Strong solution for Korteweg system in bmo−1(RN) with initial density in L∞

Abstract

In this paper we investigate the question of the existence of strong solution in finite time for the Korteweg system for small initial data provided that the initial momentum ρ 0 u 0 belongs to bmo −1 T (R N) for T > 0 and the initial density ρ 0 is in L ∞ (R N) with N ≥ 1 and far away from the vacuum. This result extends the so called Koch-Tataru theorem for the Korteweg system. It is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes Lipschitz for any ρ(t, ·) with t > 0. We also prove the existence of global strong solution for initial data (ρ 0 − 1, ρ 0 u 0) ∈ (B N 2 −1 2,∞ (R N) ∩ B N 2 2,∞ (R N)∩L ∞ (R N))×(B N 2 −1 2,∞ (R N)) N. This result allows in particular to extend the notion of Oseen solution (corresponding to particular solution of the incompressible Navier Stokes system in dimension N = 2) to the Korteweg system provided that the vorticity of the momentum ρ 0 u 0 is a Dirac mass αδ 0 with α sufficiently small. IHowever unlike the Navier Stokes equations the property of self similarity is not conserved for the Korteweg system since there is no invariance by scaling because the term of pressure.