Asymptotic Variational Wave Equations
Asymptotic Variational Wave Equations
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DOI：
10.1007/s00205-006-0014-8
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发表时间：
2005-02
影响因子：
2.5
通讯作者：
A. Bressan;Ping Zhang;Yuxi Zheng
中科院分区：
文献类型：
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作者：
A. Bressan;Ping Zhang;Yuxi Zheng
研究方程（ut+（f（u））x）x=f′ ′（u）（ux）2/2，其中f（u）是给定的光滑函数.典型的f（u）=u2/2或u3/3。该方程模拟了变分波动方程utt −c（u）（c（u）ux）x=0的单向和弱非线性波，该方程模拟了一些具有自然正弦曲线c的液晶。方程本身也是变分问题的欧拉-拉格朗日方程。两个自然类的解决方案可以与这个方程。一个保守解将保持其能量在时间，而耗散弱解失去能量时，奇异性出现。保守解是全局定义的，在时间上向前和向后，并保留有趣的几何特征，如哈密顿结构。另一方面，耗散的解决方案似乎是更自然的从物理的观点来看，我们建立了一类保守的解决方案的Cauchy问题的适定性，初始数据具有有限的能量，并假设通量函数f具有Lipschitz连续的二阶导数。当f是凸的时，Cauchy问题也在耗散解类中适定。然而，当f不是凸的，我们证明了耗散解不连续依赖于初始数据。
We investigate the equation (ut+(f(u))x)x=f′ ′(u) (ux)2/2 wheref(u) is a given smooth function. Typicallyf(u)=u2/2 oru3/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equationutt−c(u) (c(u)ux)x=0 which models some liquid crystals with a natural sinusoidalc. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux functionfhas a Lipschitz continuous second-order derivative. In the case wherefis convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, whenfis not convex, we show that the dissipative solutions do not depend continuously on the initial data.