2021/10/21 by Mikhaël Balabane, Mikhael Balabane, Balabane, Mikhael
Decision Sciences · Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics and Turbulent Flows #Lattice Boltzmann Simulation Studies #Mathematical Physics (math-ph) #Model Reduction and Neural Networks #Probabilistic and Robust Engineering Design #math-ph #math.AP #math.MP
paper · pdf · doi:10.48550/arxiv.2110.11119
arxiv created 2021/10/21 · openalex publication_date 2021/10/21 · arxiv updated 2021/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that the Burgers flow with a steady external forcing has a unique steady state which is a sink. Although this flow cannot be linearized through Cole-Hopf transforms, we prove that it has a convergent Koopman Modes decomposition. This gives an asymptotic formula for solutions of the Burgers equation with an external force. Time dependence and the coefficients of the decomposition are proved to be eigenvalues and eigenfunctionals of the Koopman operator. Convergence of the Koopman decomposition is proved for orbits close to the sink. The analysis of Burgers dynamical system relies on the properties of a nonlinear heat flow, that shows invariant sets with complete orbits, and invariant sets where blow-up in finite time do ocurr. This behaviour helps understanding some instabilities in numeric computing for fluids.