On the Local Smoothness of Solutions of the Navier–Stokes Equations

Abstract.

We consider the Cauchy problem for incompressible Navier–Stokes equations $u_t + u\nabla_xu - \Delta_xu + \nabla_xp = 0,\; {\text{div}}\,u = 0\;{\text{in}}\;\mathbb{R}^d \times \mathbb{R}^ +$ with initial data in $L^d (\mathbb{R}^d )$, and study in some detail the smoothing effect of the equation. We prove that for T < ∞ and for any positive integers n and m we have $t^{m + n/2} D_t^m D_x^n u \in L^{d + 2} (\mathbb{R}^d \times (0,T))$, as long as $||u||_{L_{x,t}^{d + 2} (R^d \times (0,T))}$ stays finite.