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 Hölder 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.

A Finsler type Lipschitz optimal transport metric for a quasilinear wave equation

“…Now, we state the uniqueness result in [13], which together with the energy conservation proved in [24] show that the problem (1.4)-(1.5) has a unique weak solution which conserves the total energy. Theorem 2.2 (Uniqueness [13] and energy conservation [24]). Let the condition (1.6) be satisfied, then there exists a unique conservative weak solution u(x, t) for (1.4)- (1.5).…”

“…Finally, we remark that the Lipschitz continuous dependence result in this paper does not direct to a uniqueness result for the original equation (1.4), because in the current paper we only consider the solution constructed in [24]. The uniqueness result in [13] rules our the possibility to construct a different solution using a method different from [24]. It also makes the Lipschitz continuous dependence result obtained in this paper working for all solutions, due to the uniqueness.…”

“…We begin, in this paper, by reviewing the existence and uniqueness of conservative weak solution to the Cauchy problem (1.4)-(1.5) in [13,24]. Theorem 2.1 (Existence [24]).…”

“…The uniqueness of energy weak solution satisfying the energy conservative condition, i.e. the uniqueness of conservative solution, was recently proved by the authors in [13]. We will review the existence and uniqueness results in the next section.…”

“…Solutions of (1.4)-(1.5) may form finite time cusp singularity, see examples in [4,9,20]. The existence and uniqueness of global-in-time energy conservative Hölder continuous (weak) solution have been established by Hu in [24] and the authors in [13], respectively. In this paper, we address the Lipschitz continuous dependence and generic regularity of solution.…”