Global-in-time existence for the two-dimensional Kuramoto–Sivashinsky equation on the torus

Prove global-in-time well-posedness for the two-dimensional Kuramoto–Sivashinsky equation ∂_t φ + (Δ^2 + Δ)φ + |∇φ|^2 = 0 on the periodic torus for general initial data, without restricting the domain geometry or the number of unstable modes.

Background

The two-dimensional Kuramoto–Sivashinsky equation exhibits long-wave instability coupled with higher-order dissipation, leading to complex spatio-temporal dynamics. Standard maximum principle methods are unavailable due to the fourth-order operator, and energy estimates do not readily close to provide global bounds.

Existing global results typically require strong structural restrictions (e.g., thin or anisotropic domains, or special advecting flows). Establishing global existence in the general 2D periodic setting remains a major unresolved problem.