a smooth closed embedded self-shrinker S with index I in (mathbb {R}^{n}), we construct an I-dimensional family of complete translators polynomially asymptotic to (Stimes mathbb {R}) at infinity, which answers a long-standing question by Ilmanen. We further prove that (mathbb {R}^{n+1}) can be decomposed in many ways into a one-parameter family of closed sets (coprod _{ain mathbb {R}} T_a), and each closed set (T_a) contains a complete translator asymptotic to (Stimes mathbb {R}) at infinity. If the closed set (T_a) fattens, namely it has nonempty interior, then there are at least two translators asymptotic to each other at an exponential rate, which can be viewed as a kind of nonuniqueness. We show that this fattening phenomenon is non-generic but indeed happens.

We study a stochastic pde model for an evolving set (mathbb {M}({t})subseteq {mathbb {R}}^{textrm{d}+1}) that resembles a continuum version of origin-excited or reinforced random walk (Benjamini and Wilson in Electron Commun Probab 8:86–92, 2003; Davis in Probab Theory Relat Fields 84(2):203–229, 1990; Kosygina and Zerner in Bull Inst Math Acad Sinica (N.S.) 8(1):105–157, 2013; Kozma in Oberwolfach Rep 27:1552, 2007; Kozma in: European congress of mathematics. European Mathematical Society, Zurich, 429–443, 2013). We show that long-time fluctuations of an associated height function are given by a regularized Kardar–Parisi–Zhang (kpz)-type pde on a hypersurface in ({mathbb {R}}^{textrm{d}+1}), modulated by a Dirichlet-to-Neumann operator. We also show that, for (textrm{d}+1=2), the regularization in this kpz-type equation can be removed after renormalization. To the best of our knowledge, this gives the first instance of kpz-type behavior in Laplacian growth, which investigated (for somewhat different models) in Parisi and Zheng (Phys Rev Lett 53:1791, 1984), Ramirez and Sidoravicius (J Eur Math Soc 6(3):293–334, 2004).

In this paper, we prove the well-posedness theory of compressible subsonic jet flows for a two-dimensional steady Euler system with general incoming horizontal velocity as long as the flux is larger than a critical value. One of the key observations is that the stream function formulation for two-dimensional compressible steady Euler system enjoys a variational structure even when the flows have nontrivial vorticity, so that the jet problem can be reformulated as a domain variation problem. This variational structure helps to adapt the framework developed by Alt, Caffarelli, and Friedman to study the jet problem, which is a Bernoulli-type free boundary problem. A major technical point for analyzing the jet flows is that the inhomogeneous terms in the rescaled equation near the free boundary are always small, even when the vorticity of the flows is big.

We deal with the inverse problem of reconstructing acoustic material properties or/and external sources for the time-domain acoustic wave model. The traditional measurements consist of repeated active (or passive) interrogations, such as the Dirichlet-Neumann map, or point sources with source points varying outside of the domain of interest. It is reported in the existing literature that based on such measurements, one can recover some (but not all) of the three parameters: mass density, bulk modulus or the external source term. In this work, we first inject isolated small-scales bubbles into the region of interest and then measure the generated pressure field at a single point outside, or at the boundary, of this region. Then we repeat such measurements by moving the bubble to scan the region of interest. Using such measurements, we show that

1. 1.

   If either the mass density or the bulk modulus is known then we can simultaneously reconstruct the other one and the source term.

2. 2.

   If the source term is known at the initial time, precisely we assume to know its first non vanishing time-derivative, at the initial time, then we reconstruct simultaneously the two parameters, namely the mass density with the bulk modulus and eventually the source function.

Here, the source term, which is space-time dependent, can be active (and hence known) or passive (and unknown). It is worth mentioning that in the induced inverse problem, we use measurements with (4=3+1) dimensions (3 in space and 1 in time) to recover 2 coefficients of 3 spatial dimensions, i.e. the mass density and the bulk modulus and the 4 = 3 + 1 dimensional source function. In addition, the result is local, meaning that we do reconstruction in any subpart, of the domain of interest, we want.