On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions
On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions
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DOI：
10.1007/bf00379259
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发表时间：
1995-12
影响因子：
2.5
通讯作者：
J. K. Hunter;Yuxi Zheng
中科院分区：
文献类型：
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作者：
J. K. Hunter;Yuxi Zheng
我们研究非线性双曲偏微分方程，(ut+uux)x=1/2ux2。该偏微分方程是从变分原理导出的一类双曲方程的弱非线性解的正则渐近方程。特别是，它描述了向列液晶的巨大指向矢场中的波。偏微分方程的全局光滑解不存在，因为它们的导数在有限时间内爆炸，而弱解不是唯一的。因此，我们定义了两类不同的可接受的弱解，我们称之为耗散解和保守解。我们证明了每种类型的可接受弱解的全局存在性，前提是初始数据的导数具有有界变化和紧支持。尽管它们的导数会爆炸，但这些解仍然是连续的。在任何 Lp 空间中，二阶导数都没有先验估计，因此不能使用 Sobolev 型论证来推断弱解的存在。相反，我们通过对偏微分方程的显式近似解的爆炸奇点建立详细估计来证明存在性。我们还描述了偏微分方程的定性性质，包括与无粘流体的 Burgers 方程的比较以及显式解的许多说明性示例。我们证明了保守的弱解是通过特征正则化方法获得的解的极限，并且我们证明了耗散解的大时间渐近行为是一种特殊的分段线性解，我们称之为扭结波。
We study the nonlinear hyperbolic partial differential equation, (ut+uux)x=1/2ux2. This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal.Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up.There are noa prioriestimates on the second derivatives in anyLpspace, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation.We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave.