Global well-posedness and self-similar solution of the inhomogeneous Navier-Stokes system

Abstract

In this paper, we study the global well-posedness of the 3-D inhomogeneous incompressible Navier-Stokes system (INS in short) with initial density $\rho_0$ being discontinuous and initial velocity $u_0$ belonging to some critical space. Firstly, if $\rho_0u_0$ is sufficiently small in the space $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$ and $\rho_0$ is close enough to a positive constant in $L^\infty$, we establish the global existence of strong solution to (INS) for $3<p<\infty$ and provide the uniqueness of the solution for $3<p<6$. This result corresponds to Cannone-Meyer-Planchon solution of the classical Navier-Stokes system. Furthermore, with the additional assumption that $u_0\in L^2(\mathbb{R}^3)$, we prove the weak-strong uniqueness between Cannone-Meyer-Planchon solution and Lions weak solution of (INS). Finally, we prove the global well-posedness of (INS) with $u_0\in \dot{B}^{\frac{1}{2}}_{2,\infty}(\mathbb{R}^3)$ being small and only an upper bound on the density. This gives the first existence result of the forward self-similar solution for (INS).