Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Time-asymptotic stability of composite waves of degenerate Oleinik shock and rarefaction for non-convex conservation laws with Cattaneo's law

Yuxi Hu, Ran Song

TL;DR

The paper proves the time-asymptotic stability of a composite wave formed by a degenerate Oleinik shock and a rarefaction for a 1D non-convex conservation law with Cattaneo's law. It constructs the composite wave using a degenerate viscous Oleinik shock and a smooth approximate rarefaction, then analyzes perturbations via a time-dependent shift and a weight function within a relative entropy framework. By establishing a sequence of a priori estimates—$L^2$ relative-entropy, high-order $H^2$, and dissipative bounds—the authors close the energy method under small shock strength and perturbations, showing global existence and convergence to the composite wave with the shift rate vanishing at infinity. This work extends stability results to relaxed hyperbolic systems with non-convex flux, illustrating how a combination of Oleinik entropy, a-contraction with time-dependent shifts, and weighted energy methods can control complex wave interactions. The findings have implications for understanding long-time behavior in non-convex conservation laws with finite-speed heat conduction and contribute to the theory of composite wave stability in relaxed models.

Abstract

This paper examines the large-time behavior of solutions to a one-dimensional conservation law featuring a non-convex flux and an artificial heat flux term regulated by Cattaneo's law, forming a 2$\times$2 system of hyperbolic equations. Under the conditions of small wave strength and sufficiently small initial perturbations, we demonstrate the time-asymptotic stability of a composite wave that combines a degenerate Oleinik shock and a rarefaction wave. The proof utilizes the Oleinik entropy condition, the a-contraction method with time-dependent shifts, and weighted energy estimates.

Time-asymptotic stability of composite waves of degenerate Oleinik shock and rarefaction for non-convex conservation laws with Cattaneo's law

TL;DR

The paper proves the time-asymptotic stability of a composite wave formed by a degenerate Oleinik shock and a rarefaction for a 1D non-convex conservation law with Cattaneo's law. It constructs the composite wave using a degenerate viscous Oleinik shock and a smooth approximate rarefaction, then analyzes perturbations via a time-dependent shift and a weight function within a relative entropy framework. By establishing a sequence of a priori estimates— relative-entropy, high-order , and dissipative bounds—the authors close the energy method under small shock strength and perturbations, showing global existence and convergence to the composite wave with the shift rate vanishing at infinity. This work extends stability results to relaxed hyperbolic systems with non-convex flux, illustrating how a combination of Oleinik entropy, a-contraction with time-dependent shifts, and weighted energy methods can control complex wave interactions. The findings have implications for understanding long-time behavior in non-convex conservation laws with finite-speed heat conduction and contribute to the theory of composite wave stability in relaxed models.

Abstract

This paper examines the large-time behavior of solutions to a one-dimensional conservation law featuring a non-convex flux and an artificial heat flux term regulated by Cattaneo's law, forming a 22 system of hyperbolic equations. Under the conditions of small wave strength and sufficiently small initial perturbations, we demonstrate the time-asymptotic stability of a composite wave that combines a degenerate Oleinik shock and a rarefaction wave. The proof utilizes the Oleinik entropy condition, the a-contraction method with time-dependent shifts, and weighted energy estimates.
Paper Structure (14 sections, 16 theorems, 222 equations)

This paper contains 14 sections, 16 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Oleinik shock wave and rarefaction wave
  4. Composite wave and main result
  5. Reformulation of the problem
  6. Construction of approximate smooth rarefaction wave
  7. Reformulation of the problem
  8. Construction of weight function and shift function
  9. The continuity arguments
  10. Time-asymptotic behavior
  11. A priori estimates
  12. $L^2$ estimates
  13. High-order energy estimates
  14. Dissipative estimates

Key Result

Lemma 2.1

For any states $u_{\pm}$ satisfying $u_{-}< 0$ and $u_{+}=-\frac{u_{-}}{2}>0$, and $\sigma>0$ satisfying R-H condition RH and Oleinik condition Oleinik, and $\tau<\frac{4}{9\sigma u_{-}^2}$, and let $\delta_{S} := |u_{+}-u_{-}|$ be the wave strength of the degenerate Oleinik shock. Then there exists and

Theorems & Definitions (28)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Proposition 3.4
  • Lemma 4.1
  • ...and 18 more