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A nonlinear instability for \(3\times 3\) systems of conservation laws. (English) Zbl 0820.35093
Summary: The phenomenon of nonlinear resonance provides a mechanism for the unbounded amplification of small solutions of systems of conservation laws. We construct spatially \(2 \pi\)-periodic solutions \(u^ N \in C^ \infty ([0, t_ N] \times \mathbb{R})\) with \(t_ N\) bounded, satisfying \(\| u^ N \|_{L^ \infty ([0,t_ N] \times \mathbb{R})} \to 0\), \(\int^{2 \pi}_ 0 | \partial_ x u^ N(0,x) | dx \leq C\), \(\int^{2 \pi}_ 0 | \partial_ x u^ N(t_ N,x) | dx \geq N\), \(\| u^ N (t_ N,x) \|_{L^ p (\mathbb{R})} \geq N \| u^ N (0,x) \|_{L^ p (\mathbb{R})}\), \(1 \leq p \leq \infty\). The variation grows arbitrarily large, and the sup norm is amplified by arbitrarily large factors.

MSC:
35L65 Hyperbolic conservation laws
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