DCDS
Well-posedness results for the Navier-Stokes equations in the rotational framework
Matthias Hieber Sylvie Monniaux
Discrete & Continuous Dynamical Systems - A 2013, 33(11&12): 5143-5151 doi: 10.3934/dcds.2013.33.5143
Consider the Navier-Stokes equations in the rotational framework either on $\mathbb{R}^3$ or on open sets $\Omega \subset \mathbb{R}^3$ subject to Dirichlet boundary conditions. This paper discusses recent well-posedness and ill-posedness results for both situations.
keywords: mild solutions. Coriolis force Navier-Stokes equations Dirichlet boundary conditions Stokes-Coriolis semigroup
DCDS-S
Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains
Sylvie Monniaux
Discrete & Continuous Dynamical Systems - S 2013, 6(5): 1355-1369 doi: 10.3934/dcdss.2013.6.1355
We present here different boundary conditions for the Navier-Stokes equations in bounded Lipschitz domains in $\mathbb{R}^3$, such as Dirichlet, Neumann or Hodge boundary conditions. We first study the linear Stokes operator associated to the boundary conditions. Then we show how the properties of the operator lead to local solutions or global solutions for small initial data.
keywords: Stokes operator Neumann Navier-Stokes equations. Dirichlet Lipschitz domains Hodge boundary conditions
CPAA
The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains
Dorina Mitrea Marius Mitrea Sylvie Monniaux
Communications on Pure & Applied Analysis 2008, 7(6): 1295-1333 doi: 10.3934/cpaa.2008.7.1295
We formulate and solve the Poisson problem for the exterior derivative operator with Dirichlet boundary condition in Lipschitz domains, of arbitrary topology, for data in Besov and Triebel-Lizorkin spaces.
keywords: Sobolev divergence equation Lipschitz domain Poisson problem Triebel-Lizorkin spaces. Exterior derivative Besov differential forms Dirichlet condition
CPAA
The maximal regularity operator on tent spaces
Pascal Auscher Sylvie Monniaux Pierre Portal
Communications on Pure & Applied Analysis 2012, 11(6): 2213-2219 doi: 10.3934/cpaa.2012.11.2213
Recently, Auscher and Axelsson gave a new approach to non-smooth boundary value problems with $L^2$ data, that relies on some appropriate weighted maximal regularity estimates. As part of the development of the corresponding $L^p$ theory, we prove here the relevant weighted maximal estimates in tent spaces $T^{p, 2}$ for $p$ in a certain open range. We also study the case $p=\infty$.
keywords: singular integral operators off-diagonal estimates tent spaces Maximal regularity

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