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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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