On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions
 John K. Hunter,
 Yuxi Zheng
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We study the nonlinear hyperbolic partial differential equation, (u _{t}+uu_{x})_{x}=1/2u _{x} ^{2} . 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 no a priori estimates on the second derivatives in any L ^{p} space, so the existence of weak solutions cannot be deduced by using Sobolevtype 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 largetime asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kinkwave.
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 Title
 On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions
 Journal

Archive for Rational Mechanics and Analysis
Volume 129, Issue 4 , pp 305353
 Cover Date
 19951201
 DOI
 10.1007/BF00379259
 Print ISSN
 00039527
 Online ISSN
 14320673
 Publisher
 SpringerVerlag
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 Authors

 John K. Hunter ^{(1)} ^{(2)}
 Yuxi Zheng ^{(1)} ^{(2)}
 Author Affiliations

 1. Department of Mathematics and Institute for Theoretical Dynamics, University of California at Davis, USA
 2. Department of Mathematics, Indiana University at Bloomington, USA