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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
##### Keywords:
blow up; nonlinear resonance
Full Text:
##### References:
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