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 $N\geq 2$ in critical spaces for the scaling of the equations. More precisely, we introduce as in \cite{ Hprepa} a new unknown,\textit{ a effective velocity} $v=u+\mu\n\ln h$ ($u$ is the classical velocity and $h$ the depth variation of the fluid) with $\mu$ 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 $m_{ 0}=h_{ 0}v_{ 0}$ is small in $B^{ \N-1}_{ 2,1}$ and $(h_{ 0}-1)$ large in $B^{ \N}_{ 2,1}$. In particular it implies that the classical momentum $m_{ 0}^{ '}=h_{ 0} u_{ 0}$ can be large in $B^{ \N-1}_{ 2,1}$, but small when we project $m_{ 0}^{ '}$ on the divergence field. These solutions are in some sense \textit{ 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.
Categories: math.AP