Radially Symmetric Weak Solutions for a Quasilinear Wave Equation in Two Space Dimensions
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João Paulo Dias ,
Hermano Frid
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
