Global strong solution for the Korteweg system with quantum pressure in dimension $$N\ge 2$$ N ≥ 2

Abstract

This work is devoted to prove the existence of global strong solution in dimension \(N\ge 2\) for a isothermal model of capillary fluids derived by Dunn and Serrin (see Arch. Ration. Mech. Anal. 88(2):95–133, 1985), which can be used as a phase transition model. We will restrict us to the case of the so called compressible Navier-Stokes system with quantum pressure (which corresponds to the Korteweg system with capillary coefficient such that \(\kappa (\rho )=\frac{\kappa _1}{\rho }\) with \(\kappa _1>0\) and \(\rho \) the density). In a first part we prove the existence of strong solution in finite time for large initial data in critical Besov spaces with precise estimate on the life span \(T^*\). The second part consists in proving the existence of global strong solution with large initial data for a specific choice on the capillary coefficient \(\kappa _1 = \mu ^2\) with \(\mu \) the viscosity coefficient. To do this we derive different energy estimates on the density and the effective velocity v which allow us to extend the strong solution beyond \(T^*\). The main difficulty consists in estimating the \(L^\infty \) norm of \(\frac{1}{\rho }\). The proof relies mostly on a method introduced by De Giorgi (Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3(3):25–43, 1957) [see also Ladyzhenskaya et al. (Linear and quasilinear equations of parabolic type. AMS translations, Providence, 1968) for the parabolic case] to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument.