Existence results in Sobolev spaces for a stationary transport equation. (English) Zbl 0691.35087

Beirão da Veiga, Hugo

Ric. Mat. 36, Suppl., 173-184 (1987).

Let \(\Omega\) be an open bounded subset of \(R^ n\) with smooth boundary \(\Gamma\). Let v(x) denote a smooth vector field always tangential to \(\Gamma\), a(x) a smooth matrix-valued function. The author considers the equation \(\lambda u+(v\cdot \nabla)u+au=f,\) where \(\lambda >0\). Given an integer \(k\geq -1\) and \(p\geq n/(k+2),\) the author proves that for any f in the Sobolev space \(W^{k,p}(\Omega)\) there exists a unique solution in the same space, provided \(\lambda\) is large enough. If \(k\geq 1\), then it is proved that the solution is zero on \(\Gamma\) if and only if the same is true for f. The paper was motivated by the study of the stationary solution of the compressible heat-conducting Navier-Stokes equations [see the author, Commun. Math. Phys. 109, 229-248 (1987; Zbl 0621.76074)] where an application of the previous result in the case \(k=-1\) is given.

Reviewer: P.Secchi

MSC:

35Q99	Partial differential equations of mathematical physics and other areas of application
35D05	Existence of generalized solutions of PDE (MSC2000)
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

stationary transport equation; Navier-Stokes equations; boundary behavior

Citations:

Zbl 0621.76074