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On the solutions of a model equation for shallow water waves of moderate amplitude. (English) Zbl 1288.35197

This paper studies the initial value problem for the equation \[ \eta_{t}+\eta_{x}+\frac{3}{2}\varepsilon\eta\eta_{x}-\frac{3}{8}\varepsilon^{2}\eta^{2}\eta_{x} +\frac{3}{16}\varepsilon^{3}\eta^{3}\eta_{x}+\mu(\alpha\eta_{xxx}+\beta\eta_{xxt}) =\varepsilon\mu(\gamma\eta\eta_{xxx}+\delta\eta_{x}\eta_{xx}). \] This equation was introduced by Constantin and Lannes in a study of the Camassa-Holm and Degasperis-Procesi equations [A. Constantin and D. Lannes, Arch. Ration. Mech. Anal. 192, No. 1, 165–186 (2009; Zbl 1169.76010)]. In the present work, the authors show that the inital value problem is well posed, locally in time, with initial data in Besov spaces. Furthermore, when the initial data is analytic, it is shown that the solution is analytic in space and time.

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs

Citations:

Zbl 1169.76010
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References:

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