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I am looking for references for the following notion. Suppose $\rho_1,\rho_2:[0,T]\times\mathbb R$ satisfy the following system of PDE: $$ \begin{align} \frac{\partial \rho_1}{\partial t} &= -\kappa c_1 (\rho_1 - \rho_2) - u \frac{\partial \rho_1}{\partial x} \\ \frac{\partial \rho_2}{\partial t} &= \kappa c_2 (\rho_1 - \rho_2) \\ \end{align} $$ Then as $\kappa \to \infty$, the solutions should converge to a common solution $\rho := \rho_1 \approx \rho_2$, which satisfies an advection equation $$ \frac{\partial \rho}{\partial t} = - \frac{c_2 u}{c_1 + c_2} \frac{\partial \rho}{\partial x} \tag1 $$ I would like to make this statement rigorous. I am particularly interested in the case when $c_1$ and $c_2$ depend upon $\rho_1$ and $\rho_2$, in which case equation (1) will be like the Burgers equation, see for example Section 3.4 of "Partial Differential Equations, 2nd Ed" by Lawrence Evans. Thus the solutions are susceptible to shocks, and I want to be sure that equation (1) should more properly be written in the weak form given in Evans, so that I can properly compute the speed of the shocks.

I am guessing someone has done this before, and I don't want to reinvent the wheel.

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The following paper should be useful:

Hyperbolic conservation laws with stiff relaxation terms and entropy, G-Q Chen, C D Levermore, and T-P Liu, Comm. Pure Appl. Math., 47 (1994), 787-–830.

There's also a review of hyperbolic relaxation problems in:

R. Natalini, Recent results on hyperbolic relaxation problems, Analysis of systems of conservation laws (Aachen, 1997), 128--198, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., 99, Chapman & Hall/CRC, Boca Raton, FL, 1999.

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