$\mathbf {L}^1$ 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.
- 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
BianchiniBressan S. Bianchini and A. Bressan. Vanishing viscosity solutions of nonlinear hyperbolic systems. Annals of Mathematics, to appear.
- Stefano Bianchini and Rinaldo M. Colombo, On the stability of the standard Riemann semigroup, Proc. Amer. Math. Soc. 130 (2002), no. 7, 1961–1973. MR 1896028, DOI https://doi.org/10.1090/S0002-9939-02-06568-1
- 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
- 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, DOI https://doi.org/10.1002/cpa.20030
- 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
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
FriedmanPDE A. Friedman. Partial differential equations. Holt, Rinehart and Winston, Inc., New York, 1969.
- 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
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
- L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162. MR 109940
Amann H. Amann. Linear and quasilinear parabolic problems. Vol. I, volume 89 of Monographs in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1995.
BianchiniBressan S. Bianchini and A. Bressan. Vanishing viscosity solutions of nonlinear hyperbolic systems. Annals of Mathematics, to appear.
BianchiniColombo S. Bianchini and R. M. Colombo. On the stability of the standard Riemann semigroup. Proc. Amer. Math. Soc., 130(7):1961–1973 (electronic), 2002.
BressanLectureNotes 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.
BressanYang A. Bressan and T. Yang. On the convergence rate of vanishing viscosity approximations. Comm. Pure Appl. Math., 57(8):1075–1109, 2004.
Brezis 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.
Evans L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.
FriedmanPDE A. Friedman. Partial differential equations. Holt, Rinehart and Winston, Inc., New York, 1969.
HormanderI 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.
LionsJLBook J.-L. Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, 1969.
Nirenberg59 L. Nirenberg. On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa (3), 13:115–162, 1959.
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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
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.