# Breakdown of the Camassa-Holm Equation

Chapter
Part of the Contemporary Mathematicians book series (CM)

## Abstract

The Camassa-Holm equation is written in its Eulerian form as
$$\displaystyle\begin{array}{rcl} \text{CH:}\quad \frac{\partial v} {\partial t} + v\frac{\partial v} {\partial x} + \frac{\partial p} {\partial x} = 0,& & {}\\ \end{array}$$
with “pressure” $$p = G[v^{2} + \frac{1} {2}(v')^{2}]$$. Here $$G = (1 - D^{2})^{-1} = \frac{1} {2}e^{-\vert x-y\vert }$$. It is easy to see that if, at time t = 0, v is odd with v(x) > 0 for x < 0 and v′(0) < 0, then the slope s(t) = v′(t, 0) satisfies $$s^{\bullet } < -\frac{1} {2}s^{2}$$ and so is driven down to $$-\infty$$ at some time $$T \leq -2/s(0)$$: in short, the flow may “break down.” This is the “steepening lemma” of Camassa and Holm [2]. Nothing worse happens: v(t, x) itself cannot jump like the usual kind of shock. Actually, it is not the obvious v but rather the auxiliary function $$m = v - v''$$ that controls this; m is a caricature of the vorticity that controls the behavior of honest Euler in dimensions 2 and 3; see Bertozzi and Majda [1] for a full account.

## Keywords

Initial Data Finite Number Simple Root Simple Proof Auxiliary Function
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## References

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Bertozzi, A; Majda, A. Vorticity and incompressible flow. Cambridge University Press, Cambridge, 2002.
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Camassa, R.; Holm, D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71 (1993), 1661–1164.
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