COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH RIPPED DENSITY

Raphaël Danchin (1) , Piotr Bogusław Mucha

1 UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12

Here we prove the all-time propagation of the Sobolev regularity for the velocity field solution of the two-dimensional compressible Navier-Stokes equations, provided the volume (bulk) viscosity coefficient is large enough. The initial velocity can be arbitrarily large and the initial density is just required to be bounded. In particular, one can consider a characteristic function of a set as an initial density. Uniqueness of the solutions to the equations is shown, in the case of a perfect gas. As a by-product of our results, we give a rigorous justification of the convergence to the inhomogeneous incompressible Navier-Stokes equations when the volume viscosity tends to infinity. Similar results are proved in the three-dimensional case, under some scaling invariant smallness condition on the velocity field.

Domaines: Equations aux dérivées partielles [math.AP]

Raphaël Danchin, Piotr Bogusław Mucha. COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH RIPPED DENSITY. Communications on Pure and Applied Mathematics, 2023, 76 (11), pp.3437-3492. ⟨10.1002/cpa.22116⟩. ⟨hal-02075922v2⟩