Abstract
Let T=−νy∂/∂x, |x|<a, |y|<1, be the streaming operator with non re-entry boundary conditions, under the assumption of plane symmetry. In an analogous way, let T1 be the free streaming operator with specular reflection boundary conditions. We study some properties of T and T1, under the assumption that the domains D(T) and D(T1) are contained in suitable asymmetric Sobolev spaces. The norm of the element f is defined in such spaces by the sum ||f||=||f||1+c||∂f/∂x||1, where ||·||1 is the usual L1 norm and c is a constant. We prove that both T and T1 generate contraction semigroups, which preserve positivity. Moreover, the semigroup generated by T1 also has a norm conservation property. The results, listed above, are of interest for applications. In fact, in several evolution problems, one has to deal at the same time with T (or with T1) and with “local interaction” operators, such as the one examined in Sec. 7. These operators are bounded in the asymmetric Sobolev spaces which contain D(T) or D(T1), whereas they are not bounded in L1-like spaces.
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