Navier-Stokes equations: local existence, uniqueness and blow-up of solutions in Sobolev-Gevrey spaces

This work establishes local existence and uniqueness as well as blow- up criteria for solutions u(x, t) of the Navier- Stokes equations in Sobolev- Gevrey spaces Hsa, s (R3). More precisely, if it is assumed that the initial data u0 belongs to Hs0 a, s (R3), with s0. (12, 32), we prove that there is a time T > 0 such that u. C([ 0, T]; Hsa, s (R3)) for a > 0, s = 1 and s = s0. If the maximal time interval of existence of solutions is finite, 0 = t < T *, then, we prove, for example, that the blow- up inequality C1 exp{C2(T * - t) p}(T * - t) - q = u(t) Hsa, s (R3), q = 2(ss + s0) + 1 6s, p = - 1 3s, holds for 0 = t < T *, s. (1 2, s0], a> 0, s > 1 (2s0 is the integer part of 2 sigma).

Automated summary

This work establishes local existence, uniqueness, and blow-up criteria for solutions of the Navier-Stokes equations in Sobolev-Gevrey spaces. It provides conditions for initial data and time intervals of solutions, proving specific properties of the solutions.