Existence and Regularity of a Class of Weak Solutions to the Navier]Stokes Equations

We construct a class of weak solutions to the Navier]Stokes equations, which have second order spatial derivatives and one order time derivatives, of p power summability for 1-pF5r4. Meanwhile, we show that ugLs 0, T;W2, r V.. with 1rsq3r2rs2 for 1-rF5r4. r can be relaxed not to exceed 3r2 if we consider only in the interior of V. In the end, we extend the classical regularity theorem. Our results show that u is a regular solution if =ugLs 0, T; Lr V.. with 1rsq3r2rs1 for V satisfying 1.3., with 1rsq1rrs5r6 for arbitrary domain in R3 and 1-sF2. For VsRn with nG3, this result was previously obtained by H. Beir˜ao da Veiga Chinese Ann. Math. Ser. B 16, 1995, 407]412..