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steady transport equation; bounded; unbounded; exterior domains; existence of solutions; estimates
We investigate the steady transport equation $$ \lambda z+w\cdot \nabla z+az=f,\quad \lambda >0 $$ in various domains (bounded or unbounded) with smooth noncompact boundaries. The functions $w,\,a$ are supposed to be small in appropriate norms. The solution is studied in spaces of Sobolev type (classical Sobolev spaces, Sobolev spaces with weights, homogeneous Sobolev spaces, dual spaces to Sobolev spaces). The particular stress is put onto the problem to extend the results to as less regular vector fields $w,\,a$, as possible (conserving the requirement of smallness). The theory presented here is well adapted for applications in various problems of compressible fluid dynamics.
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