Long-time behavior of the one-dimensional Kuramoto–Sivashinsky equation and its dependence on domain size

Characterize the long-time behavior of solutions to the one-dimensional Kuramoto–Sivashinsky equation on periodic domains and ascertain how this behavior depends on the size of the periodic box.

Background

Despite extensive progress on the one-dimensional Kuramoto–Sivashinsky equation, several aspects of its asymptotic dynamics remain unclear, particularly how invariant objects or statistical properties scale with the domain length.

Understanding this dependence is important for connecting rigorous analysis with observed spatio-temporal patterns and for clarifying scaling laws and attractor properties in the 1D setting.

References

While the analysis of the 1D KSE is well-developed, due to the special structure of the non-linearity, which is an exact derivative, although important questions remain open on the characteristic long-time behavior of solutions, especially with respect to the dependence on the size of the periodic box, much less is known in the two-dimensional case.

— Enhanced dissipation by advection and applications to PDEs (2501.17695 - Mazzucato et al., 29 Jan 2025) in Subsection 'Shear flows: applications to the Kuramoto-Sivanshinsky equation' in Section 'Applications to non-linear systems'