Existence and regularity of a class of weak solutions to the Navier-Stokes equations

[摘要] We construct a class oi weak solutions to the Navier-Stokes equations, which have second order spatial derivatives and one order time derivatives, of p power summability for 1 < p less than or equal to 5/4. Meanwhile, we show that u is an element of L-s(0, T; W-2,W-r(Omega)) with 1/s + 3/2r = 2 for 1 < r less than or equal to 5/4. r can be relaxed not to exceed 3/2 if we consider only in the interior of Omega. In the end, we extend the classical regularity theorem, Our results show that u is a regular solution if del u is an element of L-s(0, T; L-r(Omega)) with 1/s + 3/2r = 1 for Omega satisfying (1.3), with 1/s + 1/r = 5/6 for arbitrary domain in R-3 and 1 < s less than or equal to 2. For Omega = R-n with n greater than or equal to 3, this result was previously obtained by H. Beirao da Veiga (Chinese Ann. Math. Ser. B 16, 1995, 407-412). (C) 1997 Academic Press.

[发布日期] 1997-06-15