Koopman operator for Burgers's equation

Abstract

We consider the flow of Burgers's equation on an open set of (small) functions in L2([0,1]). We derive explicitly the Koopman decomposition of the Burgers flow. We identify the frequencies and the coefficients of this decomposition as eigenvalues and eigenfunctionals of the Koopman operator. We prove the convergence of the Koopman decomposition for t>0 for small Cauchy data and up to t=0 for regular Cauchy data. The convergence up to t=0 leads to a completenessproperty for the basis of Koopman modes. We construct all modes and eigenfunctionals, including the eigenspaces involved in geometric multiplicity. This goes beyond the summation formulas provided by Page and Kerswell [Phys. Rev. Fluids 3, 071901(R) (2018)2469-990X10.1103/PhysRevFluids.3.071901], where only one term per eigenvalue was given. A numeric illustration of the Koopman decomposition is given and the Koopman eigenvalues compares to the eigenvalues of a dynamic mode decomposition. © 2021 American Physical Society.