Resonant nonlinear geometric optics for weak solutions of conservation laws (Q1336324)

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 \(\varepsilon\to 0\), \[ \sup_{0\leq t\leq T_0/\varepsilon} |u(t,.,\varepsilon)- [u_0+ \varepsilon V^0(\tau, t,.)|_{\tau= \varepsilon t}]|_{L^1}= o(\varepsilon). \]