Asymptotic behavior of solutions for the 2D micropolar equations in Sobolev–Gevrey spaces

Abstract

This work guarantees the existence of a positive instant $\mathit{t}=\mathit{T}$ and a unique solution $\mathrm{(}\mathit{u}\mathrm{,}\mathit{w}\mathrm{)}\in {[\mathit{C}\mathrm{(}[0\mathrm{,}\mathit{T}];{\mathit{H}}_{\mathit{a}\mathrm{,}\mathit{\sigma }}^{\mathit{s}}\mathrm{(}{\mathbb{R}}^2\mathrm{)}\mathrm{)}]}^3$ (with $\mathit{a}\mathrm{>}0$ , $\mathit{\sigma }\mathrm{>}1$ , $\mathit{s}\mathrm{>}0$ and $\mathit{s}\ne 1$ ) for the micropolar equations. Furthermore, we consider the global existence in time of this solution in order to prove the following decay rates: $$\begin{array}{c}\underset{\mathit{t}\to \infty }{lim}{\mathit{t}}^{\frac{\mathit{s}}{2}}{\Vert \mathrm{(}\mathit{u}\mathrm{,}\mathit{w}\mathrm{)}\mathrm{(}\mathit{t}\mathrm{)}\Vert }_{{\dot{\mathit{H}}}_{\mathit{a}\mathrm{,}\mathit{\sigma }}^{\mathit{s}}\mathrm{(}{\mathbb{R}}^2\mathrm{)}}^2=\underset{\mathit{t}\to \infty }{lim}{\mathit{t}}^{\frac{\mathit{s}+1}{2}}{\Vert \mathit{w}\mathrm{(}\mathit{t}\mathrm{)}\Vert }_{{\dot{\mathit{H}}}_{\mathit{a}\mathrm{,}\mathit{\sigma }}^{\mathit{s}}\mathrm{(}{\mathbb{R}}^2\mathrm{)}}^2=\underset{\mathit{t}\to \infty }{lim}{\Vert \mathrm{(}\mathit{u}\mathrm{,}\mathit{w}\mathrm{)}\mathrm{(}\mathit{t}\mathrm{)}\Vert }_{{\mathit{H}}_{\mathit{a}\mathrm{,}\mathit{\sigma }}^{\mathit{\lambda }}\mathrm{(}{\mathbb{R}}^2\mathrm{)}}=0\mathrm{,}\forall \mathit{\lambda }⩽\mathit{s}.\end{array}$$ These limits are established by applying the estimate $$\begin{array}{c}{\Vert {\mathcal{F}}^{-1}\mathrm{(}{\mathit{e}}^{\mathit{T}|·|}\mathrm{(}\hat{\mathit{u}}\mathrm{,}\hat{\mathit{w}}\mathrm{)}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}\Vert }_{{\mathit{H}}^{\mathit{s}}\mathrm{(}{\mathbb{R}}^2\mathrm{)}}⩽{[1+2{\mathit{M}}^2]}^{\frac{1}{2}}\mathrm{,}\forall \mathit{t}⩾\mathit{T}\mathrm{,}\end{array}$$ where T relies only on $\mathit{s}\mathrm{,}\mathit{\mu }\mathrm{,}\mathit{\nu }$ and M (the inequality above is also demonstrated in this paper). Here M is a bound for $\Vert \mathrm{(}\mathit{u}\mathrm{,}\mathit{w}\mathrm{)}\mathrm{(}\mathit{t}\mathrm{)}{\Vert }_{{\mathit{H}}^{\mathit{s}}\mathrm{(}{\mathbb{R}}^2\mathrm{)}}$ (for all $\mathit{t}⩾0$ ) which results from the limits ${lim}_{\mathit{t}\to \infty }{\mathit{t}}^{\frac{\mathit{s}}{2}}\Vert \mathrm{(}\mathit{u}\mathrm{,}\mathit{w}\mathrm{)}\mathrm{(}\mathit{t}\mathrm{)}{\Vert }_{{\dot{\mathit{H}}}^{\mathit{s}}\mathrm{(}{\mathbb{R}}^2\mathrm{)}}={lim}_{\mathit{t}\to \infty }\Vert \mathrm{(}\mathit{u}\mathrm{,}\mathit{w}\mathrm{)}\mathrm{(}\mathit{t}\mathrm{)}{\Vert }_{{\mathit{L}}^2\mathrm{(}{\mathbb{R}}^2\mathrm{)}}=0$ .