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Zbl 0918.35005
Constantin, Adrian; Escher, Joachim
Global existence and blow-up for a shallow water equation. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 26, No.2, 303-328 (1998). ISSN 0391-173X

An interesting phenomenon in water channels is the appearance of waves with length much greater than the depth of the water. Recently, R. Camassa and D. Holm proposed a new model for the same phenomenon: $$\cases u_t- u_{xxt}+ 3uu_x= 2u_x u_{xx}+ uu_{xxx},\qquad t>0,\quad & x\in\bbfR,\\ u(0, x)= u_0(x),\quad & x\in\bbfR.\endcases\tag 1$$ The variable $u(t,x)$ in (1) represents the fluid velocity at time $t$ in the $x$ direction in appropriate nondimensional units (or, equivalently, the height of the water's free surface above a flat bottom).\par The aim of this paper is to prove local well-posedness of strong solutions to (1) for a large class of initial data, and to analyze global existence and blow-up phenomena. In addition, we introduce the notion of weak solutions to (1) suitable for soliton interaction.

MSC 2000:: *35A05 General existence and uniqueness theorems (PDE)
35Q35 Other equations arising in fluid mechanics
35B40 Asymptotic behavior of solutions of PDE
35Q53 KdV-like equations

Keywords: soliton interaction

Cited in: Zbl 1063.35137

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