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Comptes Rendus Mathématique
Volume 348, n° 15-16
pages 885-890 (août 2010)
Doi : 10.1016/j.crma.2010.06.025
Received : 4 January 2010 ;  accepted : 29 June 2010
  and   regularity of the solution of a steady transport equation
Régularité dans   et   de la solution d'une équation de transport stationnaire

Vivette Girault a, b , Luc Tartar c
a UPMC–Paris 6, CNRS, UMR 7598, 75005 Paris, France 
b Department of Mathematics, TAMU, College Station, TX 77843, USA 
c Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA 


We consider a steady transport system of equations in a bounded Lipschitz domain of  ,  , with a divergence-free transport velocity in  , tangential on the boundary. By means of two regularizations, first with a viscous penalty term and next with a Yosida approximation, we prove that an   data,  , yields a solution in  . We apply this result to establish that for data in   and transport velocity in  , sufficiently small, the solution of a scalar transport equation belongs to  .

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On considère une équation de transport vectorielle stationnaire dans un domaine Lipschitz borné de  ,  , avec une vitesse de transport dans  , à divergence nulle, tangentielle sur le bord. A l'aide de deux régularisations, d'abord avec un terme visqueux de pénalisation et ensuite avec une approximation de Yosida, on montre que si la donnée est dans  ,  , alors la solution est dans  . On applique ce résultat pour démontrer que si la donnée d'une équation de transport scalaire est dans   et la vitesse de transport est dans  , assez petite, alors la solution est dans  .

The full text of this article is available in PDF format.

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