On the stochastic Korteweg-de Vries equation
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 (*).
stochastic partial differential equation
scientific article; zbMATH DE number 1213985
scientific article