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Breakdown of the Camassa-Holm Equation

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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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Courant InstituteNew YorkUSA

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