51 research outputs found
On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions
We find a new class of data for which the Prandtl boundary layer equations
and the hydrostatic Euler equations are locally in time well-posed. In the case
of the Prandtl equations, we assume that the initial datum is monotone on
a number of intervals (on some strictly increasing on some strictly decreasing)
and analytic on the complement and show that the local existence and uniqueness
hold. The same is true for the hydrostatic Euler equations except that we
assume this for the vorticity
On the inviscid limit of the Navier-Stokes equations
We consider the convergence in the norm, uniformly in time, of the
Navier-Stokes equations with Dirichlet boundary conditions to the Euler
equations with slip boundary conditions. We prove that if the Oleinik
conditions of no back-flow in the trace of the Euler flow, and of a lower bound
for the Navier-Stokes vorticity is assumed in a Kato-like boundary layer, then
the inviscid limit holds.Comment: Improved the main result and fixed a number of typo
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
Ill-posedness of the hydrostatic Euler and singular Vlasov equations
In this paper, we develop an abstract framework to establish ill-posedness in
the sense of Hadamard for some nonlocal PDEs displaying unbounded unstable
spectra. We apply it to prove the ill-posedness for the hydrostatic Euler
equations as well as for the kinetic incompressible Euler equations and the
Vlasov-Dirac-Benney system
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