Rarefactive solutions to a nonlinear variational wave equation of liquid crystals

Ping Zhang; Yuxi Zheng

doi:10.1081/PDE-100002240

Rarefactive solutions to a nonlinear variational wave equation of liquid crystals

Ping Zhang, Yuxi Zheng

Mathematics

Research output: Contribution to journal › Article › peer-review

45 Scopus citations

Abstract

We study a nonlinear wave equation derived from a simplified liquid crystal model, in which the wave speed is a given function of the wave amplitude. We formulate a viscous approximation of the equation and establish the global existence of smooth solutions for the viscously perturbed equation. For a monotone wave speed function in the equation, we find an invariant region in the phase space in which we discover: (a) smooth data evolve smoothly forever; (b) both the viscous regularization and the smooth solutions obtained through data smoothing for rough initial data yield weak solutions to the Cauchy problem of the nonlinear variational wave equation. The main tool is the Young measure theory and related techniques.

Original language	English (US)
Pages (from-to)	381-419
Number of pages	39
Journal	Communications in Partial Differential Equations
Volume	26
Issue number	3-4
DOIs	https://doi.org/10.1081/PDE-100002240
State	Published - 2001

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

Access to Document

10.1081/PDE-100002240

Cite this

@article{450c5563e7db49fc8c2b08ae42d80ef0,

title = "Rarefactive solutions to a nonlinear variational wave equation of liquid crystals",

abstract = "We study a nonlinear wave equation derived from a simplified liquid crystal model, in which the wave speed is a given function of the wave amplitude. We formulate a viscous approximation of the equation and establish the global existence of smooth solutions for the viscously perturbed equation. For a monotone wave speed function in the equation, we find an invariant region in the phase space in which we discover: (a) smooth data evolve smoothly forever; (b) both the viscous regularization and the smooth solutions obtained through data smoothing for rough initial data yield weak solutions to the Cauchy problem of the nonlinear variational wave equation. The main tool is the Young measure theory and related techniques.",

author = "Ping Zhang and Yuxi Zheng",

note = "Funding Information: The work of Ping Zhang is supported by the Chinese Post-Doctor{\textquoteright}s Foundation and the Morningside Center of Mathematics, and that of Yuxi Zheng is supported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fellows award. We would also like to thank Professors Tong Zhang and Ling Hsiao for their constant help and encouragements, Professors Bob Glassey, Fanghua Lin, John Hunter, and Andy Majda for their interest in the work. This work was done during the first author{\textquoteright}s visit to Indiana University, he would like to express his appreciation of the hospitality of the Department of Mathematics.",

year = "2001",

doi = "10.1081/PDE-100002240",

language = "English (US)",

volume = "26",

pages = "381--419",

journal = "Communications in Partial Differential Equations",

issn = "0360-5302",

publisher = "Taylor and Francis Ltd.",

number = "3-4",

}

TY - JOUR

T1 - Rarefactive solutions to a nonlinear variational wave equation of liquid crystals

AU - Zhang, Ping

AU - Zheng, Yuxi

N1 - Funding Information: The work of Ping Zhang is supported by the Chinese Post-Doctor’s Foundation and the Morningside Center of Mathematics, and that of Yuxi Zheng is supported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fellows award. We would also like to thank Professors Tong Zhang and Ling Hsiao for their constant help and encouragements, Professors Bob Glassey, Fanghua Lin, John Hunter, and Andy Majda for their interest in the work. This work was done during the first author’s visit to Indiana University, he would like to express his appreciation of the hospitality of the Department of Mathematics.

PY - 2001

Y1 - 2001

N2 - We study a nonlinear wave equation derived from a simplified liquid crystal model, in which the wave speed is a given function of the wave amplitude. We formulate a viscous approximation of the equation and establish the global existence of smooth solutions for the viscously perturbed equation. For a monotone wave speed function in the equation, we find an invariant region in the phase space in which we discover: (a) smooth data evolve smoothly forever; (b) both the viscous regularization and the smooth solutions obtained through data smoothing for rough initial data yield weak solutions to the Cauchy problem of the nonlinear variational wave equation. The main tool is the Young measure theory and related techniques.

AB - We study a nonlinear wave equation derived from a simplified liquid crystal model, in which the wave speed is a given function of the wave amplitude. We formulate a viscous approximation of the equation and establish the global existence of smooth solutions for the viscously perturbed equation. For a monotone wave speed function in the equation, we find an invariant region in the phase space in which we discover: (a) smooth data evolve smoothly forever; (b) both the viscous regularization and the smooth solutions obtained through data smoothing for rough initial data yield weak solutions to the Cauchy problem of the nonlinear variational wave equation. The main tool is the Young measure theory and related techniques.

UR - http://www.scopus.com/inward/record.url?scp=0000642792&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000642792&partnerID=8YFLogxK

U2 - 10.1081/PDE-100002240

DO - 10.1081/PDE-100002240

M3 - Article

AN - SCOPUS:0000642792

SN - 0360-5302

VL - 26

SP - 381

EP - 419

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

IS - 3-4

ER -

Rarefactive solutions to a nonlinear variational wave equation of liquid crystals

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this