Representation of Dissipative Solutions to a Nonlinear Variational Wave Equation
Representation of Dissipative Solutions to a Nonlinear Variational Wave Equation
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DOI：
10.4310/cms.2016.v14.n1.a2
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发表时间：
2014-07
期刊：
影响因子：
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通讯作者：
A. Bressan;Tao Huang
中科院分区：
文献类型：
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作者：
A. Bressan;Tao Huang
介绍了一种构造二阶变分波动方程耗散解的新方法。通过变量变换，从非线性偏微分方程组出发，得到一个带源的半线性双曲组。与保守情况相比，这里的源项是不连续的，不连续项并不总是横向交叉的。根据Kolmogorov紧致性定理，通过近似论证得到了半线性系统的解。回归到原始变量，我们恢复了非线性波动方程的解，其中总能量是时间的单调递减函数。
The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with the conservative case, here the source terms are discontinuous and the discontinuities are not always crossed transversally. Solutions to the semilinear system are obtained by an approximation argument, relying on Kolmogorov's compactness theorem. Reverting to the original variables, one recovers a solution to the nonlinear wave equation where the total energy is a monotone decreasing function of time.