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Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system

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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

  1. Dipartimento di Matematica P.A., Università dell'Aquila, via Roma 33, 67100, L'Aquila, Italy

    Pierangelo Marcati

  2. Dipartimento di Matematica, Università di Torino, via C. Alberto 10, 10123, Torino, Italy

    Albert J. Milani

  3. Dipartimento di Matematica, Università di Trento, 38050, Povo (Trento), Italy

    Paolo Secchi

Authors
  1. Pierangelo Marcati
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  2. Albert J. Milani
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  3. Paolo Secchi
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Marcati, P., Milani, A.J. & Secchi, P. Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system. Manuscripta Math 60, 49–69 (1988). https://doi.org/10.1007/BF01168147

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  • DOI: https://doi.org/10.1007/BF01168147

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