On the fractional chemotaxis Navier-Stokes system in the critical spaces
The authors deal with the double chemotaxis model of fractional order under the effect of the Navier-Stokes fluid \begin{align*} &^cD^\alpha_tn+u\cdot\nabla n=\Delta n-\nabla\cdot(n\nabla c)-\nabla\cdot(n\nabla v),&&\quad \text{in } \mathbb{R}^N\times(0,\infty), \\ &^cD^\alpha_tc+u\cdot\nabla c=\Delta c-nc,&&\quad \text{in } \mathbb{R}^N\times(0,\infty), \\ &^cD^\alpha_tv+u\cdot\nabla v=\Delta v-\gamma v+n,&&\quad \text{in } \mathbb{R}^N\times(0,\infty), \\ &^cD^\alpha_tu+(u\cdot\nabla) u=\Delta u-\nabla p-nf,&&\quad\text{in } \mathbb{R}^N\times(0,\infty), \\ &\nabla\cdot u=0,&&\quad \text{in } \mathbb{R}^N\times(0,\infty), \\ &n(x,0)=n_0(x),c(x,0)=c_0(x),v(x,0)=v_0(x),u(x,0)=u_0(x),&&\quad \text{in } \mathbb{R}^N, \end{align*} where \(N\ge2\), \(\gamma\ge0\), \(^cD^\alpha_t\) is the Caputo fractional derivative of order \(\alpha\in(0, 1)\). They prove the existence of global mild solutions with the small critical initial data in Besov-Morrey spaces. The results can obtain the self-similar solutions provided the initial data are homogeneous functions with small norms and considering the case of chemical attractant without degradation rate. Moreover, they show the asymptotic stability of solutions as the time goes to infinity and obtain a class of asymptotically self- similar ones.
scientific article; zbMATH DE number 7599027
scientific article