Remarks on the existence of weak solutions to 2-D incompressible Navier-Stokes equations (Q1273114)

The existence and uniqueness of global weak solutions to 2-D incompressible Navier-Stokes equations with initial vorticity in \(L^{4/3}\) is proved. Earlier A. Majda obtained the global existence of weak solutions to such equations with initial vorticity in \(L^p\) for \(p>{4\over 3}\). The result in this paper is optimal in the sense, that if initial vorticity \(\omega_0\) is in \(L^p\) for \(p<{4\over 3}\), then by Riesz potential theory the corresponding velocity \(V_0\) is in \(L^q\), \({1\over q}={1\over p}-{1\over 2}\), but \({1\over p}+ {1\over q}>1\) and \(V_0\cdot\omega_0\) has no meaning as a distribution in general.