191 research outputs found
Existence, uniqueness and approximation of a doubly-degenerate nonlinear parabolic system modelling bacterial evolution
Accepted versio
Thermodynamic Limit Of The Ginzburg-Landau Equations
We investigate the existence of a global semiflow for the complex
Ginzburg-Landau equation on the space of bounded functions in unbounded domain.
This semiflow is proven to exist in dimension 1 and 2 for any parameter values
of the standard cubic Ginzburg-Landau equation. In dimension 3 we need some
restrictions on the parameters but cover nevertheless some part of the
Benjamin-Feijer unstable domain.Comment: uuencoded dvi file (email: [email protected]
Existence of global strong solutions in critical spaces for barotropic viscous fluids
This paper is dedicated to the study of viscous compressible barotropic
fluids in dimension . We address the question of the global existence
of strong solutions for initial data close from a constant state having
critical Besov regularity. In a first time, this article show the recent
results of \cite{CD} and \cite{CMZ} with a new proof. Our result relies on a
new a priori estimate for the velocity, where we introduce a new structure to
\textit{kill} the coupling between the density and the velocity as in
\cite{H2}. We study so a new variable that we call effective velocity. In a
second time we improve the results of \cite{CD} and \cite{CMZ} by adding some
regularity on the initial data in particular is in . In this
case we obtain global strong solutions for a class of large initial data on the
density and the velocity which in particular improve the results of D. Hoff in
\cite{5H4}. We conclude by generalizing these results for general viscosity
coefficients
A Blow-Up Criterion for Classical Solutions to the Compressible Navier-Stokes Equations
In this paper, we obtain a blow up criterion for classical solutions to the
3-D compressible Naiver-Stokes equations just in terms of the gradient of the
velocity, similar to the Beal-Kato-Majda criterion for the ideal incompressible
flow. In addition, initial vacuum is allowed in our case.Comment: 25 page
Maximal L p -regularity for the Laplacian on Lipschitz domains
We consider the Laplacian with Dirichlet or Neumann boundary
conditions on bounded Lipschitz domains ?, both with the following two domains of
definition:D1(?) = {u ? W1,p(?) : ?u ? Lp(?), Bu = 0}, orD2(?) = {u ? W2,p(?) :
Bu = 0}, where B is the boundary operator.We prove that, under certain restrictions
on the range of p, these operators generate positive analytic contraction semigroups
on Lp(?) which implies maximal regularity for the corresponding Cauchy problems.
In particular, if ? is bounded and convex and 1 < p ? 2, the Laplacian with domain
D2(?) has the maximal regularity property, as in the case of smooth domains. In the
last part,we construct an example that proves that, in general, the Dirichlet–Laplacian
with domain D1(?) is not even a closed operator
Error estimates for the discretization of the velocity tracking problem
In this paper we are continuing our work (Casas and Chrysafinos, SIAM J Numer Anal 50(5):2281–2306, 2012), concerning a priori error estimates for the velocity tracking of two-dimensional evolutionary Navier–Stokes flows. The controls are of distributed type, and subject to point-wise control constraints. The discretization scheme of the state and adjoint equations is based on a discontinuous time-stepping scheme (in time) combined with conforming finite elements (in space) for the velocity and pressure. Provided that the time and space discretization parameters, t and h respectively, satisfy t = Ch2, error estimates of order O(h2) and O(h 3/2 – 2/p ) with p > 3 depending on the regularity of the target and the initial velocity, are proved for the difference between the locally optimal controls and their discrete approximations, when the controls are discretized by the variational discretization approach and by using piecewise-linear functions in space respectively. Both results are based on new duality arguments for the evolutionary Navier–Stokes equations
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