On the Existence Theorem for Weak Solutions of Increased Smoothness of the Multidimensional Navier7ndash;Stokes Equations

Abstract

Hopf [1] defined the notion of a weak solution of the 3D Navier–Stokes equations and proved the existence of such solutions. Later, it was proved that, under certain conditions, the weak solutions (the 3D viscous fluid flow velocities) have second derivatives 5/4th-power integrable over the space-time cylinder [2–4]. The present paper deals with some development of these results. We consider viscous fluid flows in an n-dimensional space with velocities 2π-periodic in each of the space variables and show that if the initial velocity is square integrable over the period cube, its derivatives are integrable, and the mass forces satisfy similar integrability conditions over the obvious space-time parallelepiped, then there exist weak solutions whose second derivatives are (4/3 − e)th-power integrable over the space-time parallelepiped for any e > 0.