A note on Gevrey class regularity for the solutions of the Navier-Stokes equations
The Navier-Stokes equations \[ u_ t+(u\cdot\nabla)u-\nu\Delta u+\nabla p=f,\quad\text{div} u=0 \tag{1} \] are considered in \(R^ n\) \((n=2,3)\) with space periodic boundary conditions and with the initial condition (2) \(u(x,0)=u_ 0(x)\). This problem is investigated in Gevrey classes. Let \(A\) be the operator of the Stokes problem \(-\Delta u+\nabla p=f\), \(\text{div} u=0\), with space periodic boundary conditions. It is proved that if \(u_ 0\in D(A^ s)\), \(f\in D(A^{s-1}e^{\sigma A^{1/2}})\) for some \(\sigma>0\) then the problem (1), (2) has a unique regular solution on some interval \((0,L_ s)\) and this solution is analytic in time in a neighborhood of the interval \((0,L_ s)\) on the complex plane, with values in a Gevrey class \(D(A^{s+1/2}e^{\psi(t)A^{1/2}})\). Moreover if \(| A^ su(\cdot,t)|\leq \mu_ s\) for \(t\geq 0\) then \(u\) is analytic in time in a neighborhood of the interval \((L_ s,\infty)\) with values in \(D(A^{s+1/2}e^{\sigma'A^{1/2}})\) for \(\sigma'=\min(L_ s,\sigma)\). Here \(s\) is a fixed number, \(s>n/4\) or \(s=1/2\), \(\psi(t)=\min(t,\sigma)\). Results and notations of \textit{C. Foias} and \textit{R. Temam} [J. Funct. Anal. 87, No. 2, 359-369 (1989; Zbl 0702.35203)] are used.
scientific article; zbMATH DE number 91563
scientific article