Roberto Feola (Univ. Roma Tre): KAM for pure gravity 3d traveling gravity water waves.
Abstract: We consider a three dimensional incompressible and irrotational perfect fluid under the action of gravity. The fluid layer is assumed to have finite depth with flat bottom and the motion is governed by the free surface Euler equations. We discuss the existence and the stability of small amplitude, quasi-periodic in time, traveling waves on the surface (2d interface) of the fluid. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure.
The major difficulties arises from the fact that the 3D water waves system is a quasi-linear PDE in higher space dimension with ``weak dispersion relation’’. As a consequence, in the search for quasi-periodic solutions one must deal with the presence of very strong resonance phenomena.
The proof rely on a nonlinear Nash-Moser scheme. The invertibility of the linearized operator at a quasi-periodic approximate solution is obtained combining a KAM reducibility argument, tailored for PDEs in high space dimensions, pseudo-differential calculus and exploiting the conservation of momentum to remove some resonances.
Abstract: There is an interesting connection between the Uncertainty Principle (a heuristic principle in harmonic analysis which basically says that the more a function is localized, the more its Fourier transform is sparse) and the realm of PDEs. Indeed, the more a solution to some PDE (heat, Schrödinger...) is localized in the space variables at an observation instant of time, the more it is sparse at a second different observation instant of time. This second heuristic principle goes by the name of Dynamical Uncertainty Principle. There are several rigorous statements of the Uncertainty Principle, which lead to different dynamical results. In the talk, I will present our attempt to adapt an harmonic analytic result of Amrein and Berthier (a non-zero function and its Fourier transform cannot be both supported over sets of finite measure) to solutions to quite general magnetic Schrödinger PDEs in R^d_x × R_t. We have proved a quantitative result consisting in a bound of the spatial L^2 norm of the solution in terms of the L^2 norms of the localization of the solution, observed at two different instants of time, outside two compact sets. Slides are available at this link!
Abstract: Kirchhoff's equation, introduced in a monograph in 1875, has been extensively studied over the last 150 years but still presents many significant open problems. Here, we review the current state of the art concerning the existence of local and global solutions, as well as estimates for the lifespan of these solutions. Furthermore, we will outline a a potential approach to identifying a solution that exhibits finite-time blow-up. Slides are available at this link!
Abstract: We study the formation of extreme waves from a statistical viewpoint in the context of various Hamiltonian systems. Firstly, we derive an asymptotic development for the probability of appearance of an extreme wave as the amplitude of the wave tends to infinity. Secondly, we analyze the most likely mechanism of formation of such waves with the help of two toy models.
In the case of an integrable Hamiltonian system, we show that extreme waves are typically the result of a linear superposition mechanism. In the case of a highly resonant system, however, we prove that a nonlinear focusing mechanism is most likely. This nonlinear mechanism is then proved to increase the likelihood of appearance of extreme waves, as has often been conjectured in the physics literature.
Abstract: In this talk I will focus on a class of completely resonant Klein-Gordon equations on the 3 dimensional sphere with quadratic, cubic and quintic nonlinearity, which arise as toy models in General Relativity. I will show that these equations admit small amplitude, time periodic solutions.
Their existence is obtained by a variational Lyapunov-Schmidt decomposition, which reduces the problem to the search of mountain pass critical points of a restricted Euler-Lagrange action functional.
In order to gain compactness of the gradient of such a functional and smoothness of the critical points, we implement Strichartz-type estimates for the solutions of the linear Klein-Gordon equation on the three dimensional sphere.
This talk is based on a joint work with Massimiliano Berti and Diego Silimbani.
Abstract: We study the transfer of energy to high frequencies in a quasi-linear Schrödinger equation with a sublinear dispersion relation on the one-dimensional torus, a toy model for gravity water waves. We construct initial data that exhibit finite-time Sobolev norm growth: starting with arbitrarily small norms in high-regularity spaces, these norms become arbitrarily large at later times. Our analysis identifies an instability mechanism driving this energy cascade. Using para-differential normal forms, we derive an effective equation governed by a transport operator with non-constant coefficients. A positive commutator method inspired by Mourre’s theory reveals how this operator induces the instability. We believe that our work provides a foundational step toward investigating energy cascades in more complex fluid models, including gravity water waves. This is a joint work with Alberto Maspero.
Slides are available at this link!
Abstract: We consider the Euler equations within a simply-connected bounded domain. The dynamics of a single point vortex are governed by a Hamiltonian system, with most of its energy levels corresponding to time-periodic orbits. We show that, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches. We provide concrete examples of these non-degeneracy conditions, which are satisfied by a broad class of domains, including convex ones. Additionally, we will present a vortex duplication mechanism to generate synchronized time-periodic motion of multiple vortices.
Abstract: We discuss the reducibility of a linear Klein Gordon equation on the torus in presence of a pseudo-differential potential of order 2 depending quasi-periodically on time. Under suitable conditions on the frequency vector, we develop a general strategy, combining Egorov theory with straightening of vector fields, to reduce the equation to constant coefficients. Finally, we shall discuss possible generalizations to operators arising from the linearization of fluid models such as pure gravity Water Waves.
This is a joint work with M. Berti, R. Feola and M. Procesi. Slides are available at this link!
Abstract: We discuss the Cauchy theory associated with the mKdV equation for complex valued functions. Then we construct for the corresponding dynamic a sequence of invariant weighted Gaussian measures supported on Sobolev spaces more and more regular. This is a joint work with C. Kenig, A. Nahmod., N. Pavlovic, G. Staffilani.