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

Author(s): Rinaldo M. Colombo; Piotr Gwiazda
Journal: Quart. Appl. Math. 63 (2005), 509-526.
MSC (2000): Primary 35L65, 35K10
Posted: August 17, 2005
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

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.

References:

1.
H. Amann.
Linear and quasilinear parabolic problems. Vol. I, volume 89 of Monographs in Mathematics.
Birkhäuser Boston Inc., Boston, MA, 1995. MR 1345385 (96g:34088)
2.
S. Bianchini and A. Bressan.
Vanishing viscosity solutions of nonlinear hyperbolic systems.
Annals of Mathematics, to appear.
3.
S. Bianchini and R. M. Colombo.
On the stability of the standard Riemann semigroup.
Proc. Amer. Math. Soc., 130(7):1961-1973 (electronic), 2002. MR 1896028 (2002m:35140)
4.
A. Bressan.
Hyperbolic systems of conservation laws, volume 20 of Oxford Lecture Series in Mathematics and its Applications.
Oxford University Press, Oxford, 2000.
The one-dimensional Cauchy problem. MR 1816648 (2002d:35002)
5.
A. Bressan and T. Yang.
On the convergence rate of vanishing viscosity approximations.
Comm. Pure Appl. Math., 57(8):1075-1109, 2004. MR 2053759 (2005b:35180)
6.
H. Brezis.
Analyse fonctionnelle.
Théorie et applications. [Theory and applications]. Collection Mathématiques Appliquées pour la Maîtrise [Collection of Applied Mathematics for the Master's Degree]. Masson, Paris, 1983. MR 0697382 (85a:46001)
7.
L. C. Evans.
Partial differential equations, volume 19 of Graduate Studies in Mathematics.
American Mathematical Society, Providence, RI, 1998. MR 1625845 (99e:35001)
8.
A. Friedman.
Partial differential equations.
Holt, Rinehart and Winston, Inc., New York, 1969.
9.
L. Hörmander.
The analysis of linear partial differential operators. I.
Classics in Mathematics. Springer-Verlag, Berlin, 2003.
Distribution theory and Fourier analysis, Reprint of the second (1990) edition. Springer, Berlin. MR 1996773 (91m:35001a)
10.
J.-L. Lions.
Quelques méthodes de résolution des problèmes aux limites non linéaires.
Dunod, 1969. MR 0259693 (41:4326)
11.
L. Nirenberg.
On elliptic partial differential equations.
Ann. Scuola Norm. Sup. Pisa (3), 13:115-162, 1959. MR 0109940 (22:823)

Similar Articles:

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35L65, 35K10

Retrieve articles in all Journals with MSC (2000): 35L65, 35K10

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

PII: S0033-569X-05-00973-8
Keywords: Stability of multiD systems of partial differential equations
Received by editor(s): October 26, 2004
Posted: August 17, 2005
Copyright of article: Copyright 2005, Brown University
The copyright for this article reverts to public domain after 28 years from publication.


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2008 Brown University
Comments: qam-query@ams.org		 AMS Website