### 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 | |

State | Published - Jan 1 2001 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

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*Communications in Partial Differential Equations*, vol. 26, no. 3-4, pp. 381-419. https://doi.org/10.1081/PDE-100002240

**Rarefactive solutions to a nonlinear variational wave equation of liquid crystals.** / Zhang, Ping; Zheng, Yuxi.

Research output: Contribution to journal › Article

TY - JOUR

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

AU - Zhang, Ping

AU - Zheng, Yuxi

PY - 2001/1/1

Y1 - 2001/1/1

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

VL - 26

SP - 381

EP - 419

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 3-4

ER -