On the Nikolaevskiy equation and the fractal dimension of its attractor

We investigate the attractor of the Nikolaevskiy equation, a sixth-order partial differential equation (PDE) containing a small parameter whose solutions exhibit spatiotemporal chaos with strong scale separation. We first prove well-posedness and regularity of the solutions, and derive asymptotic bounds on their derivatives, to put the subsequent results on a firm footing. The rest of the work focuses on showing that the dynamical system associated with the Nikolaevskiy equation possesses an attractor with a finite fractal dimension. Bounds on this dimension are both derived analytically and computed numerically, paying particular attention to their scaling with the parameters. We describe the numerical methods, and present computational results that include the scaling of various norms of the solutions, as well as of the power spectrum and the spectrum of Lyapunov exponents of the PDE.