1,978 research outputs found
The Two Dimensional Euler Equations on Singular Exterior Domains
This paper is a follow-up of article [Gerard-Varet and Lacave, ARMA 2013], on
the existence of global weak solutions to the two dimensional Euler equations
in singular domains. In [Gerard-Varet and Lacave, ARMA 2013], we have
established the existence of weak solutions for a large class of bounded
domains, with initial vorticity in (). For unbounded domains, we
have proved a similar result only when the initial vorticity is in
() and when the domain is the exterior of a single obstacle. The goal here
is to retrieve these two restrictions: we consider general initial vorticity in
(), outside an arbitrary number of obstacles (not reduced to
points)
Blow up for the 2D Euler Equation on Some Bounded Domains
We find a smooth solution of the 2D Euler equation on a bounded domain which
exists and is unique in a natural class locally in time, but blows up in finite
time in the sense of its vorticity losing continuity. The domain's boundary is
smooth except at two points, which are interior cusps
The Inviscid Limit and Boundary Layers for Navier-Stokes Flows
The validity of the vanishing viscosity limit, that is, whether solutions of
the Navier-Stokes equations modeling viscous incompressible flows converge to
solutions of the Euler equations modeling inviscid incompressible flows as
viscosity approaches zero, is one of the most fundamental issues in
mathematical fluid mechanics. The problem is classified into two categories:
the case when the physical boundary is absent, and the case when the physical
boundary is present and the effect of the boundary layer becomes significant.
The aim of this article is to review recent progress on the mathematical
analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of
Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final
publication is available at http://www.springerlink.co
On the Yudovich solutions for the ideal MHD equations
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 Riesz transforms in the vorticity-current formulation. We shall prove
that the vortex patches with smooth boundary offer a suitable class of initial
data for which the problem can be solved. However this is only done under a
geometric constraint by assuming the boundary of the initial vorticity to be
frozen in a magnetic field line.
We shall also discuss the stationary patches for the incompressible Euler
system and the MHD system. For example, we prove that a stationary simply
connected patch with rectifiable boundary for the system is necessarily
the characteristic function of a disc.Comment: 40 page
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