Weak solutions to a nonlinear variational wave equation and some related problems |
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Authors: | Ping Zhang |
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Institution: | (1) Academy of Mathematics & Systems Science, CAS, Beijing, 100080, P. R. China |
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Abstract: | In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and
some related problems. We first introduce the main tools, the L
p
Young measure theory and related compactness results, in the first section. Then we use the L
p
Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear
wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove
the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed.
In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic
equation, which is also the so-called vortex density equation arising from sup-conductivity. |
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Keywords: | variational wave equation weak solutions L p Young measure renormalized solutions |
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