Mathematical Theory of Incompressible Flows: Local Existence, Uniqueness, and Blow-Up of Solutions in Sobolev–Gevrey Spaces

Abstract

This work establishes the local existence and uniqueness as well as the blow-up criteria for solutions of the Navier–Stokes equations in Sobolev–Gevrey spaces. More precisely, if the maximal time of existence of solutions for these equations is finite, we demonstrate the explosion, near this instant, of some limits superior and integrals involving a specific usual Lebesgue spaces and, as a consequence, we prove the lower bounds related to Sobolev–Gevrey spaces.