Regularizing effects for ut + Aϑ(u) = 0 in L1 |
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Authors: | Michael Crandall Michel Pierre |
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Affiliation: | Department of Mathematics and Mathematics Research Center, University of Wisconsin-Madison, Madison, Wisconsin 53706 USA;Université Scientifique et Médicale de Grenoble, Institut de Mathématiques Pures, B.P. 116 38402, Saint Martin-D''Hères, France |
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Abstract: | Various initial-boundary value problems and Cauchy problems can be written in the form , where ?:R→ is nondecreasing and A is the linear generator of strongly continuous nonexpansive semigroup e?tA in an L1 space. For example, if A = ?Δ (subject, perhaps, to suitable boundary conditions) we obtain equations arising in flow in a porous medium or plasma physics (depending on the choice of ?) while if acting in L1() we have a scalar conservation law. In this paper we show that if M, m > 0 and m?′2 ? ν??′' ? M?′2, where ν ? {1,?1}, then (roughly speaking), the norm of may be estimated in terms of the initial data u0 in L1. Such estimates give information about the regularity of solutions, asymptotic behaviour, etc., in applications. Side issues, such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator A?, are also dealt with. These considerations are of independent interest. |
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