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Unique continuation and decay for the Korteweg-de Vries equation with localized damping

Ademir Fernando Pazoto (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This work is devoted to prove the exponential decay for the energy of solutions of the Korteweg-de Vries equation in a bounded interval with a localized damping term. Following the method in Menzala (2002) which combines energy estimates, multipliers and compactness arguments the problem is reduced to prove the unique continuation of weak solutions. In Menzala (2002) the case where solutions vanish on a neighborhood of both extremes of the bounded interval where equation holds was solved combining...

Variable depth KdV equations and generalizations to more nonlinear regimes

Samer Israwi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study here the water waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known that, for such regimes, a generalization of the KdV equation (somehow linked to the Camassa-Holm equation) can be derived and justified [Constantin and Lannes, Arch. Ration. Mech. Anal. 192 (2009) 165–186] when the bottom is flat. We generalize here this result with a new class of equations taking into account...

Well-posedness of a class of non-homogeneous boundary value problems of the Korteweg-de Vries equation on a finite domain

Eugene Kramer, Ivonne Rivas, Bing-Yu Zhang (2013)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study a class of Initial-Boundary Value Problems proposed by Colin and Ghidaglia for the Korteweg-de Vries equation posed on a bounded domain (0,L). We show that this class of Initial-Boundary Value Problems is locally well-posed in the classical Sobolev space Hs(0,L) for s > -3/4, which provides a positive answer to one of the open questions of Colin and Ghidaglia [Adv. Differ. Equ. 6 (2001) 1463–1492].

Zero-Dissipation Limit for Nonlinear Waves

Jerry L. Bona, Jiahong Wu (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Evolution equations featuring nonlinearity, dispersion and dissipation are considered here. For classes of such equations that include the Korteweg-de Vries-Burgers equation and the BBM-Burgers equation, the zero dissipation limit is studied. Uniform bounds independent of the dissipation coefficient are derived and zero dissipation limit results with optimal convergence rates are established.

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