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In this paper we investigate the question of the existence of global weaksolution for the compressible Navier Stokes equations provided that the initialmomentum $\rho_0 u_0$ belongs to $\mbox{bmo}^{-1}(\mathbb{R}^N)$ with $N= 2,3$and is radially symmetric. More precisely we deal with the so called viscousshallow water system when the viscosity coefficients verify$\mu(\rho)=\mu\rho$, $\lambda(\rho)=0$ with $\mu>0$. We prove then a equivalentof the so called Koch-Tataru theorem for the compressible Navier-Stokesequations. In addition we assume that the initial density $\rho_0$ is onlybounded in $L^\infty(\mathbb{R}^N)$, it allows us in particular to considerinitial density admitting shocks. Furthermore we show that if the couplingbetween the density and the velocity is sufficiently strong, then the initialdensity which admits initially shocks is instantaneously regularizing inasmuchas the density becomes Lipschitz. This coupling is expressed via the regularityof the so called effective velocity $v=u+2\mu\nabla\ln\rho$. In our case $v_0$belongs to $L^2(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, it is important topoint out that this choice on the initial data implies that we work in asetting of infinite energy on the initial data $(\rho_0,u_0)$, it extends inparticular the results of \cite{V}. In a similar way, we consider also the caseof the dimension $N=1$ where the momentum $\rho_0 u_0$ belongs to$bmo^{-1}(\mathbb{R})$ without any geometric restriction.\\ To finish we provethe global existence of strong solution for large initial data provided thatthe initial data are radially symmetric and sufficiently regular in dimension$N=2,3$ for $\gamma$ law pressure.

Global bmo$^{-1}(\mathbb{R}^N)$ radially symmetric solution for  compressible Navier-Stokes equations with initial density in  $L^\infty(\mathbb{R}^N)$