Weak solutions of the Navier-Stokes equation and their Euler limits

The existence of a weak solution of the incompressible isothermal Navier-Stokes equation with given initial conditions in the Sobolev space ℋ

−2

for energy fluctuations and in ℋ

−3

for enstrophy fluctuations is assumed. The existence, if it exists, is uniform with respect to the Euler limit of zero viscosity ν. Thus, the existence of weak solutions to the Euler equation with given initial conditions is established. These Euler solutions are the zero-viscosity limit of Navier-Stokes solutions, provided the Navier-Stokes solutions exist.