ON THE LIOUVILLE TYPE THEOREMS WITH WEIGHTS FOR THE NAVIER-STOKES EQUATIONS AND EULER EQUATIONS

摘要

We deduce Liouville type theorems for the Navier-Stokes and the Euler the Navier-Stokes and the Euleron R~N, N≥ 2. Specifically, we prove that if a weak solution (v; p) satisfies |v|~2 + |p| ? L~1(0, T;L~1(R~N, w_1(x)dx)) and ∫_RN p(x, t)w_2(x)dx ≥ 0 for some weight functions w_1(x) and w_2(x), then the solution is trivial, namely v = 0 almost everywhere on R~N × (0, T). Similar results hold for the MHD equations on R~N, N ≥ 3.