Abstract
We show that for every initial dataa εL 2(Ω) there exists a weak solutionu of the Navier-Stokes equations satisfying the generalized energy inequality introduced by Caffarelli-Kohn-Nirenberg forn=3. We also show that if a weak solutionu εL s(0,T;L q(Ω)) with 2/q + 2/s ≤ 1 and 3/q + 1/s ≤ 1 forn=3, or 2/q + 2/s ≤ 1 andq ≥ 4 forn ≥ 4, thenu satisfies both the generalized and the usual energy equalities. Moreover we show that the generalized energy equality holds only under the local hypothesis thatu εL s (ε, T; L q (K)) for all compact setsK ⊂ ⊂ Ω and all 0 <ε <T with the same (q, s) as above when 3 ≤n ≤ 10.
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Taniuchi, Y. On generalized energy equality of the Navier-Stokes equations. Manuscripta Math 94, 365–384 (1997). https://doi.org/10.1007/BF02677860
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DOI: https://doi.org/10.1007/BF02677860