Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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$\mathbf{L}^1$ 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
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.


References [Enhancements On Off] (What's this?)

  • 1. Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory. MR 1345385
  • 2. S. Bianchini and A. Bressan.
    Vanishing viscosity solutions of nonlinear hyperbolic systems.
    Annals of Mathematics, to appear.
  • 3. Stefano Bianchini and Rinaldo M. Colombo, On the stability of the standard Riemann semigroup, Proc. Amer. Math. Soc. 130 (2002), no. 7, 1961–1973 (electronic). MR 1896028, 10.1090/S0002-9939-02-06568-1
  • 4. Alberto Bressan, Hyperbolic systems of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR 1816648
  • 5. Alberto Bressan and Tong Yang, On the convergence rate of vanishing viscosity approximations, Comm. Pure Appl. Math. 57 (2004), no. 8, 1075–1109. MR 2053759, 10.1002/cpa.20030
  • 6. Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
  • 7. Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
  • 8. A. Friedman.
    Partial differential equations.
    Holt, Rinehart and Winston, Inc., New York, 1969.
  • 9. Lars 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; MR1065993 (91m:35001a)]. MR 1996773
  • 10. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
  • 11. L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115–162. MR 0109940

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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: http://dx.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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