no. 1-2
Partial Differential Equations
Stability estimates on general scalar balance laws
[Estimation de la variation totale et stabilité pour des lois de conservations scalaires généralisées]

Présenté par : Jean-Michel Bony

Colombo, Rinaldo M.1 ; Mercier, Magali1 ; Rosini, Massimiliano D.1
1 Department of Mathematics, Brescia University, Via Branze 38, 25133 Brescia, Italy
Comptes Rendus. Mathématique, Tome 347 (2009) no. 1-2, pp. 45-48.

Nous considérons ici une loi de conservation généralisée en dimension N : ∂tu+Divf(t,x,u)=F(t,x,u). Sous des hypothèses adaptées pour f et F, nous obtenons une borne de la variation totale de la solution. À partir de ce résultat, il est alors possible de donner une estimation de la dépendance des solutions au flot f et au terme source F. Dans les cas particuliers déjà étudiés, notre résultat se réduit à ceux déjà connus.

Consider the general scalar balance law in N space dimensions ∂tu+Divf(t,x,u)=F(t,x,u). Under suitable assumptions on f and F, we provide bounds on the total variation of the solution. Based on this first result, we establish estimates on the dependence of the solutions from f and F. In the more particular cases considered in the literature, the present estimate reduces to the known ones.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.11.005
Colombo, Rinaldo M. 1 ; Mercier, Magali 1 ; Rosini, Massimiliano D. 1

1 Department of Mathematics, Brescia University, Via Branze 38, 25133 Brescia, Italy 
@article{CRMATH_2009__347_1-2_45_0,
     author = {Colombo, Rinaldo M. and Mercier, Magali and Rosini, Massimiliano D.},
     title = {Stability estimates on general scalar balance laws},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {45--48},
     publisher = {Elsevier},
     volume = {347},
     number = {1-2},
     year = {2009},
     doi = {10.1016/j.crma.2008.11.005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2008.11.005/}
}
TY  - JOUR
AU  - Colombo, Rinaldo M.
AU  - Mercier, Magali
AU  - Rosini, Massimiliano D.
TI  - Stability estimates on general scalar balance laws
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 45
EP  - 48
VL  - 347
IS  - 1-2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2008.11.005/
DO  - 10.1016/j.crma.2008.11.005
LA  - en
ID  - CRMATH_2009__347_1-2_45_0
ER  - 
%0 Journal Article
%A Colombo, Rinaldo M.
%A Mercier, Magali
%A Rosini, Massimiliano D.
%T Stability estimates on general scalar balance laws
%J Comptes Rendus. Mathématique
%D 2009
%P 45-48
%V 347
%N 1-2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2008.11.005/
%R 10.1016/j.crma.2008.11.005
%G en
%F CRMATH_2009__347_1-2_45_0
Colombo, Rinaldo M.; Mercier, Magali; Rosini, Massimiliano D. Stability estimates on general scalar balance laws. Comptes Rendus. Mathématique, Tome 347 (2009) no. 1-2, pp. 45-48. doi : 10.1016/j.crma.2008.11.005. http://www.numdam.org/articles/10.1016/j.crma.2008.11.005/

[1] Ambrosio, L.; Fusco, N.; Pallara, D. Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000

[2] Bianchini, S.; Colombo, R.M. On the stability of the standard Riemann semigroup, Proc. Amer. Math. Soc., Volume 130 (2002) no. 7, pp. 1961-1973 (electronic)

[3] Bouchut, F.; Perthame, B. Kružkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., Volume 350 (1998) no. 7, pp. 2847-2870

[4] Chen, G.-Q.; Karlsen, K.H. Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients, Comm. Pure Appl. Anal., Volume 4 (2005) no. 2, pp. 241-266

[5] R.M. Colombo, M. Mercier, M.D. Rosini, Stability and total variation estimates on general scalar balance laws, Comm. Math. Sci., submitted for publication

[6] Dafermos, C.M. Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., Volume 38 (1972), pp. 33-41

[7] Karlsen, K.H.; Risebro, N.H. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients, Discrete Contin. Dynam. Syst., Volume 9 (2003) no. 5, pp. 1081-1104

[8] Kružkov, S.N. First order quasilinear equations with several independent variables, Mat. Sb., Volume 81 (1970) no. 123, pp. 228-255

[9] Lucier, B.J. A moving mesh numerical method for hyperbolic conservation laws, Math. Comp., Volume 46 (1986) no. 173, pp. 59-69

Cité par Sources :