Liouville type theorems for the Euler and the Navier–Stokes equations

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