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

2011/10/27 by Haspot, Boris
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1110.6100

Abstract

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 \citeHprepa a new unknown,a effective velocity v=u+μ\nln h (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-1_2,1 and (h₀-1) large in B^\N_2,1. In particular it implies that the classical momentum m₀^'=h₀ u₀ can be large in B^\N-1_2,1, but small when we project m₀^' 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.

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