相关论文: Breakdown of smoothness for the Muskat problem

The susceptibility of timestepping algorithms to numerical instabilities is an important consideration when simulating partial differential equations (PDEs). Here we identify and analyze a pernicious numerical instability arising in…

In this paper we describe the metastable behavior of solutions to a class of parabolic systems. In particular, we improve some results contained in \cite{MS} by using different techniques to describe the slow motion of the internal layers.…

In this paper, we establish the global well-posedness of the one-phase Muskat problem with surface tension for small initial data. This problem describes the motion of the interface separating a wet region from a dry region within a porous…

We prove that the 3D stable Muskat problem is globally well-posed in the critical Sobolev space $\dot H^2 \cap \dot W^{1,\infty}$ provided that the semi-norm $\Vert f_0 \Vert_{\dot H^{2}}$ is small enough. Consequently, this allows the…

We give an explicit arithmetical condition which guarantees the existence of the unstable manifold of the MacKay approximate renormalisation scheme for the breakup of invariant tori in one and a half degrees of freedom Hamiltonian systems,…

We introduce a generalized version of the famous Stable Marriage problem, now based on multi-modal preference lists. The central twist herein is to allow each agent to rank its potentially matching counterparts based on more than one…

This paper is denoted to the study of dynamical behavior near explicit finite time blowup solutions for three dimensional incompressible Magnetohydrodynamics (MHD) equations. More precisely, we find a family of explicit finite time blowup…

This letter presents a scaling theory of the coalescence of two viscous spherical droplets. An initial value problem was formulated and analytically solved for the evolution of the radius of a liquid neck formed upon droplet coalescence.…

Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in smooth domains. Existence and uniqueness results are given in weighted Sobolev spaces allowing the derivatives of the…

We prove that homological stability fails for the moduli space of any simply-connected closed smooth 4-manifold in any degree of homology, unlike what happens in all dimensions $\neq 4$. We detect also the homological discrepancy between…

In this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $W^s_p(\mathbb{R})$, where ${p\in(1,2]}$ and ${s\in(1+1/p,2)}$. This is achieved by showing…

The bifurcation of an incompressible neo-Hookean thick block with a ratio of thickness to length {eta}, subject to pure bending, is considered. The two incremental equilibrium equations corresponding to a nonlinear pre-buckling state of…

For the $\mathfrak{so}(4)$ free rigid body the stability problem for the isolated equilibria has been completely solved using Lie-theoretical and topological arguments. For each case of nonlinear stability previously found we construct a…

A linear stability analysis of the hydrodynamic equations with respect to the homogeneous cooling state is carried out to identify the conditions for stability of a granular gas of rough hard spheres. The description is based on the results…

What characterises a solid is its way to respond to external stresses. Ordered solids, such crystals, display an elastic regime followed by a plastic one, both well understood microscopically in terms of lattice distortion and dislocations.…

We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and…

In this note we discuss the global dynamics of an integrable nonlocal NLS on $\mathbb{R}$, which has been the object of recent investigation by integrable systems methods. We prove two results which are in striking contrast with the case of…

The purpose of this paper is to show how the combination of the well-known results for convergence to equilibrium and conditional regularity, in addition to a short-time existence result, lead to a quick proof of the existence of global…

The Cauchy problem for the two-dimensional incompressible Euler equation is globally well-posed for smooth initial data. In this paper, we show that for a dense $G_\delta$ set of initial data, the solutions lose regularity in infinite time,…

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(\R)$ maximum principle, in the form of a new…