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Uniqueness of conservative solutions to a one-dimensional general quasilinear wave equation through variational principle

Hong Cai, Geng Chen, Yi Du, Yannan Shen

Abstract

In this paper, we prove the uniqueness of energy conservative Holder continuous weak solution to a general quasilinear wave equation by the analysis of characteristics. This result has no restriction on the size of solutions, i.e. it is a large data result.

Uniqueness of conservative solutions to a one-dimensional general quasilinear wave equation through variational principle

Abstract

In this paper, we prove the uniqueness of energy conservative Holder continuous weak solution to a general quasilinear wave equation by the analysis of characteristics. This result has no restriction on the size of solutions, i.e. it is a large data result.

Paper Structure

This paper contains 12 sections, 9 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Existing existence result
  4. Main result of this paper
  5. The existence and uniqueness of characteristics
  6. An equivalent semi-linear system
  7. (i).
  8. (ii).
  9. (iii).
  10. (iv).
  11. (v).
  12. Proof of Theorem \ref{['thm_un']}

Key Result

Theorem 2.1

Let the condition con be satisfied, then the Cauchy problem vwl--ID admits a global weak solution$u=u(t,x)$ defined for all $(t,x)\in\mathbb{R}^+\times\mathbb{R}$, as follows:

Figures (1)

  • Figure 1: Left: The support of the test function $\phi^\epsilon$ in (\ref{['pen11']}). Right: An enlarged picture of $\Gamma_{34}^\varepsilon$, which is used in \ref{['lime2']} where we only do the calculation in the shaded region, becuase the unshaded region can be omitted as $\varepsilon\rightarrow 0$.

Theorems & Definitions (15)

  • Theorem 2.1: Global existence H
  • Definition 2.1: Energy conservation H
  • Theorem 2.2: Energy conservation H
  • Theorem 2.3: Uniqueness
  • Lemma 3.1
  • proof : Proof
  • Remark 3.1
  • Lemma 3.2
  • proof : Proof
  • Lemma 4.1
  • ...and 5 more