914 research outputs found
Reducibility of 1-d Schrödinger equation with unbounded time quasiperiodic perturbations. III
No full textIn this paper, we study the reducibility of time quasiperiodic perturbations of the quantum harmonic or anharmonic oscillator in one space dimension. We modify known algorithms obtaining a reducibility result which allows us to deal with perturbations of order strictly larger than the ones considered in all the previous paper
Attivita’ agricola, valorizzazione enologica e sostenibilità ambientale nel sottobacino Iudeo – Bucari (bacino del fiume Mazaro – TP).
No full textQuasi-periodic solutions for the forced Kirchhoff equation on T^d
No full textIn this paper we prove the existence of small-amplitude quasi-periodic solutions with Sobolev regularity, for the d-dimensional forced Kirchhoff equation with periodic boundary conditions. This is the first result of this type for a quasi-linear equation in high dimension. The proof is based on a Nash–Moser scheme in Sobolev class and a regularization procedure combined with a multiscale analysis in order to solve the linearized problem at any approximate solutio
A Reducibility Result for a Class of Linear Wave Equations on T-d
No full textWe prove a reducibility result for a class of quasi-periodically forced linear wave equations on the d-dimensional torus Td of the form ∂ttv − v + εP(ωt)[v] = 0, where the perturbation P(ωt) is a second order operator of the form P(ωt) = −a(ωt) − R(ωt), the frequency ω ∈ Rν is in some Borel set of large Lebesgue measure, the function a : Tν → R (independent of the space variable) is sufficiently smooth and R(ωt) is a time-dependent finite rank operator. This is the first reducibility result for linear wave equations with unbounded perturbations on the higher dimensional torus Td. As a corollary, we get that the linearized Kirchhoff equation at a smooth and sufficiently small quasi-periodic function is reducible
Reducibility of Non-Resonant Transport Equation on Td with Unbounded Perturbations
No full textWe prove reducibility of a transport equation on the d-dimensional torus Td with a time quasiperiodic unbounded perturbation. As far as we know, this is one of the few reducibility results for an equation in more than one dimension with unbounded perturbations. Furthermore, the unperturbed problem has eigenvalues whose differences are dense on the real axis
On the growth of Sobolev norms for a class of linear Schrödinger equations on the torus with superlinear dispersion
No full textIn this paper we consider time dependent Schrödinger equations on the one-dimensional torus T:=R/(2πZ) of the form ∂tu=iV(t)[u] where V(t) is a time dependent, self-adjoint pseudo-differential operator of the form V(t)=V(t,x)|D|M+W(t), M>1, |D|:=−∂xx, V is a smooth function uniformly bounded from below and W is a time-dependent pseudo-differential operator of order strictly smaller than M. We prove that the solutions of the Schrödinger equation ∂tu=iV(t)[u] grow at most as tε, t→+∞ for any ε>0. The proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field iV(t) which uses Egorov type theorems and pseudo-differential calculu
Almost Global Existence for Some Hamiltonian PDEs with Small Cauchy Data on General Tori
No full textIn 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 quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain’s Lemma which provides a partition of the “resonant sites” of the Laplace operator on irrational tori
The Navier–Stokes Equation with Time Quasi-Periodic External Force: Existence and Stability of Quasi-Periodic Solutions
No full textWe prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier-Stokes equation on the d-dimensional torus T-d, with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in H-s (for s large enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for t ->+infinity, with an exponential rate of convergence O(e(-alpha t)) for any arbitrary alpha is an element of(0, 1)
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