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Entropy solutions in $BV^s$ for a class of triangular systems involving a transport equation

Christian Bourdarias, Anupam Pal Choudhury, Billel Guelmame, Stéphane Junca

Abstract

In this article, we consider a class of strictly hyperbolic triangular systems involving a transport equation. Such systems are known to create measure solutions for the initial value problem. Adding a stronger transversality assumption on the fields, we are able to obtain solutions in $L^\infty$ under optimal fractional $BV$ regularity of the initial data. Our results show that the critical fractional regularity is $s=1/3$. We also construct an initial data that is not in $BV^{1/3}$ but for which a blow-up in $L^\infty$ occurs, proving the optimality of our results.

Entropy solutions in $BV^s$ for a class of triangular systems involving a transport equation

Abstract

In this article, we consider a class of strictly hyperbolic triangular systems involving a transport equation. Such systems are known to create measure solutions for the initial value problem. Adding a stronger transversality assumption on the fields, we are able to obtain solutions in under optimal fractional regularity of the initial data. Our results show that the critical fractional regularity is . We also construct an initial data that is not in but for which a blow-up in occurs, proving the optimality of our results.
Paper Structure (18 sections, 8 theorems, 88 equations, 5 figures)

This paper contains 18 sections, 8 theorems, 88 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The hyperbolic triangular system
  3. Weak and entropy solutions
  4. $BV^s$ functions
  5. Main results
  6. The uniformly strictly hyperbolic system
  7. Riemann invariants
  8. The Riemann problem
  9. The Lax curves
  10. The Lax cubic estimate on the Rankine-Hugoniot curve
  11. Cubic estimates for the Riemann problem
  12. Existence in $BV^{s}$
  13. The Wave Front Tracking algorithm
  14. Uniform estimates
  15. Passage to the limit in the weak formulation
  16. ...and 3 more sections

Key Result

Theorem 1.1

MO For all $s \in ]0,1[$, $BV^s$ functions are regulated functions.

Figures (5)

  • Figure 1: Nonlinear interaction of two waves. The 1-wave on the left crosses the interaction point with the same speed and the same value $u_-$ on the left of the 1-wave ($u_m=u_-$) and $u_+$ on the right. On the other hand, the second wave is affected by the interaction: the speed of the 2-wave and a new value $Z_m$ appears.
  • Figure 2: Interaction of a shock with a $2$-contact discontinuity. The interacting waves are represented by dotted lines, a 1-wave in black followed by a 2-wave in red. The full lines represent the resulting waves.
  • Figure 3: Interaction between a $1$-contact discontinuity and a $2$-contact discontinuity. The interacting waves are represented by dotted lines, a 1-wave in black (horizontal line) followed by a 2-wave in red (vertical line). The full lines represent the resulting waves.
  • Figure 4: Wave front tracking algorithm. For the picture, $f'(u)>0 > a(u)$. Thus the 1-waves in blue go to the right and the 2-waves in black to the left. Notice that these 1-waves are not affected by the interaction with the 2-waves, but the 2-waves are affected by the interaction.
  • Figure 5: A typical building-block

Theorems & Definitions (17)

  • Definition 1.1: weak solution
  • Definition 1.2: entropy solution
  • Definition 1.3
  • Theorem 1.1
  • Lemma 1.1
  • Theorem 2.1: Existence in $L^\infty$ with $(u_0,v_0) \in BV^{1/3} \times L^\infty$
  • Theorem 2.2: Blow-up in $L^\infty$ at $t=0+$ for $u_0 \in BV^{1/3-0}$
  • Lemma 4.1: Cubic flatness of the global Rankine-Hugoniot curve
  • proof
  • Proposition 4.1: Variation of $Z$ through a composite wave
  • ...and 7 more