Weak solutions to a nonlinear variational wave equation and some related problems

Ping Zhang

Applications of Mathematics (2006)

  • Volume: 51, Issue: 4, page 427-466
  • ISSN: 0862-7940

Abstract

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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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Zhang, Ping. "Weak solutions to a nonlinear variational wave equation and some related problems." Applications of Mathematics 51.4 (2006): 427-466. <http://eudml.org/doc/33260>.

@article{Zhang2006,
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.},
author = {Zhang, Ping},
journal = {Applications of Mathematics},
keywords = {variational wave equation; weak solutions; $L^p$ Young measure; renormalized solutions; variational wave equation; weak solutions; Young measure; renormalized solutions},
language = {eng},
number = {4},
pages = {427-466},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Weak solutions to a nonlinear variational wave equation and some related problems},
url = {http://eudml.org/doc/33260},
volume = {51},
year = {2006},
}

TY - JOUR
AU - Zhang, Ping
TI - Weak solutions to a nonlinear variational wave equation and some related problems
JO - Applications of Mathematics
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 4
SP - 427
EP - 466
AB - 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.
LA - eng
KW - variational wave equation; weak solutions; $L^p$ Young measure; renormalized solutions; variational wave equation; weak solutions; Young measure; renormalized solutions
UR - http://eudml.org/doc/33260
ER -

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