Dynamics for the stochastic nonlocal Kuramoto-Sivashinsky equation
The paper deals with \(du+\bigl(u_{xxxx}+u_{xx}+uu_x+\alpha H(u_{xxx})\bigr)\, dt =\sigma\, d W(t)\), \(u\) periodic on \((-L,L)\), \(L>0\). \(H\) denotes the Hilbert transform \(\psi\mapsto {1\over{2L}}\int_{-L}^L \cot{{\pi(x-y)}\over{2L}}\psi(y)\, dy\), \(\alpha\in (0,\infty)\), and \(W\) belongs to a special class of infinite-dimensional Wiener processes related to the eigenvectors of the periodic negative Laplacian on \((-L,L)\). The author establishes existence of a finite-dimensional random attractor. The work relies on a result by Debussche on the Hausdorff dimension of random attractors and a gauge function technique. Difficulties due to the presence of the Hilbert transform are circumvented by using Galerkin's method in proving existence of a weak solution.
scientific article; zbMATH DE number 5141600
scientific article