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 $N\ge 2$ in critical spaces for the scaling of the equations. More precisely, we introduce as in \cite{Hprepa} a new unknown,$\mathit{\text{a}}\mathit{\text{effective}}\mathit{\text{velocity}}$ $v=u+\mu \text{n}\ln h$ ($u$ 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 $B_{2,1}^{\mathbb{N}-1}$ and $(h_0-1)$ large in $B_{2,1}^{\mathbb{N}}$. In particular it implies that the classical momentum $m_0^{{}^{\mathrm{\prime }}}=h_0{u}_0$ can be large in $B_{2,1}^{\mathbb{N}-1}$, but small when we project $m_0^{{}^{\mathrm{\prime }}}$ on the divergence field. These solutions are in some sense $\mathit{\text{purely}}\mathit{\text{compressible}}$. 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.