Boundary-value problems for a class of first order partial differential equations in Sobolev spaces and applications to the Euler flow (Q921265)

The differential operator, \(A(t)u\equiv (v\cdot \nabla)u+au\), is studied as it acts in the distributional sense. That is to say for any test function, \(\phi\), the differential operator is considered as \(<A(t)u,\phi >\). Several very nice results are proven with clarity and careful motivation. The equation \[ (*)\quad \lambda u+(v\cdot \nabla)u+au=f\text{ in } \Omega \] is shown to have a unique solution, \(u\in W_{\ell}^{k,p}\) whenever \(f\in W_{\ell}^{k,p}\) provided \(\lambda\) satisfies some technical constraints. Here \(W_{\ell}^{k,p}:=W^{k,p}\cap W_ 0^{\ell,p}\). For several wonderful applications the following theorem is proven: whenever \(v,a\in W^{k,p}\) then the equation (*) has a unique solution, \(u\in W_{\ell}^{k,p}\) whenever \(f\in W_{\ell}^{k,p}\). Again many technical constraints must be satisfied. The paper concludes with some very interesting stability results. I very much enjoyed the style with which the paper was written discussing separately the stationary and evolution problem.