| Abstrakt: |
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 H a , σ s (R 3). More precisely, if it is assumed that the initial data u 0 belongs to H a , σ s 0 (R 3) , with s 0 ∈ (1 2 , 3 2) , we prove that there is a time T>0 such that u ∈ C ([ 0 , T ] ; H a , σ s (R 3)) for a > 0 , σ ≥ 1 and s ≤ s 0 . If the maximal time interval of existence of solutions is finite, 0 ≤ t < T ∗ , then, we prove, for example, that the blow-up inequality C 1 exp { C 2 (T ∗ − t) p } (T ∗ − t) − q ≤ ∥ u (t) ∥ H a , σ s (R 3) , q = 2 (s σ + σ 0) + 1 6 σ , p = − 1 3 σ , holds for 0 ≤ t < T ∗ , s ∈ (1 2 , s 0 ] , a>0, σ > 1 ( 2 σ 0 is the integer part of 2 σ). [ABSTRACT FROM AUTHOR] |
