On energy inequality, smoothness and large time behavior in L 2 for weak solutions of the Navier-Stokes equations in exterior domains
The present paper deals with the exterior nonstationary problem for the Navier-Stokes equations: \[ u_ t-\Delta u+u\cdot \nabla u+\nabla p=f\quad in\quad \Omega \times (0,\infty), \] \[ \nabla \cdot u=0\quad in\quad \Omega \times (0,\infty);\quad u=0\quad on\quad \partial \Omega \times (0,\infty);\quad u(x,0)=a(x), \] where \(\Omega\) is an exterior domain in R n \((n=3,4)\) with smooth boundary \(\partial \Omega.\) It is shown that for any L 2 data a and f there is a weak solution u satisfying the following form of energy inequality: \[ (E)\quad \| u(t)\| ^ 2_{L^ 2}+2\int ^{t}_{s}\| \nabla u\| ^ 2_{L^ 2} d\tau \leq \| u(s)\| ^ 2_{L^ 2} + 2\int ^{t}_{s}(f,u)_{L^ 2} d\tau, \] for \(s=0\), a.e. \(s>0\), and all \(t\geq s.\) This result improves those of \textit{G. P. Galdi} and \textit{P. Maremonti} [Arch. Ration. Mech. Anal. 94, 253-266 (1986; Zbl 0617.35108)] and of \textit{H. Sohr}, \textit{W. von Wahl} and \textit{M. Wiegner} [Nachr. Akad. Wiss. Gött., II. Ser.-Phys. Kl. 1986, 59 S. (1986; Zbl 0629.35100)], and gives an affirmative answer to a question raised by \textit{K. Masuda} [Tohoku Math. J., II. Ser. 36, 623-646 (1984; Zbl 0568.35077)]. Energy inequality (E) is then applied to show the large time smoothness and L 2 decay property for the weak solutions. It is also proved that the stationary solutions obtained by \textit{R. Finn} [Arch. Ration. Mech. Anal. 19, 363-406 (1965; Zbl 0149.446)] are stable (in L 2) under any L 2 perturbation, if the associated Reynolds numbers are small.
scientific article; zbMATH DE number 4046279
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