On a stationary transport equation. (English) Zbl 0641.35006

Beirão da Veiga, Hugo

Ann. Univ. Ferrara, Nuova Ser., Sez. VII 32, 79-91 (1986).

Let X be a Banach space of real functions defined in an open subset \(\Omega \subset {\mathbb{R}}^ n\), and consider the problem of finding a solution \(y\in X\) of the stationary equation \(\mu y+v\cdot \nabla y+ay=g\), for each \(g\in X\). Here, \(X=W^{1,p}\), \(X=W_ 0^{1,p}\), \(X=L^ p\), or \(X=W^{-1,p}\), \(p\in]1,+\infty [\). Under suitable hypothesis on the coefficients \(v(x)=(v_ 1(x),...,v_ n(x))\) and a(x), it is shown that the above problem is solvable in X, and that \(\| y\|_ x\leq (| \mu | -c)\| g\|_ x\), \(\forall g\in X\), if \(| \mu | >c\) (c is a suitable positive constant). For similar results in spaces \(X=W^{k,p}\), \(X=W^{k,p}\cap W_ 0^{\ell,p}\) (0\(\leq \ell \leq k)\), and \(X=W^{-k,p}\), for related results for the evolution problem \(D_ tv+v(x,t)\cdot \nabla y+a(x,t)y=f(t,x)\), and for applications to nonlinear evolution equations see (by the same author) “Boundary value problems for a class of first order partial differential equations in Sobolev spaces and applications to the Euler flow”, Rend. Semin. Mat. Univ. Padova 79 (1988), and references [9], [25], therein.

Reviewer: H.Beirão Da Veiga

MSC:

35F15	Boundary value problems for linear first-order PDEs
47B44	Linear accretive operators, dissipative operators, etc.
47D03	Groups and semigroups of linear operators

Keywords:

generators; groups of operators; stationary equation; nonlinear evolution equations