Existence results in Sobolev spaces for a stationary transport equation (Q582483)

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.