We prove Liouville type of theorems for weak solutions of the Navier-Stokes and the Euler equations. In particular, if the pressure satisfies p ∈ L1 (0,T;L 1 (RN)) with ∫ RN p(x,t)dx ≥ 0, then the corresponding velocity should be trivial, namely v = 0 on R N × (0,T). In particular, this is the case when p ∈ L 1 (0,T; H 1 (R N)), where H 1 (R N) the Hardy space. On the other hand, we have equipartition of energy over each component, if p ∈ L1 (0,T;L 1 (RN)) with ∫ RN p(x,t)dx < 0. Similar results hold also for the magnetohydrodynamic equations.
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