Global Solutions to 2-D Inhomogeneous Navier-Stokes System with General 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 are concerned with the global wellposedness of 2-D density-dependent incompressible Navier-Stokes equations with variable viscosity, in a critical functional frame- work which is invariant by the scaling of the equations and under a non-linear smallness condition on fluctuation of the initial density which has to be doubly exponential small compared with the size of the initial velocity. In the second part of the paper, we apply our methods combined with the techniques of R. Danchin and P. B. Mucha to prove the global existence of solutions to inhomogeneous Navier-Stokes system with piecewise constant initial density which has small jump at the interface and is away from vacuum. In particular, this latter result removes the smallness condition for the initial velocity in a corresponding theorem of R. Danchin and P. B. Mucha.

Mots clés: Inhomogeneous Navier-Stokes Equations; Littlewood-Paley Theory; Wellposedness

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

Jingchi Huang, Marius Paicu, Ping Zhang. Global Solutions to 2-D Inhomogeneous Navier-Stokes System with General Velocity. Journal de Mathématiques Pures et Appliquées, 2013, 100 (6), pp.806-831. ⟨hal-00765696⟩