Global Solution to the Incompressible Inhomogeneous Navier–Stokes Equations with Some Large Initial Data

Abstract

In this paper, we prove that the incompressible inhomogeneous Navier–Stokes equations have a unique global solution with initial data ${(a_0,u_0)}$ in critical Besov spaces ${\dot{B}_{q,1}^{\frac{n}{q}}(\mathbb{R}^{n})\times\dot{B}_{p,1}^{\frac{n}{p}-1}(\mathbb{R}^{n})}$ satisfying a nonlinear smallness condition for all ${(p,q)\in[1,2n)\times[1,\infty)}$, ${-\frac1n\leq\frac1p-\frac1q\leq\frac1n}$ and ${\frac1p+\frac1q > \frac1n}$. We also construct an initial data satisfying that nonlinear smallness condition, but the norm of each component of the initial velocity field can be arbitrarily large in ${\dot{B}_{p,1}^{\frac{n}{p}-1}(\mathbb{R}^{n})}$ with ${n < p < 2n}$.