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)

 
 

 

A mixed type system of three equations modelling reacting flows


Authors: Yun-guang Lu and Christian Klingenberg
Journal: Proc. Amer. Math. Soc. 131 (2003), 3511-3516
MSC (2000): Primary 35L65; Secondary 35M10
DOI: https://doi.org/10.1090/S0002-9939-03-06958-2
Published electronically: April 1, 2003
MathSciNet review: 1991763
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we contrast two approaches for proving the validity of relaxation limits $\alpha \rightarrow \infty$ of systems of balance laws

\begin{displaymath}u_t +{f(u)}_x = \alpha g(u) \quad . \end{displaymath}

In one approach this is proven under some suitable stability condition; in the other approach, one adds artificial viscosity to the system

\begin{displaymath}u_t +{f(u)}_x = \alpha g(u) + \epsilon u_{xx} \end{displaymath}

and lets $\alpha \rightarrow \infty$ and $\epsilon \rightarrow 0$ together with $M \alpha \leq \epsilon $ for a suitable large constant $M$. We illustrate the usefulness of this latter approach by proving the convergence of a relaxation limit for a system of mixed type, where a subcharacteristic condition is not available.


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

  • [D] R.J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal., 82(1983), 27-70. MR 84k:35091
  • [KL] C. Klingenberg, Y.G. Lu, Cauchy problem for Hyperbolic conservation laws with a relaxation term, Proc. Roy. Soc. of Edinb., Series A, 126(1996), 821-828. MR 97f:35130
  • [Liu] T.P. Liu, Hyperbolic Conservation Laws with Relaxation, Comm. Math. Phys., 108(1987), 153-175. MR 88f:35092
  • [L1] Y.G. Lu, Cauchy problem for an extended model of combustion, Proc. Roy. Soc. Edin., 120A(1992), 349-360. MR 93f:80021
  • [L2] Y.G. Lu, Cauchy problem for a hyperbolic model, Nonlinear Analysis, TMA, 23(1994), 1135-1144. MR 95i:35177
  • [L3] Y.G. Lu, Singular limits of stiff relaxation and dominant diffusion for nonlinear systems, J. Diff. Equs., 179(2002), 687-713. MR 2003b:35134
  • [Lu1] Y.G. Lu and C. Klingenberg, The Cauchy problem for hyperbolic conservation laws with three equations, J. Math. Appl.Anal., 202(1996), 206-216. MR 97h:35147
  • [Lu2] Y.G. Lu and C. Klingenberg, The relaxation limit for systems of Broadwell type, Diff. Int. Equations, 14(2001), 117-127. MR 2001k:35200
  • [M] A. Majda, A qualitative model for dynamic combustion, SIAM J. Appl. Math., 41(1981), 70-93. MR 82j:35096
  • [S] J. Shearer, Global existence and compactness in $L^{p}$ for the quasilinear wave equation, Comm. PDE, 19(1994), 1829-1877. MR 95m:35120
  • [T] A. Tzavaras, Materials with internal variables and relaxation to conservation laws, Arch. Rational Mech., 146 (1999), 129-155. MR 2000i:74004

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35L65, 35M10

Retrieve articles in all journals with MSC (2000): 35L65, 35M10


Additional Information

Yun-guang Lu
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, People’s Republic of China – and – Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
Email: yglu@matematicas.unal.edu.co

Christian Klingenberg
Affiliation: Applied Mathematics, Würzburg University, Am Hubland, Würzburg 97074, Germany
Email: Christian.Klingenberg@iwr.uni-heidelberg.de

DOI: https://doi.org/10.1090/S0002-9939-03-06958-2
Received by editor(s): November 1, 2000
Received by editor(s) in revised form: June 6, 2002
Published electronically: April 1, 2003
Communicated by: Suncica Canic
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society