Existence of global strong solutions for the shallow-water equations with large initial data

This work is devoted to the study of a viscous shallow-water system with friction and capillarity term. We prove in this paper the existence of global strong solutions for this system with some choice of large initial data when N ≥ 2 in critical spaces for the scaling of the equations. More precisely, we introduce as in [14] a new unknown,a effective velocity v = u+μ∇ lnh (u is the classical velocity and h the depth variation of the fluid) with μ the viscosity coefficient which simplifies the system and allow us to cancel out the coupling between the velocity u and the depth variation h. We obtain then the existence of global strong solution if m0 = h0v0 is small in B N 2 −1 2,1 and (h0 − 1) large in B N 2 2,1. In particular it implies that the classical momentum m ′ 0 = h0u0 can be large in B N 2 −1 2,1 , but small when we project m ′ 0 on the divergence field. These solutions are in some sense purely compressible. We would like to point out that the friction term term has a fundamental role in our work inasmuch as coupling with the pressure term it creates a damping effect on the effective velocity.