Smooth solutions and singularity formation for the inhomogeneous nonlinear wave equation

被引：13

作者：

Chen, Geng ^{[2
]}

Young, Robin ^{[1
]}

机构：

[1] Univ Massachusetts, Dept Math & Stat, Amherst, MA 01003 USA

[2] Penn State Univ, Dept Math, University Pk, PA 16802 USA

来源：

JOURNAL OF DIFFERENTIAL EQUATIONS | 2012年 / 252卷 / 03期

基金：

美国国家科学基金会;

关键词：

Wave equation; Conservation laws; Shock formation; Nonlinear elasticity; Compressible Euler equations; PARTIAL-DIFFERENTIAL-EQUATIONS; COMPRESSIBLE FLUIDS;

D O I：

10.1016/j.jde.2011.09.004

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We study the nonlinear inhomogeneous wave equation in one space dimension: nu(tt) - T (nu, x)(xx) = 0. By constructing some "de coupled" Riccati type equations for smooth solutions, we provide a singularity formation result without restrictions on the total variation of the data, which generalizes earlier singularity results of Lax and the first author. We apply these results to compressible Euler flows with a general pressure law and elasticity in an inhomogeneous medium. (C) 2011 Elsevier Inc. All rights reserved.

引用

页码：2580 / 2595

页数：16

共 14 条

[1]

CHEN G, J HYPERBOLI IN PRESS

[2]

CHEN G, 2011, SHOCK FORMATIO UNPUB

[3]

JOHN F, 1984, LECT NOTES PHYS, V195, P194

[4] FORMATION OF SINGULARITIES IN ONE-DIMENSIONAL NONLINEAR-WAVE PROPAGATION [J].

JOHN, F .

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1974, 27 (03) :377-405

[5] DEVELOPMENT OF SINGULARITIES OF SOLUTIONS OF NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS [J].

LAX, PD .

JOURNAL OF MATHEMATICAL PHYSICS, 1964, 5 (05) :611-&

[6]

LAX RD, 1973, CBMS REG C SER MATH, V11

[7] DEVELOPMENT OF SINGULARITIES IN THE NON-LINEAR WAVES FOR QUASILINEAR HYPERBOLIC PARTIAL-DIFFERENTIAL EQUATIONS [J].

LIU, TP .

JOURNAL OF DIFFERENTIAL EQUATIONS, 1979, 33 (01) :92-111

[8] FORMATION OF SINGULARITIES IN COMPRESSIBLE FLUIDS IN 2-SPACE DIMENSIONS [J].

RAMMAHA, MA .

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 1989, 107 (03) :705-714

[9] FORMATION OF SINGULARITIES IN 3-DIMENSIONAL COMPRESSIBLE FLUIDS [J].

SIDERIS, TC .

COMMUNICATIONS IN MATHEMATICAL PHYSICS, 1985, 101 (04) :475-485

[10]

Temple B, 2010, METHODS APPL ANAL, V17, P225

← 1 2 →