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Resonant nonlinear geometric optics for weak solutions of conservation laws. (English) Zbl 0856.35080
The work in this paper concerns the evolution of solutions to systems of conservation laws of the form \(u_t+ f(u)_x= 0\) subject to periodic initial data specified along the real line. Assuming that \(f\in C^2\) and that the system is stricty hyperbolic, that the initial data \(u(0, x)= u_0+ \varepsilon v_0(x)\) are nearly constant and of bounded variation over a period, and that each field is genuinely nonlinear or linearly degenerate, the author obtains a periodic solution \(u(t, x, \varepsilon)\) existing for at least a time \(T_0/\varepsilon\), satisfying \(|u(t, x, \varepsilon)- u_0|_{L^\infty}\leq c\varepsilon|v_0|_{L^\infty}\) and \(|u(t, x, \varepsilon)|_{BV}\leq c\varepsilon|v_0|_{BV}\), the \(BV\) estimate taken over a period. Moreover, if \(V^0(\tau, t, x)= \sum_k \sigma^k(\tau, x- \lambda_k t) r_k(u_0)\), where the \(\sigma^k\) denote the unique entropy solution of the corresponding resonant geometric optics modulation equations, with \(\lambda_k\) and \(r_k\) denoting the eigenvalues and right eigenvectors of \(\partial f/\partial u\), then as \(\varepsilon\to 0\), \[ \sup_{0\leq t\leq T_0/\varepsilon} |u(t,.,\varepsilon)- [u_0+ \varepsilon V^0(\tau, t,.)|_{\tau= \varepsilon t}]|_{L^1}= o(\varepsilon). \]

MSC:
35L65 Hyperbolic conservation laws
78A05 Geometric optics
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