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Shock interactions for the Burgers-Hilbert Equation

Alberto Bressan, Sondre T. Galtung, Katrin Grunert, Khai T. Nguyen

Abstract

This paper provides an asymptotic description of a solution to the Burgers-Hilbert equation in a neighborhood of a point where two shocks interact. The solution is obtained as the sum of a function with $H^2$ regularity away from the shocks plus a corrector term having an asymptotic behavior like |x|ln|x| close to each shock. A key step in the analysis is the construction of piecewise smooth solutions with a single shock for a general class of initial data.

Shock interactions for the Burgers-Hilbert Equation

Abstract

This paper provides an asymptotic description of a solution to the Burgers-Hilbert equation in a neighborhood of a point where two shocks interact. The solution is obtained as the sum of a function with regularity away from the shocks plus a corrector term having an asymptotic behavior like |x|ln|x| close to each shock. A key step in the analysis is the construction of piecewise smooth solutions with a single shock for a general class of initial data.
Paper Structure (8 sections, 15 theorems, 324 equations, 3 figures)

This paper contains 8 sections, 15 theorems, 324 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Solutions with one shock and general initial data
  3. Preliminary estimates
  4. Proof of Theorem \ref{['t:21']}
  5. Two interacting shocks
  6. Constructing a solution with two interacting shocks
  7. Some preliminary estimates
  8. Proof of Theorem \ref{['t:61']}

Key Result

Theorem 2.1

For every $\overline{w}\in H^2\bigl(\mathbb{R}\setminus \{0\}\bigr)$ satisfying $\overline w(0-)-\overline w(0+)>0$ and every $c_1$, $c_2\in \mathbb{R}$, the Cauchy problem for the Burgers-Hilbert equation (BH), with initial condition as in (id2)-(o-vp), admits a unique piecewise regular solution de

Figures (3)

  • Figure 1: Decomposing a solution in the form (\ref{['uwp']})
  • Figure 2: The characteristics for a solution to Burgers' equation with two shocks at $x_1(t)<x_2(t)$.
  • Figure 3: Positions of the two shocks in the original variables (left), and in the adapted variables (right).

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Lemma 3.1
  • Remark 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 9 more