Propagating fronts for a viscous Hamer-type system

Giada Cianfarani Carnevale; Corrado Lattanzio; Corrado Mascia

doi:10.3934/dcds.2021130

Article Contents

2022, Volume 42, Issue 2: 605-621. Doi: 10.3934/dcds.2021130

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$d$

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Propagating fronts for a viscous Hamer-type system

1.
Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università degli Studi dell'Aquila, L'Aquila 67100, ITALY
2.
Dipartimento di Matematica "Guido Castelnuovo", Sapienza Università di Roma, Roma 00185, ITALY

^* Corresponding author: Giada Cianfarani Carnevale
^* Corresponding author: Giada Cianfarani Carnevale

Received: February 2021

Revised: July 2021

Early access: September 2021

Published: February 2022

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Abstract

Motivated by radiation hydrodynamics, we analyse a $ 2\times2 $ system consisting of a one-dimensional viscous conservation law with strictly convex flux –the viscous Burgers' equation being a paradigmatic example– coupled with an elliptic equation, named viscous Hamer-type system. In the regime of small viscosity and for large shocks, namely when the profile of the corresponding underlying inviscid model undergoes a discontinuity –usually called sub-shock– it is proved the existence of a smooth propagating front, regularising the jump of the corresponding inviscid equation. The proof is based on Geometric Singular Perturbation Theory (GSPT) as introduced in the pioneering work of Fenichel [5] and subsequently developed by Szmolyan [21]. In addition, the case of small shocks and large viscosity is also addressed via a standard bifurcation argument.

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Mathematics Subject Classification: Primary: 35C07, 34E15, 35B25, 76N30, 35A24.

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