Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



$\mathbf{L}^1$ stability of semigroups with respect to their generators

Authors: Rinaldo M. Colombo and Piotr Gwiazda
Journal: Quart. Appl. Math. 63 (2005), 509-526
MSC (2000): Primary 35L65, 35K10
DOI: https://doi.org/10.1090/S0033-569X-05-00973-8
Published electronically: August 17, 2005
MathSciNet review: 2169031
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Abstract: This note is concerned with the $\mathbf{L}^1$-theory for the system $\partial_t u = \operatorname{div}_x A(u) + B \cdot \Delta u + C(u)$ in several space dimensions. First, an existence result is proved for data in $\mathbf{L}^1\cap \mathbf{L}^\infty \cap \mathbf{BV}$. Then, the $\mathbf{L}^1$-Lipschitz dependence of the solutions with respect to the natural norms of $A$, $B$ and $C$ is achieved. As a corollary, the vanishing viscosity limit for conservation laws in 1D recently obtained in a work by Bianchini and Bressan is slightly extended.

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Additional Information

Rinaldo M. Colombo
Affiliation: Department of Mathematics, University of Brescia, Italy
Email: rinaldo@ing.unibs.it

Piotr Gwiazda
Affiliation: Institute of Applied Mathematics, Warsaw University
Email: pgwiazda@hydra.mimuw.edu.pl

DOI: https://doi.org/10.1090/S0033-569X-05-00973-8
Keywords: Stability of multiD systems of partial differential equations
Received by editor(s): October 26, 2004
Published electronically: August 17, 2005
Article copyright: © Copyright 2005 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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