Extreme Behavior in the 2D Kuramoto-Sivashinsky Equation on a Periodic Domain (with Bartosz Protas)

Complex spatiotemporal dynamics arise in nonlinear PDE models across a wide range of scientific contexts. In fluid mechanics, several such phenomena are modeled by the two-dimensional Kuramoto-Sivashinsky (2D KS) equation. Despite nearly five decades of study, fundamental questions regarding basic mathematical properties of this system remain unresolved. In particular, the question of global existence of solutions in standard spaces such as L² is still open. Thus, quantifying the maximal transient energy amplification attainable by the system, as well as the subsequent evolution of these highly amplified states, is central to understanding the long-time behavior of solutions. We probe these questions by formulating a family of variational PDE optimization problems where the L² energy at some prescribed time T is maximized with respect to the initial condition subject to suitable constraints. These problems are solved numerically with a state-of-the-art adjoint-based Riemannian gradient method. While no evidence for unbounded growth of this norm that would signal singularity formation is found, by systematically examining how the maximal energy scales with domain size, initial energy, and time window length, we aim to formulate conjectures about bounds on the energy growth and the structure of extreme transient states in solutions to the 2D KS system.

• github.com/zigicjovan/2DKS_Solver contains code to reproduce the results described in this report.

[Link to slides]

My research interests lie broadly in nonlinear partial differential equations (PDEs), with an emphasis on pattern formation and fluid dynamics. My work combines computational and theoretical approaches to understand the evolution, optimization, and long-time behavior of PDE models.  

1. My recent research has focused on the Kuramoto–Sivashinsky and Burgers systems. During my master’s research, I investigated sensitivity and numerical stability in optimization-based PDE frameworks, where I observed limitations associated with discretize-then-optimize methodologies near dynamically sensitive regimes. This motivated my doctoral research, in which I developed an adjoint-based optimization framework for maximizing finite-time energy growth in higher-dimensional nonlinear PDE systems.

2. Within optimization, I am interested in continuation methods, regularity, and extreme dynamical behavior. My recent work has been in understanding theoretical bounds, transient growth mechanisms, and sensitivity to perturbations on problems restricted to Riemannian manifolds (i.e. Riemannian optimization). More broadly, I am interested in developing methods for improving predictive accuracy in nonlinear PDE models by exploiting geometric, spectral, or topological structure in the underlying dynamics. 

3. Within dynamical systems theory, my interests include bifurcation analysis, coherent structures, and attractor characterization in both periodic and chaotic regimes. My work has examined how variations in spatial domain parameters can induce discrete symmetry and pattern formation in dynamical behavior, particularly in the 2D Kuramoto–Sivashinsky equation. Moving forward, I am interested in developing structure-informed diagnostics and reduced representations for complex attractors arising in nonlinear PDE models.