Unique Conservative Solutions to a Variational Wave Equation

Abstract

Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $u_{tt}-c(u)(c(u)u_x{)}_x=0$. Given a solution $u(t,x)$, even if the wave speed $c(u)$ is only Hölder 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 $u(0,\cdot )\in H^1(\mathbb{R})$, $u_t(0,\cdot )\in L^2(\mathbb{R})$.

References in corpus (1)

• Global Conservative Solutions to a Nonlinear Variational Wave Equation

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• On the degenerate Cauchy problem for a nonlinear variational wave system, Part I: The same wave speed case