\(L^ 2\) decay for the compressible Navier-Stokes equations in unbounded domains. (English) Zbl 0798.35124

Deckelnick, Klaus

Commun. Partial Differ. Equations 18, No. 9-10, 1445-1476 (1993).

The author considers the equations of motion for a viscous, compressible, heat-conducting fluid that occupies the complement of a bounded domain in \(\mathbb{R}^ 3\) or the half-space. If the data are sufficiently small, global existence in time has been proved by A. Matsumura and T. Nishida [Commun. Math. Phys. 89, 445-464 (1983; Zbl 0543.76099)]. Such solutions are shown to tend to a stationary state for \(t\to\infty\) in the \(L_ 2\)-norm. If the initial data are chosen in such a way that the solution to the homogeneous linear problem decays like \(t^{- \alpha}\) for some \(\alpha>0\) then the decay of the solution to the nonlinear problem can be estimated by \(t^{-\beta}\), \(\beta= \min(\alpha,1/8)\).

Reviewer: J.Bemelmans (Aachen)

MSC:

35Q30	Navier-Stokes equations
35B40	Asymptotic behavior of solutions to PDEs
76N10	Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics

Keywords:

Navier-Stokes equation; viscous, compressible, heat-conducting fluid; decay of the solution