Henry P. McKean Jr. Selecta pp 189-193 | Cite as

# Breakdown of the Camassa-Holm Equation

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## Abstract

The Camassa-Holm equation is written in its Eulerian form as 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

$$\displaystyle\begin{array}{rcl} \text{CH:}\quad \frac{\partial v} {\partial t} + v\frac{\partial v} {\partial x} + \frac{\partial p} {\partial x} = 0,& & {}\\ \end{array}$$

*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

- [1]Bertozzi, A; Majda, A. Vorticity and incompressible flow. Cambridge University Press, Cambridge, 2002.zbMATHGoogle Scholar
- [2]Camassa, R.; Holm, D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71 (1993), 1661–1164.MathSciNetCrossRefzbMATHGoogle Scholar
- [3]Camassa, R.; Holm, D; Hyman, M. A new integrable shallow water equation. Adv. Appl. Math. 31 (1994), 1–33.Google Scholar
- [4]McKean, H. P. Breakdown of a shallow water equation. Asian J. Math. 2 (1998), 767–774.Google Scholar
- [5]McKean, H. P. Correction to ‘Breakdown of a shallow water equation’. Asian J. Math. 3 (1999).Google Scholar
- [6]McKean, H. P. Fredholm determinants and the Camassa-Holm hierarchy. Comm. Pure Appl. Math. 61 (2003), 638–680.Google Scholar

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