Related papers: Euler Equations on General Planar Domains
In this paper, we address the problem of weak solutions of Yudovich type for the inviscid MHD equations in two dimensions. The local-in-time existence and uniqueness of these solutions sound to be hard to achieve due to some terms involving…
A well-known result of Carrillo, Choi, Tadmor, and Tan states that the 1D Euler Alignment model with smooth interaction kernels possesses a 'critical threshold' criterion for the global existence or finite-time blowup of solutions,…
In this paper, we study two-dimensional steady incompressible Euler flows in which the vorticity is sharply concentrated in a finite number of regions of small diameter in a bounded domain. Mathematical analysis of such flows is an…
We study the stationary Navier--Stokes equations in the whole plane with a compactly supported force term and with a prescribed constant spatial limit. Prior to this work, existence of solutions to this problem was only known under special…
We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and non-trivial solutions in bounded domains. Roughly speaking, the boundary of the domain serves as an outer obstacle, and the evolving…
In this article we examine the interaction of incompressible 2D flows with compact material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow…
We give a short proof of local well-posedness for the initial boundary value problem in general relativity with sole boundary condition the requirement that the boundary is umbilic. This includes as a special case the totally geodesic…
We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the…
We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with…
In this paper, we establish the asymptotic stability of the steady-state for a 1-D stochastic Euler-Poisson equations with Ohmic contact boundary conditions forced by the Wiener process. We utilize Banach's fixed point theorem and the a…
We are concerned with the (stochastic) Lagrangian trajectories associated with Euler or Navier-Stokes equations. First, in the vanishing viscosity limit, we establish sharp non-uniqueness results for positive solutions to transport…
This paper contributes to the study of large data problems for $C^1$ solutions of the relativistic Euler equations. In the $(1+1)$-dimensional spacetime setting, if the initial data are away from vacuum, a key difficulty in proving the…
In a smooth bounded domain we obtain existence, uniqueness, regularity and boundary behavior for a class of singular quasi-linear elliptic equations.
We study global regularity for solutions of quasilinear elliptic equations of the form $\div \A(x,u,\nabla u) = \div \F $ in rough domains $\Omega$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed…
We show that given an initial vorticity which is bounded and $m$-fold rotationally symmetric for $m \ge 3$, there is a unique global solution to the 2D Euler equation on the whole plane. That is, in the well-known $L^1 \cap L^\infty$ theory…
This article is concerned with the local well-posedness problem for the compressible Euler equations in gas dynamics. For this system we consider the free boundary problem which corresponds to a physical vacuum. Despite the clear physical…
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in $\R^n$ with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat…
We consider rotational initial data for the two-dimensional incompressible Euler equations on an annulus. Using the convex integration framework, we show that there exist infinitely many admissible weak solutions (i.e. such with…
We consider incompressible Euler equations in any dimension $ d\geq3 $ imposing axisymmetric symmetry without swirl. While the global regularity of smooth flows in this setting has been well-known in $ d=3 $, the same question in higher…
We consider the three-dimensional incompressible Navier-Stokes equations in a bounded domain with Navier boundary conditions. We provide a sufficient condition for the absence of anomalous energy dissipation without making assumptions on…