Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system

Abstract

We construct a unique local regular solution in L q (0, T; L p ) for a class of semilinear parabolic equations which includes the semilinear heat equation u t − Δu = ¦u¦ α u (α > 0) and the Navier-Stokes system. Here p and q are so chosen that the norm of L q (0, T; L p ) is dimensionless or scaling invariant. The main relation between p and q for the semilinear heat equation is 1 q =( 1 p − 1 q ) n 2 ) , provided that initial data are in L r with r = nα/2 > 1, where n is the space dimension. Applying our regular solutions to the Navier-Stokes system, we show that the k/2-dimensional Hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in L q(0, T; L p), where k = 2 − q + nq p , p ⩾ n, 1 ⩽ q < α . We show, moreover, that a turbulent solution is regular if it is in C((0, T); L n ).