Archive for Rational Mechanics and Analysis最新文献

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.

In the limit of vanishing lattice spacing we provide a rigorous variational coarse-graining result for a next-to-nearest neighbor lattice model of a simple crystal. We show that the (Gamma )-limit of suitable scaled versions of the model leads to an energy describing a continuum mechanical model depending on partial dislocations and stacking faults. Our result highlights the necessary multiscale character of the energies setting the groundwork for more comprehensive models that can better explain and predict the mechanical behavior of materials with complex defect structures.

We rigorously construct the first steady traveling wave solutions of the 2D incompressible Euler equation that take the form of a contiguous vortex-patch dipole, which can be viewed as the vortex-patch counterpart of the well-known Lamb–Chaplygin dipole. Our construction is based on a novel fixed-point approach that determines the patch boundary as the fixed point of a certain nonlinear map. Smoothness and other properties of the patch boundary are also obtained.

We consider a family of isolated inhomogeneous steady states of the gravitational Vlasov–Poisson system with a point mass at the centre. These are parametrised by the polytropic index (k>1/2), so that the phase space density of the steady state is (C^1) at the vacuum boundary if and only if (k>1). We prove the following sharp dichotomy result: if (k>1), the linear perturbations Landau damp and if (1/2< kle 1) they do not. The above dichotomy is a new phenomenon and highlights the importance of steady state regularity at the vacuum boundary in the discussion of the long-time behaviour of the perturbations. Our proof of (nonquantitative) gravitational relaxation around steady states with (k>1) is the first such result for the gravitational Vlasov–Poisson system. The key novelty of this work is the proof that no embedded eigenvalues exist in the essential spectrum of the linearised system.

We study a dilute system of N interacting bosons coupled to an impurity particle via a pair potential in the Gross–Pitaevskii regime. We derive an expansion of the ground state energy up to order one in the boson number, and show that the difference of excited eigenvalues to the ground state is given by the eigenvalues of the renormalized Bogoliubov–Fröhlich Hamiltonian in the limit (Nrightarrow infty ).