A KdV-type Boussinesq system: From the energy level to analytic spaces

Jerry L. Bona; Zoran Grujić; Henrik Kalisch

doi:10.3934/dcds.2010.26.1121

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2010, Volume 26, Issue 4: 1121-1139. Doi: 10.3934/dcds.2010.26.1121

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A KdV-type Boussinesq system: From the energy level to analytic spaces

1.
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607
2.
Department of Mathematics, University of Virginia, Charlottesville, VA 22904
3.
Department of Mathematics, University of Bergen, 5008 Bergen

Received: February 28, 2009

Revised: April 30, 2009

Published: November 2010

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Abstract

Considered here is the well-posedness of a KdV-type Boussinesq system modeling two-way propagation of small-amplitude long waves on the surface of an ideal fluid when the motion is sensibly two dimensional. Solutions are obtained in a range of Sobolev-type spaces, from the energy level to the analytic Gevrey spaces. In addition, a criterion for detecting the possibility of blow-up in finite time in terms of loss of analyticity is derived.

Keywords:

Boussinesq systems,
two-way propagation of water waves,
well-posedness for nonlinear dispersive equations,
analyticity,
Gevrey-space analysis.

Mathematics Subject Classification: Primary: 35Q35, 35Q53, 35A07; Secondary: 76B07, 76B15.

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