Well-posedness for the Navier-Stokes equations with datum in Sobolev-Fourier-Lorentz spaces

Abstract

In this note, for $s\in \mathbb{R}$ and $1\le p,r\le \infty$, we introduce and study Sobolev-Fourier-Lorentz spaces ${\dot{H}}_{{\mathcal{L}}^{p,r}}^s({\mathbb{R}}^d)$. In the family spaces ${\dot{H}}_{{\mathcal{L}}^{p,r}}^s({\mathbb{R}}^d)$, the critical invariant spaces for the Navier-Stokes equations correspond to the value $s=\frac{d}{p}-1$. When the initial datum belongs to the critical spaces ${\dot{H}}_{{\mathcal{L}}^{p,r}}^{\frac{d}{p}-1}({\mathbb{R}}^d)$ with $d\ge 2,1\le p<\infty$, and $1\le r<\infty$, we establish the existence of local mild solutions to the Cauchy problem for the Navier-Stokes equations in spaces $L^{\infty }([0,T];{\dot{H}}_{{\mathcal{L}}^{p,r}}^{\frac{d}{p}-1}({\mathbb{R}}^d))$ with arbitrary initial value, and existence of global mild solutions in spaces $L^{\infty }([0,\infty );{\dot{H}}_{{\mathcal{L}}^{p,r}}^{\frac{d}{p}-1}({\mathbb{R}}^d))$ when the norm of the initial value in the Besov spaces ${\dot{B}}_{{\mathcal{L}}^{\tilde{p},\infty }}^{\frac{d}{\tilde{p}}-1,\infty }({\mathbb{R}}^d)$ is small enough, where $\tilde{p}$ may take some suitable values.

35 pages