A nonlinear instability for 3×3 systems of conservation laws

Abstract

The phenomenon of nonlinear resonance provides a mechanism for the unbounded amplification of small solutions of systems of conservation laws. We construct spatially 2π-periodic solutionsu ^N ∈C ^∞ ([0,t _N ] × ∝ witht _N bounded, satisfying

$$\begin{array}{*{20}c} {\left\| {u^N } \right\|L^\infty ([0,t_N ] \times \mathbb{R}) \to 0,} & {\int\limits_0^{2\pi } {\left| {\partial _x u^N (0,x)} \right|dx \leqq C,} } \\ {\int\limits_0^{2\pi } {\left| {\partial _x u^N (t_N ,x)} \right|dx \geqq N, \left\| {u^N (t_N ,x)} \right\|L^p (\mathbb{R}) \geqq N\left\| {u^N (0,x)} \right\|L^p (\mathbb{R})} } & {1 \leqq p \leqq \infty .} \\ \end{array} $$

The variation grows arbitrarily large, and the sup norm is amplified by arbitrarily large factors.