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

Abstract

We consider the global well-posedness of weak energy conservative solution to a general quasilinear wave equation through variational principle, where the solution may form finite time cusp singularity, when energy concentrates. As a main result in this paper, we construct a Finsler type optimal transport metric, then prove that the solution flow is Lipschitz under this metric. We also prove a generic regularity result by applying Thom's transversality theorem, then find piecewise smooth transportation paths among a dense set of solutions. The results in this paper are for large data solutions, without restriction on the size of solutions.

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

“…Remark 3.1. The estimates in the proof of this Lemma, particularly (3.14), are crucial for the Lipschitz metric result in [12]. Now, we show in the next lemma that for a conservative solution the characteristics can be uniquely determined by combining the characteristic equations (3.1)-(3.2) and the balance laws (3.5)- (3.6).…”

“…In particular, system (1.1) has direct applications on nematic liquid crystals [2,8], which will be introduced in the next part. System (1.1) is also realted to the O(3) σ-model, see the introduction of [12]. Here the O(3) σ-model has applications on many physical areas, including the general relativity and Yang-Mills fields, [19].…”

“…Some estimates in this paper will also serve as crucial preparations in another paper [12] addressing the Lipschitz continuous dependence of solution. In fact, in [12], we will construct a new distance which renders Lipschitz continuous the solution flow of (1.6) constructued in [18], although the solution does not depend continuously on the initial data with respect to the natural Sobolev space corresponding to energy.…”

“…Some estimates in this paper will also serve as crucial preparations in another paper [12] addressing the Lipschitz continuous dependence of solution. In fact, in [12], we will construct a new distance which renders Lipschitz continuous the solution flow of (1.6) constructued in [18], although the solution does not depend continuously on the initial data with respect to the natural Sobolev space corresponding to energy. Because the uniqueness result in the current paper rules out the possibility of constructing a different solution through any method other than the one used in [18], one can fairly well concludes the global well-posedness of conservative solution of (1.6)-(1.7) by the current paper and [12,18].…”