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

In this note, for $s\in \mathbb{R}$ and $1\le p,r\le \mathrm{\infty }$, we introduce and study Sobolev-Fourier-Lorentz spaces $\dot{H}^s_{$\mathcal{L}$^{p, r}}($\mathbb{R}$^d)$\mathrm{.}Inthefamilyspaces$\dot{H}^s_{$\mathcal{L}$^{p, r}}($\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 $\dot{H}^{\frac{d}{p} - 1}{$\mathcal{L}$^{p,r}}($\mathbb{R}$^d)$with$d \geq 2, 1 \leq p <\infty$,and$1 \leq r < \infty$, we establish the existence of local mild solutions to the Cauchy problem for the Navier-Stokes equations in spaces $L^\infty([0, T]; \dot{H}^{\frac{d}{p} - 1}{$\mathcal{L}$^{p, r}}($\mathbb{R}$^d))$ with arbitrary initial value, and existence of global mild solutions in spaces $L^\infty([0, \infty); \dot{H}^{\frac{d}{p} - 1}_{$\mathcal{L}$^{p, r}}($\mathbb{R}$^d))$ when the norm of the initial value in the Besov spaces $\dot{B}^{\frac{d}{\tilde p}

• 1, \infty}_{$\mathcal{L}$ ^{\tilde p,\infty}}($\mathbb{R}$^d)$ is small enough, where $\tilde{p}$ may take some suitable values.