Roberto Feola
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View article: Time Quasi-Periodic Three-dimensional Traveling Gravity Water Waves
Time Quasi-Periodic Three-dimensional Traveling Gravity Water Waves Open
Starting with the pioneering computations of Stokes in 1847, the search of traveling waves in fluid mechanics has always been a fundamental topic, since they can be seen as building blocks to determine the long time dynamics (which is a wi…
View article: Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow
Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow Open
In this paper we prove an abstract result of almost global existence for small and smooth solutions of some semilinear PDEs on Riemannian manifolds with globally integrable geodesic flow. Some examples of such manifolds are Lie groups (inc…
View article: On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori
On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori Open
We consider a completely resonant nonlinear Schrödinger equation on the d-dimensional torus, for any d≥1, with polynomial nonlinearity of any degree 2p+1, p≥1, which is gauge and translation invariant. We study the behaviour of high Sobole…
View article: Long Time Dynamics of Quasi-linear Hamiltonian Klein–Gordon Equations on the Circle
Long Time Dynamics of Quasi-linear Hamiltonian Klein–Gordon Equations on the Circle Open
We consider a class of Hamiltonian Klein–Gordon equations with a quasilinear, quadratic nonlinearity under periodic boundary conditions. For a large set of masses, we provide a precise description of the dynamics for an open set of small i…
View article: Non-resonant conditions for the Klein-Gordon equation on the circle
Non-resonant conditions for the Klein-Gordon equation on the circle Open
We consider the infinite dimensional vector of frequencies $ω(m)=( \sqrt{j^2+m})_{j\in \mathbb{Z}}$, $m\in [1,2]$ arising form a linear Klein-Gordon equation on the one dimensional torus and prove that there exists a positive measure set o…
View article: Quasi-Periodic Traveling Waves on an Infinitely Deep Perfect Fluid Under Gravity
Quasi-Periodic Traveling Waves on an Infinitely Deep Perfect Fluid Under Gravity Open
We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the f…
View article: Reducibility of Klein-Gordon equations with maximal order perturbations
Reducibility of Klein-Gordon equations with maximal order perturbations Open
We prove that all the solutions of a quasi-periodically forced linear Klein-Gordon equation $ψ_{tt}-ψ_{xx}+\mathtt{m}ψ+Q(ωt)ψ=0 $ where $ Q(ωt) := a^{(2)}(ωt, x) \partial_{xx} + a^{(1)}(ωt, x)\partial_x + a^{(0)}(ωt, x) $ is a differential…
View article: Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow
Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow Open
In this paper we prove an abstract result of almost global existence for small and smooth solutions of some semilinear PDEs on Riemannian manifolds with globally integrable geodesic flow. Some examples of such manifolds are Lie groups (inc…
View article: Long time dynamics of quasi-linear Hamiltonian Klein-Gordon equations on the circle
Long time dynamics of quasi-linear Hamiltonian Klein-Gordon equations on the circle Open
We consider a class of Hamiltonian Klein-Gordon equations with a quasilinear, quadratic nonlinearity under periodic boundary conditions. For a large set of masses, we provide a precise description of the dynamics for an open set of small i…
View article: Long time solutions for quasilinear Hamiltonianperturbations of Schrödinger and Klein–Gordon equations on tori
Long time solutions for quasilinear Hamiltonianperturbations of Schrödinger and Klein–Gordon equations on tori Open
We consider quasi-linear, Hamiltonian perturbations of the cubic Schrödinger and of the cubic (derivative) Klein-Gordon equations on the $d$ dimensional torus. If $\varepsilon\ll1$ is the size of the initial datum, we prove that the lifesp…
View article: Local well posedness for a system of quasilinear pdes modelling suspension bridges
Local well posedness for a system of quasilinear pdes modelling suspension bridges Open
In this paper we provide a local well posedness result for a quasilinear beam-wave system of equations on a one-dimensional spatial domain under periodic and Dirichlet boundary conditions. This kind of systems provides a refined model for …
View article: On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori
On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori Open
We consider a completely resonant nonlinear Schrödinger equation on the $d$-dimensional torus, for any $d\geq 1$, with polynomial nonlinearity of any degree $2p+1$, $p\geq1$, which is gauge and translation invariant. We study the behaviour…
View article: Almost global existence for some Hamiltonian PDEs with small Cauchy data on general tori
Almost global existence for some Hamiltonian PDEs with small Cauchy data on general tori Open
In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a q…
View article: Sub-exponential stability for the Beam equation
Sub-exponential stability for the Beam equation Open
We consider a one-parameter family of beam equations with Hamiltonian non-linearity in one space dimension under periodic boundary conditions. In a unified functional framework we study the long time evolution of initial data in two catego…
View article: Long time NLS approximation for the quasilinear Klein-Gordon equation on large domains under periodic boundary conditions
Long time NLS approximation for the quasilinear Klein-Gordon equation on large domains under periodic boundary conditions Open
We provide the rigorous justification of the NLS approximation, in Sobolev regularity, for a class of quasilinear Hamiltonian Klein Gordon equations with quadratic nonlinearities on large one-dimensional tori $\T_L:=\mathbb{R}/(2πL \mathbb…
View article: Birkhoff Normal Form and Long Time Existence for Periodic Gravity Water Waves
Birkhoff Normal Form and Long Time Existence for Periodic Gravity Water Waves Open
We consider the gravity water waves system with a periodic one‐dimensional interface in infinite depth and give a rigorous proof of a conjecture of Dyachenko‐Zakharov [16] concerning the approximate integrability of these equations. More p…
View article: Quadratic lifespan and growth of Sobolev norms for derivative Schrödinger equations on generic tori
Quadratic lifespan and growth of Sobolev norms for derivative Schrödinger equations on generic tori Open
We consider a family of Schrödinger equations with unbounded Hamiltonian quadratic nonlinearities on a generic tori of dimension $d\geq1$. We study the behaviour of high Sobolev norms $H^{s}$, $s\gg1$, of solutions with initial conditions …
View article: Time quasi-periodic traveling gravity water waves in infinite depth
Time quasi-periodic traveling gravity water waves in infinite depth Open
We present the recent result [9] concerning the existence of quasi-periodic in time traveling waves for the 2d pure gravity water waves system in infinite depth. We provide the first existence result of quasi-periodic water waves solutions…
View article: Long time solutions for quasi-linear Hamiltonian perturbations of\n Schr\\"odinger and Klein-Gordon equations on tori
Long time solutions for quasi-linear Hamiltonian perturbations of\n Schr\\"odinger and Klein-Gordon equations on tori Open
We consider quasi-linear, Hamiltonian perturbations of the cubic\nSchr\\"odinger and of the cubic (derivative) Klein-Gordon equations on the $d$\ndimensional torus. If $\\varepsilon\\ll1$ is the size of the initial datum, we\nprove that th…
View article: Reducibility of Schrödinger equation on a Zoll manifold with unbounded potential
Reducibility of Schrödinger equation on a Zoll manifold with unbounded potential Open
In this article, we prove a reducibility result for the linear Schrödinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less than or equal to 1/2. As far as we know, this is the first re…
View article: Reducible KAM Tori for the Degasperis–Procesi Equation
Reducible KAM Tori for the Degasperis–Procesi Equation Open
We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis–Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash–Moser …
View article: A non-linear Egorov theorem and Poincaré-Birkhoff normal forms for quasi-linear pdes on the circle
A non-linear Egorov theorem and Poincaré-Birkhoff normal forms for quasi-linear pdes on the circle Open
In this paper we consider an abstract class of quasi-linear para-differential equations on the circle. For each equation in the class we prove the existence of a change of coordinates which conjugates the equation to a diagonal and constan…
View article: Quasi-periodic traveling waves on an infinitely deep perfect fluid under gravity
Quasi-periodic traveling waves on an infinitely deep perfect fluid under gravity Open
We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the f…