Generalized KdV equation subject to a stochastic perturbation

Annie Millet; Svetlana Roudenko

doi:10.3934/dcdsb.2018147

Article Contents

2018, Volume 23, Issue 3: 1177-1198. Doi: 10.3934/dcdsb.2018147

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Generalized KdV equation subject to a stochastic perturbation

Annie Millet^1, , and
Svetlana Roudenko^2,

1.
SAMM, EA 4543, Université Paris 1 Panthéon Sorbonne, 90 Rue de Tolbiac, 75634 Paris Cedex, France and LPSM, Universités Paris 6-Paris 7
2.
The George Washington University, Department of Mathematics, Phillips Hall, 801 H St NW, Washington, DC 20052, USA

* Corresponding author: Annie Millet
* Corresponding author: Annie Millet

Dedication: In the memory of Igor Chueshov

Received: February 28, 2017

Revised: May 31, 2017

Early access: February 2018

Published: June 2018

The second author is supported by NSF CAREER grant # 1151618

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Abstract

We prove global well-posedness of the subcritical generalized Korteweg-de Vries equation (the mKdV and the gKdV with quartic power of nonlinearity) subject to an additive random perturbation. More precisely, we prove that if the driving noise is a cylindrical Wiener process on $L^2(\mathbb{R})$ and the covariance operator is Hilbert-Schmidt in an appropriate Sobolev space, then the solutions with $H^1(\mathbb{R})$ initial data are globally well-posed in $H^1(\mathbb{R})$. This extends results obtained by A. de Bouard and A. Debussche for the stochastic KdV equation.

Keywords:

Generalized Korteweg de Vries (gKdV) equation,
initial value problem,
well-posedness,
stochastic additive noise.

Mathematics Subject Classification: Primary: 60H15, 35R60, 35Q53; Secondary: 35L75, 37K10.

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References

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