Localized energy equalities for the Navier–Stokes and the Euler equations

Let (v,p)(v,p) be a smooth solution pair of the velocity and the pressure for the Navier–Stokes (Euler) equations on RN×(0,T)RN×(0,T), N⩾3N⩾3. We set the Bernoulli function Q=12|v|2+p. Under suitable decay conditions at infinity for (v,p)(v,p) we prove that for almost all α(t)α(t) and β(t)β(t) defined on (0,T)(0,T) there holds∫{α(t)