Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the stability of the standard Riemann semigroup


Authors: Stefano Bianchini and Rinaldo M. Colombo
Journal: Proc. Amer. Math. Soc. 130 (2002), 1961-1973
MSC (2000): Primary 35L65, 76N10
DOI: https://doi.org/10.1090/S0002-9939-02-06568-1
Published electronically: February 27, 2002
MathSciNet review: 1896028
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the dependence of the entropic solution of a hyperbolic system of conservation laws

\begin{displaymath}\left\{ \begin{array}{c} u_t + f(u)_x = 0, \\ u(0,\cdot) = u_0 \end{array} \right. \end{displaymath}

on the flux function $f$. We prove that the solution is Lipschitz continuous w.r.t. the $C^0$ norm of the derivative of the perturbation of $f$. We apply this result to prove the convergence of the solution of the relativistic Euler equation to the classical limit.


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

  • 1. F. Ancona, A. Marson, Well posedness for general $2\times2$ systems of conservation laws, preprint.
  • 2. P. Baiti, A. Bressan, The semigroup generated by a Temple class system with large data, Differential Integral Equations 10 (1997), no. 3, p. 401-418. MR 2000k:35180
  • 3. S. Bianchini, On the shift differentiability of the flow generated by a hyperbolic system of conservation laws, Discr. Cont. Dyn. Sys. 6-2 (2000), pag. 329-350. MR 2000m:35120
  • 4. A. Bressan, The unique limit of the Glimm scheme, Arch. Rat. Mech. Anal. 130 (1995), 205-230. MR 96d:65143
  • 5. A. Bressan, On the Cauchy problem for nonlinear hyperbolic systems, preprint S.I.S.S.A., 1998.
  • 6. A. Bressan, ``Hyperbolic Systems of Conservation Laws: the One Dimensional Cauchy Problem'', Oxford Univ. Press (2000).
  • 7. A. Bressan, R.M. Colombo, The semigroup generated by $2 \times 2$ conservation laws, Arch. Ration. Mech. Anal. 133, No.1, 1-75 (1995). MR 96m:35198
  • 8. A. Bressan, G. Crasta, B. Piccoli, Well posedness of the Cauchy problem for $n\times n$ systems of conservation laws, Mem. Amer. Math. Soc. 694 (2000). MR 2000m:35122
  • 9. F. Bouchut, B. Perthame, Kruzkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc. 350, 7 (1998), 2847-2870. MR 98m:65156
  • 10. R.M. Colombo, N.H. Risebro, Continuous Dependence in the Large for some Equations of Gas Dynamics, Comm. in P.D.E., 23l, 9&10 (1998), 1693-1718. MR 99h:35124
  • 11. C.M. Dafermos, Generalized characteristics in hyperbolic systems of conservation laws, Arch. Rational Mech. Anal., 107, 2 (1989), 127-155. MR 90h:35150
  • 12. P.D. Lax, Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math., 10 (1957), 537-566. MR 20:176
  • 13. T.P. Liu, The Riemann problem for general systems of conservation laws, J. Differential Equations, 18 (1975), 218-234. MR 51:6168
  • 14. L. Min, S. Ukai, Non-relativistic global limit of weak solutions of the relativistic Euler equations, J. Math. Kyoto Univ., 38-3 (1998), 525-537. MR 99i:76134
  • 15. A.I. Panasyuk, Quasidifferential equations in a metric space, Differentsialnye Uravneniya, 21 (1985), no 8. MR 87b:34078
  • 16. J. Smoller, B. Temple, Global Solutions of the Relativistic Euler Equations, Comm. Math. Phys. 156 (1993), 67-99. MR 94h:35216
  • 17. A.H. Taub, Relativistic Rankine-Hugoniot Equations, Physical Review 74 (1948), 328-334. MR 10:72f

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35L65, 76N10

Retrieve articles in all journals with MSC (2000): 35L65, 76N10


Additional Information

Stefano Bianchini
Affiliation: Istituto per le Applicazioni del Calcolo, Viale del Policlinico 137, 00161 Roma, Italy
Email: bianchin@iac.rm.cnr.it

Rinaldo M. Colombo
Affiliation: Department of Mathematics, University of Brescia, Via Valotti 9, 25133 Brescia, Italy
Email: rinaldo@ing.unibs.it

DOI: https://doi.org/10.1090/S0002-9939-02-06568-1
Keywords: Hyperbolic systems, conservation laws, well posedness
Received by editor(s): July 1, 2000
Published electronically: February 27, 2002
Additional Notes: We thank Alberto Bressan for useful discussions.
Communicated by: Suncica Canic
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society