ON GENERALIZED ENERGY EQUALITY OF THE NAVIER-STOKES EQUATIONS

We show that for every initial data a is an element of L-2(Omega) ther e exists a weak solution u of the Navier-Stokes equations equations sa tisfying the generalized energy inequality introduced by Caffarelli-Ko hn-Nirenberg for n=3. We also show that if a weak solution u is an ele ment of L-s(0,T;L-q(Omega)) with 2/q+2/s less-than-or-equal-to 1 and 3 /q+1/s less-than-or-equal-to 1 for n=3, or 2/q+2/s less-than-or-equal- to 1 and q greater-than-or-equal-to 4 for n greater-than-or-equal-to 4 , then u satisfies both the generalized and the usual energy equalitie s. Moreover we show that the generalized energy equality holds only un der the local hypothesis that u is an element of L-s(Epsilon, T;L-q(K) ) for all compact sets K subset of subset of Omega and all 0 less-than epsilon less-than T with the same (q,s) as above when 3 less-than-or- equal-to n less-than-or-equal-to 10.