On the uniqueness and the regularity of solutions to the Navier Stokes in the Lorentz space $L^{3,\mathrm{\infty }}({\mathbb{R}}^3)$

We prove uniqueness of solutions to the Navier-Stokes equations in the space $$L^p([0,T],{\tilde{L}}^{3,\mathrm{\infty }}({\mathbb{R}}^3))\cap C([0,T],B_{\mathrm{\infty }}^{-1,\mathrm{\infty }}({\mathbb{R}}^3)),p>2,$$ where ${\tilde{L}}^{3,\mathrm{\infty }}({\mathbb{R}}^3)$ denotes the closure of the Schwartz space $\mathcal{S}({\mathbb{R}}^3)$ in the Lorentz space $L^{3,\mathrm{\infty }}({\mathbb{R}}^3)$. Moreover, we prove that any solution $u$ in this class, if it exists, is smooth for positive time, namely, $u\in C^{\mathrm{\infty }}((0,T]\times {\mathbb{R}}^3)$. In addition, we establish precise estimates for the behavior of $u(t)$ as $t\to 0^{+}$ in various Banach spaces.