manuscripta mathematica

, Volume 60, Issue 1, pp 49–69

# Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system

• Pierangelo Marcati
• Albert J. Milani
• Paolo Secchi
Article

## Summary

We show that the weak solutions of the nonlinear hyperbolic system
$$\left\{ \begin{gathered} \varepsilon u_t^\varepsilon + p(v^\varepsilon )_x = u^\varepsilon \hfill \\ v_t^\varepsilon - u_x^\varepsilon = 0 \hfill \\ \end{gathered} \right.$$
converge, as ε tends to zero, to the solutions of the reduced problem
$$\left\{ \begin{gathered} u + p(v)_x = 0 \hfill \\ v_t - u_x = 0 \hfill \\ \end{gathered} \right.$$
. Then they satisfy the nonlinear parabolic equation
$$v_t + p(v)_{XX} = 0$$
. The limiting procedure is carried out using the techniques of “Compensated Compactness”. Some connections with the theory of nonlinear heat conduction and the theory of nonlinear diffusion in a porous medium are suggested. The main result is stated in th. (2.9).

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## Authors and Affiliations

• Pierangelo Marcati
• 1
• Albert J. Milani
• 2
• Paolo Secchi
• 3
1. 1.Dipartimento di Matematica P.A.Università dell'AquilaL'AquilaItaly
2. 2.Dipartimento di MatematicaUniversità di TorinoTorinoItaly
3. 3.Dipartimento di MatematicaUniversità di TrentoPovo (Trento)Italy