Optimal L² Energy Growth in the 2D Kuramoto-Sivashinsky Equation (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 energetic bounds in the 2D KS dynamics remain unresolved. In particular, 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 potential for finite-time singularity formation. Using adjoint-based optimization and high-resolution spectral methods, we compute optimal initial conditions on periodic domains that maximize the L² energy at a prescribed terminal time. By systematically examining how the maximal energy scales with domain size, initial energy magnitude, and time window length, we aim to provide new insight on the limits of energy growth and the structure of extreme transient states in the 2D Kuramoto-Sivashinsky system.

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

[Link to slides]

My research interests generally pertain to problems involving nonlinear PDE models, and more specifically to pattern formation and fluid mechanics. Although my master’s thesis research was more focused on computational techniques, my current motivation is to also address theoretical aspects of nonlinear PDE models.  

1. Within the optimization field, I am interested in continuation methods and regularity of nonlinear PDE models. During my master’s research, I tested the sensitivity of my models and observed limitations to the discretize-then-optimize approach I was using. During my PhD research, I designed an adjoint-based optimization procedure to maximize the finite-time energy growth of a nonlinear PDE model. I am interested in researching methods that provide future-state accuracy (i.e. "learning") of PDE models by leveraging geometric or topological features of such models. Moreover, I am interested in researching methods that investigate the theoretical bounds of PDE models under different conditional inputs. 

2. Within the dynamical systems field, I am interested in center manifold theory and bifurcation analysis. During my master’s research, I observed that the time-steps in my models produced the greatest approximation errors near points of phase transition, so I chose to handle states on either side of these transitive points separately in order to improve my results. I also examined sensitivity in domain parameters leading to vastly different system behavior. During my PhD research, I worked on the higher-dimensional form of the nonlinear PDE model from my master's research and studied the associated dynamical characteristics. I am interested in researching methods that leverage information from coherent structures in PDE models to design metrics for attractors of both periodic and chaotic systems.