The Gevrey analyticity and decay for the micropolar system in the critical Besov space
The author is interested in the 3-D incompressible micropolar fluid system in \(\mathbb R^+\times\mathbb R^3\): \[ \begin{cases} \partial_tu-(\chi+\nu)\Delta u+u\cdot\nabla u+\nabla\pi-2\chi\nabla\times\omega=0,\\ \partial_t\omega-\mu\Delta\omega+u\cdot\nabla\omega+4\chi\omega-\kappa\nabla\operatorname{div}\omega-2\chi\nabla\times u=0,\\ \operatorname{div}u=0,\\ (u,\omega)_{|t=0}=(u_0,\omega_0), \end{cases} \] where \(u = u(x, t)\) denotes the fluid velocity, \(\omega(x, t)\) denotes the velocity field of rotation for the particle of the fluid, \(\pi(x, t)\) represents the scalar pressure, \(\nu\) denotes the Newtonian kinematic viscosity, \(\chi\) denotes the micro-rotation viscosity, and where \(\kappa,\mu\) are the angular viscosities. He first proves the global in-time existence of the strong solutions in the Besov spaces \(\dot{B}_{p,q}^{\frac{3}{p}-1}\) with \(1\leq p<\infty\) and \(1\leq q\leq\infty\). Then, he establishes the global Gevrey analyticity in these spaces and the time-decay estimates on any derivatives of the solutions for large time. To do that, the approach developed by the author is inspired, on one hand, by the work of \textit{J.-Y. Chemin} [J. Anal. Math. 77, 27--50 (1999; Zbl 0938.35125)] and, on the other hand, by the work of \textit{H. Bae} et al. [Arch. Ration. Mech. Anal. 205, No. 3, 963--991 (2012; Zbl 1317.76029)] for the incompressible Navier-Stokes equations.
scientific article; zbMATH DE number 7452686
scientific article