A Lagrangian approach for the compressible Navier-Stokes equations

Danchin, Raphaël

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 753-791.

On étudie le problème de Cauchy pour le système de Navier-Stokes barotrope dans ${ℝ}^n,$ avec régularité Besov critique. On affaiblit la condition d’unicité, ce qui permet d’établir entre autres que des vitesses initiales ayant une régularité Besov (pas trop) négative génèrent une solution unique. La densité initiale est à régularité critique et doit juste être strictement positive et tendre vers une constante à l’infini. Les coefficients de viscosité peuvent dépendre de la densité. L’usage de coordonnées lagrangiennes est la clef de toutes ces améliorations car il permet de résoudre le système par itérations de Picard. Comme corollaire immédiat, on obtient que les conditions pour l’unicité sont les mêmes que pour l’existence, ainsi que la continuité de l’opérateur solution (pour le système écrit en coordonnées lagrangiennes).

Here we investigate the Cauchy problem for the barotropic Navier-Stokes equations in ${ℝ}^n$, in the critical Besov spaces setting. We improve recent results as regards the uniqueness condition: initial velocities in critical Besov spaces with (not too) negative indices generate a unique local solution. Apart from (critical) regularity, the initial density just has to be bounded away from $0$ and to tend to some positive constant at infinity. Density-dependent viscosity coefficients may be considered. Using Lagrangian coordinates is the key to our statements as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence, and Lipschitz continuity of the flow map (in Lagrangian coordinates) is established.

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