Unique Conservative Solutions to a Variational Wave Equation

By Alberto Bressan, Geng Chen, Qingtian Zhang. 2014

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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).

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a0458a55687d2e48520c84cb7bb5b64237e2cce2

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Generic regularity of conservative solutions to a nonlinear wave equation (2015, background): “The analysis in [7, 3] shows that, for any initial data”