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, ofppower summability for 1 < p ≤ 5/4. Meanwhile, we show thatu L^s(0, T; W^{2, r}(Ω)) with 1/s + 3/2r = 2 for 1 < r ≤ 5/4.rcan be relaxed not to exceed 3/2 if we consider only in the interior of Ω. In the end, we extend the classical regularity theorem. Our results show thatuis a regular solution if u L^s(0, T; L^r(Ω)) with 1/s + 3/2r = 1 for Ω satisfying (1.3), with 1/s + 1/r = 5/6 for arbitrary domain inR³and 1 < s ≤ 2. For Ω = Rⁿwithn ≥ 3, this result was previously obtained byH. Beirão da Veiga (Chinese Ann. Math. Ser. B16, 1995, 407–412).