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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...

Variance decay for functionals of the environment viewed by the particle

Jean-Christophe Mourrat (2011)

Annales de l'I.H.P. Probabilités et statistiques

For the random walk among random conductances, we prove that the environment viewed by the particle converges to equilibrium polynomially fast in the variance sense, our main hypothesis being that the conductances are bounded away from zero. The basis of our method is the establishment of a Nash inequality, followed either by a comparison with the simple random walk or by a more direct analysis based on a martingale decomposition. As an example of application, we show that under certain conditions,...

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