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

We deduce Lionville type theorems for the Navier-Stokes and the Euler equations on R-N, N >= 2. Specifically, we prove that if a weak solution (v, p) satisfies vertical bar v vertical bar(2) + vertical bar p vertical bar is an element of L-1(0, T; L-1(R-N; w(1)(x)dx)) and integral(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 x (0, T). Similar results hold for the MHD equations on R-N, N >= 3.