Periodic Reporting for period 1 - HamDyWWa (Hamiltonian Dynamics, Normal Forms and Water Waves)
Reporting period: 2022-03-01 to 2024-08-31
The main general purpose of the ERC project HamDyWWa is to understand qualitative properties of fluids such as the formation of invariant and coherent structures, formations of periodic and multi-periodic waves, formation of vortices, stability and instability phenomena. Tipically, physical phenomena in Fluid Mechanics are described by nonlinear Partial Differential Equations (PDEs) on spatial domains in higher space dimension and with very degenerate nonlinearities (typically they contain derivatives in the nonlinearity). This research project tries to develop general methods to understand asymptotic behavior of solutions and analyze stability and instability phenomena arising in Fluid Mechanics.
After the first 24 months of the project, the main achievements have been the following. We developed methods in order to analyze long time stability of nonlinear asymmetric dispersive waves. We investigate the problem of the approximation of a weakly viscous fluid with an inviscid fluid and we constructed multi-periodic nonlinear waves for which this approximation holds for all times. We investigated the transition form shear flows to turbulent flows for 2D fluids in a channel and we showed that close to shear flows there exist family of multi-periodic traveling waves, in low regularity. We started to develop novel methods in order to prove existence of periodic and multi-periodic waves of arbitrarily large amplitude in Fluid Mechanics. We studied this problem in Magneto-Hydro-Dynamics.
We shall describe in this section four results that we obtained that we think are beyond the state of the art.
1) We prove a long time existence result for a large class of semi-linear Schrödinger equations on irrational tori, namely we showed that for small data, the solutions remains small for very large time. In this problem the resonances are very bad. In order to overcome this problem, we actually connected number theoretical arguments (like the celebrated Bourgain Lemma) with Birkhoff normal form techniques. This is new in the field of dispersive PDEs and I think that it is the first case in which Birkhoff normal form with very bad small denominators is implemented for PDEs in higher space dimension. This paper opens the perspectives to analyze PDEs on more general manifolds.
2) We analyzed a very important and classical problem in Mathematical Fluid Mechanics, namely the rigorous justification of the fact that the Euler equations approximate the Navier-Stokes equations for small values of the viscosity. This is a very hard problem, since it is a singular limit problem, namely there is a small parameter in front of the highest order derivative. Indeed all the previous results justified the approximation only locally in time. We developed a novel method that allowed to justify the approximation, in the class of quasi-periodic solutions, globally in time, with a rate of convergence which is uniform w.r. to time and the viscosity parameter. The proof is based on a novel normal form method which allowed to perform uniform
estimates w.r. to the viscosity. Moreover the paper is also the first KAM result (construction of global in time quasi-periodic solutions) in the framework of singular limit problems.
3) We constructed quasi-periodic traveling waves bifurcating from the Couette flow, which is the simplest example of shear flow (namely any fluid particle stays on the same layer for
all times). The dynamics in the vicinity of shear flows is very important for applications to turbulence and for the problem of transition from laminar flows to turbulent flows. Indeed the experiments show that there is this transition. On the other hand it was proved that for sufficiently regular initial data close to the Couette flow, this phenomenon does not occur since, for large time the fluid flow is close to a shear flow. In this project, we understood that this is a problem of regularity and that in low regularity, one does not have damping phenomena and one has oscillations for all time (quasi-periodic traveling waves).
4) We constructed bi-periodic traveling waves of arbitrarily large amplitude for the non-resistive Magneto-Hydro-Dynamics (MHD) system in dimension 2). This is the first existence result for large amplitude solutions for the MHD equation and it is also the first existence result of large amplitude quasi-periodic solutions in PDEs in dimension greater or equal than two. The novel approach that we propose is based on the usage of micro-local analysis and spectral theory for dealing with linear operators with small divisors and large amplitude perturbations.
1) We prove a long time existence result for a large class of semi-linear Schrödinger equations on irrational tori, namely we showed that for small data, the solutions remains small for very large time. In this problem the resonances are very bad. In order to overcome this problem, we actually connected number theoretical arguments (like the celebrated Bourgain Lemma) with Birkhoff normal form techniques. This is new in the field of dispersive PDEs and I think that it is the first case in which Birkhoff normal form with very bad small denominators is implemented for PDEs in higher space dimension. This paper opens the perspectives to analyze PDEs on more general manifolds.
2) We analyzed a very important and classical problem in Mathematical Fluid Mechanics, namely the rigorous justification of the fact that the Euler equations approximate the Navier-Stokes equations for small values of the viscosity. This is a very hard problem, since it is a singular limit problem, namely there is a small parameter in front of the highest order derivative. Indeed all the previous results justified the approximation only locally in time. We developed a novel method that allowed to justify the approximation, in the class of quasi-periodic solutions, globally in time, with a rate of convergence which is uniform w.r. to time and the viscosity parameter. The proof is based on a novel normal form method which allowed to perform uniform
estimates w.r. to the viscosity. Moreover the paper is also the first KAM result (construction of global in time quasi-periodic solutions) in the framework of singular limit problems.
3) We constructed quasi-periodic traveling waves bifurcating from the Couette flow, which is the simplest example of shear flow (namely any fluid particle stays on the same layer for
all times). The dynamics in the vicinity of shear flows is very important for applications to turbulence and for the problem of transition from laminar flows to turbulent flows. Indeed the experiments show that there is this transition. On the other hand it was proved that for sufficiently regular initial data close to the Couette flow, this phenomenon does not occur since, for large time the fluid flow is close to a shear flow. In this project, we understood that this is a problem of regularity and that in low regularity, one does not have damping phenomena and one has oscillations for all time (quasi-periodic traveling waves).
4) We constructed bi-periodic traveling waves of arbitrarily large amplitude for the non-resistive Magneto-Hydro-Dynamics (MHD) system in dimension 2). This is the first existence result for large amplitude solutions for the MHD equation and it is also the first existence result of large amplitude quasi-periodic solutions in PDEs in dimension greater or equal than two. The novel approach that we propose is based on the usage of micro-local analysis and spectral theory for dealing with linear operators with small divisors and large amplitude perturbations.