Global wellposdeness to incompressible inhomogeneous fluid system with bounded density and non-Lipschitz velocity

Jingchi Huang (1) , Marius Paicu (2) , Ping Zhang (1)

1 AMSS - Academy of Mathematics and Systems Science
2 IMB - Institut de Mathématiques de Bordeaux

In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier-Stokes equations with initial data in critical Besov spaces, which satisfies a non-linear smallness condition. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity field. Furthermore, with additional regularity assumption on the initial velocity or on the initial density, we can also prove the uniqueness of such solution. We should mention that the classical maximal regularity theorem for the heat kernel plays an essential role in this context.

Mots clés: Inhomogeneous Navier-Stokes equations; maximal regularity for heat kernel; Littlewood-Paley theory.; Littlewood-Paley theory

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]

Jingchi Huang, Marius Paicu, Ping Zhang. Global wellposdeness to incompressible inhomogeneous fluid system with bounded density and non-Lipschitz velocity. 2012. ⟨hal-00765697⟩