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

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

Zbl 0621.76074