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

In this paper we investigate the question of the local existence of strong solution for the Korteweg system in critical spaces in dimension N⩾1 provided that the initial data are small (we restrict our analysis to the case of the so‐called compressible quantum Navier–Stokes equations which corresponds to the capillary coefficient κ(ρ)=κ2ρ with κ2>0 ). More precisely the initial momentum ρ0u0 belongs to bmoT−1(RN) for T>0 and the initial density ρ0 is in L∞(RN) and far away from the vacuum. This result extends the so‐called Koch–Tataru theorem for the incompressible Navier–Stokes equations to the case of 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 small initial data (ρ0−1,ρ0u0) in the homogeneous Besov spaces (Ḃ2,∞N2−1(RN)∩Ḃ2,∞N2(RN)∩L∞(RN))×(Ḃ2,∞N2−1(RN))N . This result allows in particular to extend in dimension N=2 the notion of Oseen solutions defined for incompressible Navier–Stokes equations to the case of the Korteweg system when the vorticity of the momentum ρ0u0 is a Dirac mass αδ0 with α sufficiently small.