# Preprint Série A 260/2003

Radially Symmetric Weak Solutions for a Quasilinear Wave Equation in Two Space Dimensions

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Keywords:
quasilinear hyperbolic systems, conservation laws, compensated compactness, entropies, Young measures

Abstract:
We prove the convergence of the radially symmetric solutions to the Cauchy problem for the viscoelasticity equations $$\phi_{tt}-\Delta\phi-\div(\frac13|\nabla\phi|^2\nabla\phi)=\ve\Delta\phi_t,$$ as $\ve\to0$, with radially symmetric initial data $\phi^\ve(x,0)=\phi_0^\ve(r)$, $\phi_t^\ve(x,0)=\phi_1^\ve(r)$, $r=(x_1^2+x_2^2)^{1/2}$, where $\phi_{0r}^\ve\wto{\phi}_{0r}$, $\phi_1^\ve\wto{\phi}_1$, to a weak solution of the Cauchy problem for the corresponding limit equation with $\ve=0$, and initial data $\phi(x,0)=\phi_0(r)$, $\phi_t(x,0)=\phi_1(r)$. Our analysis is based on energy estimates and the method of compensated compactness closely following D.~Serre and J.~Shearer (1993).

MSC 2000:
35L35    Boundary value problems for higher-order, hyperbolic equations
35L70    Nonlinear second-order PDE of hyperbolic type
35L45    Initial value problems for hyperbolic systems of first-order PDE
35L65    Conservation laws