Existence of weak solutions for the Navier-Stokes equations with initial data in $L\sp p$

The existence of weak solutions for the Navier-Stokes equations for the infinite cylinder with initial data in ${L^p}$ is considered in this paper. We study the case of initial data in ${L^p}({R^n})$, $2 < p < n$, and $n = 3,4$. An existence theorem is proved covering these important cases and therefore, the "gap" between the Hopf-Leray theory $(p = 2)$ and that of Fabes-Jones-Riviere $(p > n)$ is bridged. The existence theorem gives a new method of constructing global solutions. The cases $p = n$ are treated at the end of the paper.