2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74089We consider the Cauchy problem for incompressible Navier-Stokes equations $u_t+u\nabla_xu-Δu+\nabla p=0, div u=0 in R^d \times R^+$ with initial data $a\in L^d(R^d)$, and study in some detail the smoothing effect of the equation. We prove that for $T<\infty$ and for any positive integers $n$ and $m$ we have $t^{m+n/2}D^m_tD^{n}_x u\in L^{d+2}(R^d\times (0,T))$, as long as the $\|u\|_{L^{d+2}_{x,t}(R^d\times (0,T))}$ stays finite.14 pagesAnalysis of PDEs35Q30; 76D03; 76D05text