On the stochastic Korteweg-de Vries equation
type: Item
instance of: scholarly article
title: On the stochastic Korteweg-de Vries equation
zbMATH Open document ID: 0912.60074
published in: Journal of Functional Analysis
publication date: 1999-05-18T00:00:00Z
review text: The authors study the following stochastic partial differential equation \[ {\partial u\over \partial t}+ {\partial^3 u\over\partial x^3} +u{\partial u\over \partial x} =f+ \Phi(u) {\partial^2 B\over \partial t\partial x}, \tag{*} \] where \(u\) is a random process defined on \((x,t)\in \mathbb{R}\times \mathbb{R}^+\), \(f\) is a deterministic forcing term, \(\Phi(u)\) is a linear operator depending on \(u\) and \(B\) is a two-parameter Brownian motion on \(\mathbb{R} \times \mathbb{R}^+\). The authors prove the existence and uniqueness of solutions in \(H^1(\mathbb{R})\) in the case of additive noise and the existence of martingale solutions in \(L^2 (\mathbb{R})\) in the case of multiplicative noise for stochastic Korteweg-de Vries equation of the form (*).
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 1213985
zbMATH Keywords: stochastic partial differential equation
zbMATH Keywords: existence and uniqueness
zbMATH Keywords: multiplicative noise
zbMATH Keywords: Korteweg-de Vries equation
author: A. de Bouard
author: A. Debussche
reviewed by: Stanisław Wedrychowicz
MaRDI profile type: Publication
full work available at URL: jfan.1997.3184
OpenAlex ID: W2045776586
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DOI: 10.1006/JFAN.1997.3184
description: scientific article; zbMATH DE number 1213985