TY - JOUR

T1 - Unique Conservative Solutions to a Variational Wave Equation

AU - Bressan, Alberto

AU - Chen, Geng

AU - Zhang, Qingtian

N1 - Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2015/9/17

Y1 - 2015/9/17

N2 - Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $${u_{tt} - c (u) (c(u)u_{x}) x = 0}$$utt-c(u)(c(u)ux)x=0. Given a solution u(t, x), even if the wave speed c(u) is only Hör continuous in the t – x plane, one can still define forward and backward characteristics in a unique way. Using a new set of independent variables X, Y, constant along characteristics, we prove that t, x, u, together with other variables, satisfy a semilinear system with smooth coefficients. From the uniqueness of the solution to this semilinear system, one obtains the uniqueness of conservative solutions to the Cauchy problem for the wave equation with general initial data $${u(0, \cdot) \in H^{1}(I\!R), u_{t} (0, \cdot) \in L^{2}(I\!R).}$$u(0,·)∈H1(IR),ut(0,·)∈L2(IR).

AB - Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $${u_{tt} - c (u) (c(u)u_{x}) x = 0}$$utt-c(u)(c(u)ux)x=0. Given a solution u(t, x), even if the wave speed c(u) is only Hör continuous in the t – x plane, one can still define forward and backward characteristics in a unique way. Using a new set of independent variables X, Y, constant along characteristics, we prove that t, x, u, together with other variables, satisfy a semilinear system with smooth coefficients. From the uniqueness of the solution to this semilinear system, one obtains the uniqueness of conservative solutions to the Cauchy problem for the wave equation with general initial data $${u(0, \cdot) \in H^{1}(I\!R), u_{t} (0, \cdot) \in L^{2}(I\!R).}$$u(0,·)∈H1(IR),ut(0,·)∈L2(IR).

UR - http://www.scopus.com/inward/record.url?scp=84931011144&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84931011144&partnerID=8YFLogxK

U2 - 10.1007/s00205-015-0849-y

DO - 10.1007/s00205-015-0849-y

M3 - Article

AN - SCOPUS:84931011144

VL - 217

SP - 1069

EP - 1101

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 3

ER -