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Abstract time-dependent transport equations. (English) Zbl 0657.45007
The following type of time dependent linear initial-boundary value problems is investigated: \[ \partial u/\partial t+\xi \cdot \partial u/\partial x+a\cdot \partial u/\partial \xi +hu+Au=f\quad for\quad x\in \Omega,\quad \xi \in S,\quad 0<t<T; \] u(x,\(\xi\),0)\(=u_ 0(x,\xi)\) for \(x\in \Omega\), \(\xi\in S\); \(u_-(x,\xi,t)=Ku_+(x,\xi,t)+g(x,\xi,t)\) for \((x,\xi)\in B_-\), \(0<t<T\). Here the solution u describing a particle density is to be found, on \(\Omega \times S\times (0,T)\subset {\mathbb{R}}^ n\times {\mathbb{R}}^ n\times {\mathbb{R}}\). The internal sources f and the incident current g are known; \(u_{\pm}\) denote the traces of u on the “incoming” and “outgoing” parts \(B_{\pm}\) of the boundary, and K is an operator describing the boundary process; the vector field a, the function h and the operator A are also given. There are certain suitable conditions on the data, and in fact the main results are obtained for a more general form of the problem.
The problem is treated in \(L_ p\)-setting, \(1\leq p<\infty\), and in the space of measures. It is shown that the problem is well-posed under suitable assumptions, among which \(\| K\| <1\). One of the technical tools is a general Gauss-Green-identity. In the more delicate case of conservative boundary conditions \((\| K\| =1)\) the notion of solution has to be generalized. In the case that the data do not depend on time, the authors show how their results apply to the theory of \(C_ 0\)-semigroups.
Reviewer: J.Voigt

MSC:
45N05 Abstract integral equations, integral equations in abstract spaces
45K05 Integro-partial differential equations
47D03 Groups and semigroups of linear operators
82C70 Transport processes in time-dependent statistical mechanics
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