A Finsler type Lipschitz optimal transport metric for a quasilinear wave equation
A Finsler type Lipschitz optimal transport metric for a quasilinear wave equation
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DOI：
10.1016/j.jde.2023.01.035
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发表时间：
2020-07
影响因子：
2.4
通讯作者：
H. Cai;Geng Chen;Y. Shen
中科院分区：
文献类型：
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作者：
H. Cai;Geng Chen;Y. Shen
我们通过变分原理考虑一般拟线性波动方程弱能量保守解的全局适定性，当能量集中时，解可能形成有限时间尖点奇点。作为本文的主要结果，我们构造了一个Finsler型最优传输度量，然后证明了该度量下的解流是Lipschitz。我们还通过应用汤姆横向定理证明了通用正则性结果，然后在一组密集的解中找到分段平滑的运输路径。本文的结果针对大数据解决方案，对解决方案的大小没有限制。
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.