Conservative solutions to a one-dimensional nonlinear variational wave equation
Conservative solutions to a one-dimensional nonlinear variational wave equation
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DOI：
10.1016/j.jde.2015.02.006
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发表时间：
2015-07
影响因子：
2.4
通讯作者：
Yan-bo Hu
中科院分区：
文献类型：
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作者：
Yan-bo Hu
本文研究了一个非线性变分波动方程，它是一个变分原理的欧拉-拉格朗日方程，其作用量是场的导数的二次函数。对于有限能量的初值，我们建立了其Cauchy问题能量守恒弱解的整体存在性。该方法非常接近Bressan，Zhang和Zheng [6]，[7]提出的能量相关坐标方法。通过引入一组新的变量，解决了由于可能的能量集中而导致的所有奇异性，该方程可以重写为半线性系统。通过将半线性方程组的解表示为原变量，构造了整体弱解。
This paper is focused on a nonlinear variational wave equation which is the Euler–Lagrange equation of a variational principle whose action is a quadratic function of the derivatives of the field. We establish the global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy. The approach follows very closely the method of energy-dependent coordinates proposed by Bressan, Zhang and Zheng [6], [7]. By introducing a new set of variables, which resolve all singularities due to the possible concentration of energy, the equation can be rewritten as a semilinear system. We construct the global weak solution by expressing the solution of the semilinear system in terms of the original variables.