TY - JOUR

T1 - Weak solutions to a nonlinear variational wave equation with general data

AU - Zhang, Ping

AU - Zheng, Yuxi

N1 - Funding Information:
Ping Zhang is supported by NSF of China under Grant 10131050 and 10276036, the innovation grants from Chinese Academy of Sciences. Yuxi Zheng is supported in part by NSF DMS-0305497 and 0305114. This work was done when Ping Zhang visited Penn State University. He would like to thank the department for its hospitality. Both authors would also like to thank Professors Andy Majda, Craig Evans, Fanghua Lin, Bob Glassey, and John Hunter for their constant interest in our work.

PY - 2005

Y1 - 2005

N2 - We establish the existence of global weak solutions to the initial value problem for a nonlinear variational wave equation utt-c(u)(c(u)ux)x=0 with general initial data (u(0),ut(0))=(u0,u1)∈W1,2×L2 under the assumptions that the wave speed c(u) satisfies c′(·)≥0 and c′(u0(·))>0. Moreover, we obtain high regularity for the spatial derivative ∂xu of the wave amplitude u away from where c′(u)=0. This equation arises from studies in nematic liquid crystals, long waves on a dipole chain, and a few other fields. We use Young measure method in the setting of Lp spaces and method of renormalization to overcome the difficulty that oscillations in a sequence of approximations get amplified by the quadratic growth term of the equation. We use a high space-time estimate for ∂xu to handle possible concentrations. This result improves our earlier existence result for initial data in the space W1,∞×L∞ to the natural space W1,2×L2.

AB - We establish the existence of global weak solutions to the initial value problem for a nonlinear variational wave equation utt-c(u)(c(u)ux)x=0 with general initial data (u(0),ut(0))=(u0,u1)∈W1,2×L2 under the assumptions that the wave speed c(u) satisfies c′(·)≥0 and c′(u0(·))>0. Moreover, we obtain high regularity for the spatial derivative ∂xu of the wave amplitude u away from where c′(u)=0. This equation arises from studies in nematic liquid crystals, long waves on a dipole chain, and a few other fields. We use Young measure method in the setting of Lp spaces and method of renormalization to overcome the difficulty that oscillations in a sequence of approximations get amplified by the quadratic growth term of the equation. We use a high space-time estimate for ∂xu to handle possible concentrations. This result improves our earlier existence result for initial data in the space W1,∞×L∞ to the natural space W1,2×L2.

UR - http://www.scopus.com/inward/record.url?scp=13844308241&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=13844308241&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2004.04.001

DO - 10.1016/j.anihpc.2004.04.001

M3 - Article

AN - SCOPUS:13844308241

VL - 22

SP - 207

EP - 226

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 2

ER -