分数阶不可压缩Navier-Stokes方程解的爆破性准则

On the blow-up criterion of the fractional incompressible Navier-Stokes equations

摘要: 采用Fourier分析及其标准技巧，研究分数阶不可压缩Navier-Stokes方程在齐次Sobolev-Gevrey空间 \dotH_a,\sigma ^s (\bfR^3) \left(a > 0, \sigma > 1, \dfrac52 - 2\alpha < s < \dfrac32, 1 \leqslant \alpha \leqslant \dfrac54\right)中的初值问题.首先证明当初值u_0 \in \dot H_a,\sigma ^s(\bfR^3)方程存在唯一解u \in C(0,T^ * );\dot H_a,\sigma ^s(\bfR^3))；其次证明当T^ * < \infty 时，解的指数型爆破准则.

Abstract: We proved the fractional incompressible Navier-Stokes equations for the initial data by the Fourier analysis and standard techniques in the Sobolev-Gevrey space \dotH_a,\sigma ^s (\bfR^3) \left(a > 0, \sigma > 1, \dfrac52 - 2\alpha < s < \dfrac32, 1 \leqslant \alpha \leqslant \dfrac54\right). We show the unique existence of solution u \in C(0,T^ * );\dot H_a,\sigma ^s(\bfR^3)) for the initial data u_0 \in \dot H_a,\sigma ^s(\bfR^3) and the exponential type blow-up criteria considering these same spaces with T^ * < \infty .