the velocity. In particular it was impossible to distinguish the index of integration for the Besov spaces.\
However we can remark that this change of variable $v_{1}$ is interesting only in the case of low frequencies, indeed heuristically in low frequencies ${\nabla}P(\rho)$ is small in Fourier variable so it is not a matter.\
It is natural in this case to study the variable $u$ in low frequencies. Moreover as explained in the introduction, the system (\[0.1\]) has a hyperbolic behavior, which means that we can work only with spaces builtet on $L^{2}$ (indeed classicaly the hyperbolic system are ill-posed in space constructed on general $L^{p}$). In the following ssection, we will explain how to treat the case in low frequencies and how we will use the fact that $q$ behaves in low frequencies as an heat equation.

A linear model with convection
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In this section, we will explain how we treat the low frequency regime by following the approch of Charve and Danchin in [@CD]. In low frequencies, the first order terms predominate and the viscous term ${\Delta}u$ may be neglected so that (\[0.1\]) has to be treated by means of hyperbolic energy methods. It means that we can 