64 research outputs found
Incompressible flows with piecewise constant density
We investigate the incompressible Navier-Stokes equations with variable
density. The aim is to prove existence and uniqueness results in the case of
discontinuous ini- tial density. In dimension n = 2, 3, assuming only that the
initial density is bounded and bounded away from zero, and that the initial
velocity is smooth enough, we get the local-in-time existence of unique
solutions. Uniqueness holds in any dimension and for a wider class of velocity
fields. Let us emphasize that all those results are true for piecewise constant
densities with arbitrarily large jumps. Global results are established in
dimension two if the density is close enough to a positive constant, and in
n-dimension if, in addition, the initial velocity is small. The Lagrangian
formula- tion for describing the flow plays a key role in the analysis that is
proposed in the present paper.Comment: 32 page
New maximal regularity results for the heat equation in exterior domains, and applications
This paper is dedicated to the proof of new maximal regularity results
involving Besov spaces for the heat equation in the half-space or in bounded or
exterior domains of R^n. We strive for time independent a priori estimates in
regularity spaces of type L^1(0,T;X) where X stands for some homogeneous Besov
space. In the case of bounded domains, the results that we get are similar to
those of the whole space or of the half-space. For exterior domains, we need to
use mixed Besov norms in order to get a control on the low frequencies. Those
estimates are crucial for proving global-in-time results for nonlinear heat
equations in a critical functional framework.Comment: 28 page
- …