[Paper Review] Determining form and data assimilation algorithm for weakly damped and driven Korteweg-de Vries equaton- Fourier modes case

Michael S. Jolly, Tural Sadigov+1 more|arXiv (Cornell University)|Oct 9, 2015

Fluid Dynamics and Turbulent Flows11 references3 citations

TL;DR

This paper establishes that the global attractor of the weakly damped and driven Korteweg-de Vries (KdV) equation is embedded in the long-term dynamics of a determining form—an ordinary differential equation whose steady states correspond exactly to trajectories in the attractor. It further introduces a data assimilation algorithm that recovers full reference solutions from low Fourier modes at an exponential convergence rate.

ABSTRACT

We show that the global attractor of a weakly damped and driven Korteweg-de Vries equation (KdV) is embedded in the long-time dynamics of an ordinary differential equation called a determining form. In particular, there is a one-to-one identification of the trajectories in the global attractor of the damped and driven KdV and the steady state solutions of the determining form. Moreover, we analyze a data assimilation algorithm (down-scaling) for the weakly damped and driven KdV. We show that given a certain number of low Fourier modes of a reference solution of the KdV equation, the algorithm recovers the full reference solution at an exponential rate in time.

Motivation & Objective

• To establish a determining form for the weakly damped and driven KdV equation in the $ H^2 $ space, showing that its global attractor is embedded in the long-time dynamics of an ODE.
• To analyze a data assimilation (downscaling) algorithm that reconstructs full solutions from low Fourier mode observations.
• To demonstrate exponential convergence of the data assimilation algorithm to the true solution using feedback control based on projected modes.
• To extend techniques from determining modes and compound functionals to dispersive, weakly dissipative PDEs like KdV, where inertial manifolds do not exist.
• To provide explicit bounds on the number of determining Fourier modes $ N $ in terms of damping $ \gamma $ and forcing $ f $.

Proposed method

• Construct a determining form as an ODE in the space of trajectories, where steady states correspond to global attractor trajectories of the KdV equation.
• Use a feedback control mechanism in the data assimilation algorithm that projects the solution onto the first $ N $ Fourier modes and drives the model solution toward the reference solution.
• Apply compound functionals inspired by Hamiltonian structure to derive energy estimates and control nonlinear terms in the KdV equation.
• Establish uniform bounds on solutions of the Galerkin approximation via $ L^2 $ and $ H^3 $ energy estimates, using interpolation and Sobolev embedding.
• Prove existence of a solution to the projected (Galerkin) system via Brouwer’s fixed-point theorem on a closed ball in $ H_n $.
• Use weak and strong convergence of Galerkin approximations to show that the limit solution satisfies the KdV equation in $ H^5 $, hence classically.

Experimental results

Research questions

• RQ1 Can the global attractor of the weakly damped and driven KdV equation be embedded in the long-term dynamics of an ordinary differential equation (determining form) despite the absence of strong dissipation?
• RQ2 What is the minimal number $ N $ of Fourier modes required to determine the long-term dynamics of the KdV equation, and how does $ N $ depend on $ \gamma $ and $ f $?
• RQ3 Can a data assimilation algorithm that uses only low Fourier modes of a reference solution synchronize exponentially fast to the true solution?
• RQ4 How do compound functionals and feedback control terms enable the analysis of determining forms and data assimilation in dispersive PDEs like KdV?
• RQ5 Is it possible to establish $ H^5 $ regularity and classical solution behavior for the limiting solution of the Galerkin approximation?

Key findings

• The global attractor of the weakly damped and driven KdV equation is in one-to-one correspondence with the steady states of a determining form, which is an ODE in the space of trajectories.
• The number of determining Fourier modes $ N $ is estimated in terms of the damping parameter $ \gamma $ and the forcing term $ f $, with explicit bounds derived via energy estimates.
• The data assimilation algorithm recovers the full reference solution from $ N $ low Fourier modes at an exponential rate in time, with convergence independent of initial error.
• The solution of the Galerkin approximation system satisfies uniform bounds in $ L^2 $ and $ H^3 $, with $ |u_n| \leq R_0 $ and $ |(u_n)_{xxx}| \leq R_3 $, uniformly in $ n $.
• The limit solution $ u^* $ of the Galerkin sequence belongs to $ H^5 $, and the KdV equation holds in the classical sense, ensuring regularity and strong convergence.
• The feedback control term in the data assimilation algorithm ensures synchronization with the reference solution, with the proof structure analogous to extending the $ W $-map to forward time.