arXiv subjects

Mikhael Balabane

Publications and source records attributed to Mikhael Balabane.

3 recordsLinked to original sources

arXiv2021

On the Burgers dynamical system with an external force and its Koopman decomposition

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.

math.AP↗

arXiv2021

On Koopman Operator for Burgers' Equation

We consider the flow of Burgers' equation on an open set of (small) functions in $L^2([0,1])$. 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 `completeness' property 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 & Kerswell, 2018), where only one term per eigenvalue was given. A numeric illustration of the Koopman decomposition is given and the Koopman eigenvalues compared to the eigenvalues of a Dynamic Mode Decomposition (DMD).

math.DS↗

arXiv2009

Functional Analysis for Helmholtz Equation in the Framework of Domain Decomposition

This paper gives a geometric description of functional spaces related to Domain Decomposition techniques for computing solutions of Laplace and Helmholtz equations. Understanding the geometric structure of these spaces leads to algorithms for solving the equations. It leads also to a new interpretation of classical algorithms, enhancing convergence. The algorithms are given and convergence is proved. Numerical tests are given. This is done by building tools enabling geometric interpretations of the operators related to Domain Decomposition technique. The Despres operators, expressing conservation of energy for Helmholtz equation, are defined on the fictitious boundary and their spectral properties proved.It turns to be the key for proving convergence of the given algorithm for Helmholtz equation in a non-dissipating cavity. Using these tools, one can prove that the Domain Decomposition setting for the Helmholtz equation leads to an ill-posed problem. Nevertheless, one can prove that if a solution exists, it is unique. And that the algorithm do converge to the solution.

math.AP↗