On the stochastic Kuramoto-Sivashinsky equation (Q5934266)

The authors study the stochastic Kuramoto-Sivashinsky equation NEWLINE\[NEWLINE du+(u_{xxxx}+u_{xx}+uu_{x})dt-dw=0,NEWLINE\]NEWLINE where \(w\) is a Wiener process with values in a Hilbert space. Here \(u\) is restricted to a symmetric interval and is subject to the given initial condition \(u_{0}\) and homogeneous Dirichlet boundary conditions. Their approach is similar to that used to establish existence and uniqueness for parabolic differential equations. The authors use a change of variables which enables to consider a related deterministic equation instead of the stochastic equation. Local existence and uniqueness are established using a fixed point argument. Then global existence is proved by showing that the local solution remains bounded.