Fujita–Kato solution for compressible Navier–Stokes equations with axisymmetric initial data and zero Mach number limit

In this paper, we investigate the question of the existence of global strong solution for the compressible Navier–Stokes equations for small initial data such that the rotational part of the velocity [Formula: see text] belongs to [Formula: see text] (in dimension [Formula: see text]). We show then an equivalent of the so-called Fujita–Kato theorem to the case of the compressible Navier–Stokes equations when we consider axisymmetric initial data. The main difficulty is linked to the fact that in this case the velocity is not Lipschitz, as a consequence we have to study carefully the coupling between the rotational and irrotational part of the velocity. In a second part, we address the question of convergence to the incompressible model (for ill-prepared initial data) when the Mach number goes to zero.

BV Solutions to 1D isentropic Euler equations in the zero mach number limit

Two compressible immiscible fluids in 1D and in the isentropic approximation are considered. The first fluid is surrounded and in contact with the second one. As the Mach number of the first fluid vanishes, we prove the rigorous convergence for the fully nonlinear compressible to incompressible limit of the coupled dynamics of the two fluids. A key role is played by a suitably refined wave front tracking algorithm, which yields precise [Formula: see text], [Formula: see text] and weak* convergence estimates, either uniform or explicitly dependent on the Mach number.