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Finite time BV blowup for Liu-admissible solutions to $p$-system via computer-assisted proof

Sam G. Krupa

TL;DR

Finite time blowup of the BV-norm for exact solutions to genuinely nonlinear hyperbolic systems in one space dimension, in particular the p- system, which provably does not admit a strictly convex entropy.

Abstract

In this paper, we consider finite time blowup of the $BV$-norm for exact solutions to genuinely nonlinear hyperbolic systems in one space dimension, in particular the $p$-system. We consider solutions verifying shock admissibility criteria such as the Lax E-condition and the Liu E-condition. In particular, we present Riemann initial data which admits infinitely many bounded solutions, each of which experience, not just finite time, but in fact instantaneous blowup of the $BV$ norm. The Riemann initial data is allowed to come from an open set in state space. Our method provably does not admit a strictly convex entropy. The main results in this article compare to Jenssen [SIAM J. Math. Anal., 31(4):894--908, 2000], who shows $BV$ blowup for bounded solutions, or alternatively, blowup in $L^\infty$, for an artificial $3\times 3$ system which is not genuinely nonlinear. Baiti-Jenssen [Discrete Contin. Dynam. Systems, 7(4):837--853, 2001] improves upon this Jenssen result and can consider a genuinely nonlinear system, but then the blowup is only in $L^\infty$ and they cannot construct bounded solutions which blowup in $BV$. Moreover, their system is non-physical and provably does not admit a global, strictly convex entropy. Our result also shows sharpness of the recent Bressan-De Lellis result [Arch. Ration. Mech. Anal., 247(6):Paper No. 106, 12, 2023] concerning well-posedness via the Liu E-condition. The proof of our theorem is computer-assisted, following the framework of Székelyhidi [Arch. Ration. Mech. Anal., 172(1):133--152, 2004]. Our code is available on the GitHub.

Finite time BV blowup for Liu-admissible solutions to $p$-system via computer-assisted proof

TL;DR

Finite time blowup of the BV-norm for exact solutions to genuinely nonlinear hyperbolic systems in one space dimension, in particular the p- system, which provably does not admit a strictly convex entropy.

Abstract

In this paper, we consider finite time blowup of the -norm for exact solutions to genuinely nonlinear hyperbolic systems in one space dimension, in particular the -system. We consider solutions verifying shock admissibility criteria such as the Lax E-condition and the Liu E-condition. In particular, we present Riemann initial data which admits infinitely many bounded solutions, each of which experience, not just finite time, but in fact instantaneous blowup of the norm. The Riemann initial data is allowed to come from an open set in state space. Our method provably does not admit a strictly convex entropy. The main results in this article compare to Jenssen [SIAM J. Math. Anal., 31(4):894--908, 2000], who shows blowup for bounded solutions, or alternatively, blowup in , for an artificial system which is not genuinely nonlinear. Baiti-Jenssen [Discrete Contin. Dynam. Systems, 7(4):837--853, 2001] improves upon this Jenssen result and can consider a genuinely nonlinear system, but then the blowup is only in and they cannot construct bounded solutions which blowup in . Moreover, their system is non-physical and provably does not admit a global, strictly convex entropy. Our result also shows sharpness of the recent Bressan-De Lellis result [Arch. Ration. Mech. Anal., 247(6):Paper No. 106, 12, 2023] concerning well-posedness via the Liu E-condition. The proof of our theorem is computer-assisted, following the framework of Székelyhidi [Arch. Ration. Mech. Anal., 172(1):133--152, 2004]. Our code is available on the GitHub.
Paper Structure (26 sections, 8 theorems, 54 equations, 2 figures)

This paper contains 26 sections, 8 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Finite time blowup
  3. Joly-Métivier-Rauch (1994) and Young (1999): magnification
  4. Jenssen (2000) and Baiti-Jenssen (2001): blowup for artificial systems
  5. Bressan-Chen-Zhang (2018): perturbation of wave speeds for the $p$-system
  6. Bressan-De Lellis (2023): well-posedness via Liu E-condition
  7. The $L^2$ theory (2022)
  8. Chiodaroli-De Lellis-Kreml (2015): convex integration solutions for 2-D Riemann problems
  9. Genericity and a question of Dafermos
  10. Method of proof: convex integration
  11. Large $T_5$ configurations
  12. A computational approach
  13. Plan for the paper
  14. Solutions via convex integration
  15. $T_N$ configurations and rank-one convexity
  16. ...and 11 more sections

Key Result

Theorem 1.1

There exists a smooth, strictly increasing, and strictly concave function $p\colon\mathbb{R}\to\mathbb{R}$ such that for the $p$-system psystem with this $p$, the following holds: There exists open sets $\mathcal{U}_L,\mathcal{U}_R\subset\mathbb{R}$ such that for all Riemann initial data with $(v_L,u_L)\in \mathcal{U}_L$ and $(v_R,u_R)\in \mathcal{U}_R$, there exists infinitely many different sol

Figures (2)

  • Figure 1: A schematic of a $T_5$ configuration.
  • Figure 2: A diagram of values of $DS$, the Jacobian of the subsolution $S$.

Theorems & Definitions (13)

  • Theorem 1.1: Main theorem -- BV blowup for the $p$-system
  • Definition 1.2: In-approximations
  • Theorem 1.3: MR1728376 and MR3740399 and hab_thesis
  • Definition 2.1: $T_N$-configuration)
  • Lemma 2.2
  • Theorem 2.3: Algebraic criterion MR2118899
  • Lemma 2.4: Stability of $T_5$ configurations in $\mathbb{R}^{2\times2}$ MR3740399
  • Definition 2.5: Large $T_5$
  • Theorem 2.6: Existence of in-approximations for large $T_5$s configurations MR3740399
  • Proposition 4.1: Main Proposition -- The MATLAB Proposition
  • ...and 3 more