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Cauchy problem for hyperbolic conservation laws with a relaxation term

  • Christian Klingenberg (a1) and Yun-guang Lu (a2)

This paper considers the Cauchy problem for hyperbolic conservation laws arising in chromatography:

with bounded measurable initial data, where the relaxation term g(δ, u, v) converges to zero as the parameter δ > 0 tends to zero. We show that a solution of the equilibrium equation

is given by the limit of the solutions of the viscous approximation

of the original system as the dissipation ε and the relaxation δ go to zero related by δ = O(ε). The proof of convergence is obtained by a simplified method of compensated compactness [2], avoiding Young measures by using the weak continuity theorem (3.3) of two by two determinants.

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1 G. Q. Chen and L. P. Liu , Zero relaxation and dissipation limits for conservation laws. Comm. Pure Appl. Math. 46 (1993), 755–81.

7 Y. G. Lu , The Cauchy problem for a hyperbolic model. Nonlinear Anal. TMA, 23 (1994), 1135–44.

8 A. Majda , A qualitative model for dynamic combustion. SIAM J. Appl. Math. 41 (1981), 7993.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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