On generalized energy equality of the Navier-Stokes equations |
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Authors: | Yasushi Taniuchi |
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Affiliation: | (1) Graduate School of Polymathematics, Nagoya University, 464-01 Nagoya, Japan |
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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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