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November  2021, 41(11): 5057-5085. doi: 10.3934/dcds.2021068

Transfers of energy through fast diffusion channels in some resonant PDEs on the circle

Departament de Matematiques, Universitat Politecnica de Catalunya, ETSEIB, Avinguda Diagonal 647, 08028 Barcelona, Spain

Received  September 2020 Revised  March 2021 Published  November 2021 Early access  April 2021

Fund Project: The author is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 757802

In this paper we consider two classes of resonant Hamiltonian PDEs on the circle with non-convex (respect to actions) first order resonant Hamiltonian. We show that, for appropriate choices of the nonlinearities we can find time-independent linear potentials that enable the construction of solutions that undergo a prescribed growth in the Sobolev norms. The solutions that we provide follow closely the orbits of a nonlinear resonant model, which is a good approximation of the full equation. The non-convexity of the resonant Hamiltonian allows the existence of fast diffusion channels along which the orbits of the resonant model experience a large drift in the actions in the optimal time. This phenomenon induces a transfer of energy among the Fourier modes of the solutions, which in turn is responsible for the growth of higher order Sobolev norms.

Citation: Filippo Giuliani. Transfers of energy through fast diffusion channels in some resonant PDEs on the circle. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5057-5085. doi: 10.3934/dcds.2021068
References:
[1]

P. BaldiM. Berti and R. Montalto, KAM for autonomous quasi-linear perturbations of KdV, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 33 (2016), 1589-1638.  doi: 10.1016/j.anihpc.2015.07.003.

[2]

D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations, Math. Z., 230 (1999), 345-387.  doi: 10.1007/PL00004696.

[3]

D. Bambusi and N. N. Nekhoroshev, Long time stability in perturbations of completely resonant PDE's, Acta Applicandae Mathematica, 70 (2002), 1-22.  doi: 10.1023/A:1013943111479.

[4]

M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Equations, 31 (2006), 959-985.  doi: 10.1080/03605300500358129.

[5]

L. BiascoL. Chierchia and D. Treschev, Stability of nearly integrable, degenerate Hamiltonian systems with two degrees of freedom, J. Nonlinear Sci., 16 (2006), 79-107.  doi: 10.1007/s00332-005-0692-7.

[6]

L. BiascoJ. E. Massetti and M. Procesi, An abstract birkhoff normal form theorem and exponential type stability of the 1d NLS, Communications in Mathematical Physics, 375 (2020), 2089-2153.  doi: 10.1007/s00220-019-03618-x.

[7]

A. Bounemoura, Generic perturbations of linear integrable hamiltonian systems, Regul. Chaotic Dyn., 21 (2016), 665-681.  doi: 10.1134/S1560354716060071.

[8]

A. Bounemoura and V. Kaloshin, Generic fast diffusion for a class of non-convex hamiltonians with two degrees of freedom, Mosc. Math. J., 14 (2014), 181-203.  doi: 10.17323/1609-4514-2014-14-2-181-203.

[9]

J. Bourgain, Aspects of long time behaviour of solutions of nonlinear hamiltonian evolution equations, Geom. Funct. Anal., 5 (1995), 105-140.  doi: 10.1007/BF01895664.

[10]

J. Bourgain, On the growth in time of higher sobolev norms of smooth solutions of hamiltonian PDE, Internat. Math. Res. Notices, (1996), 277–304. doi: 10.1155/S1073792896000207.

[11]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math., 181 (2010), 39-113.  doi: 10.1007/s00222-010-0242-2.

[12]

R. FeolaF. Giuliani and M. Procesi, Reducible kam tori for the Degasperis-Procesi equation, Communications in Mathematical Physics, 377 (2020), 1681-1759.  doi: 10.1007/s00220-020-03788-z.

[13]

P. Gérard and S. Grellier, The cubic Szegö equation, Ann. Sci. Éc. Norm. Supér, 43 (2010), 761-810.  doi: 10.24033/asens.2133.

[14]

P. Gérard and S. Grellier, Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDE, 5 (2012), 1139-1155.  doi: 10.2140/apde.2012.5.1139.

[15]

F. Giuliani, M. Guardia, P. Martin and S. Pasquali, Chaotic-like transfers of energy in hamiltonian pdes, Communications in Mathematical Physics, (2021).

[16]

B. Grébert and L. Thomann, Resonant dynamics for the quintic nonlinear Schrödinger equation, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 29 (2012), 455-477.  doi: 10.1016/j.anihpc.2012.01.005.

[17]

B. Grébert and C. Villegas-Blas, On the energy exchange between resonant modes in nonlinear Schrödinger equations, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 28 (2011), 127-134.  doi: 10.1016/j.anihpc.2010.11.004.

[18]

M. Guardia, E. Haus, Z. Hani, A. Maspero and M. Procesi, Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap tori for the 2d cubic nls equation, J. Eur. Math. Soc., (2020).

[19]

M. GuardiaE. Haus and M. Procesi, Growth of Sobolev norms for the analytic nls on $\mathbb{T}^2$, Adv. Math., 301 (2016), 615-692.  doi: 10.1016/j.aim.2016.06.018.

[20]

M. Guardia and V. Kaloshin, Growth of sobolev norms in the cubic defocusing nonlinear Schrödinger equation, J. Eur. Math. Soc., 17 (2015), 71-149.  doi: 10.4171/JEMS/499.

[21]

Z. Hani, Long-time instability and unbounded sobolev orbits for some periodic nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 211 (2014), 929-964.  doi: 10.1007/s00205-013-0689-6.

[22]

E. Haus and M. Procesi, Kam for beating solutions of the quintic NLS, Communications in Mathematical Physics, 354 (2017), 1101-1132.  doi: 10.1007/s00220-017-2925-7.

[23]

S. B. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations, Geom. Funct. Anal., 7 (1997), 338-363.  doi: 10.1007/PL00001622.

[24]

A. Maspero, Lower bounds on the growth of sobolev norms in some linear time dependent Schrödinger equations, Math. Res. Lett., 26 (2019), 1197-1215.  doi: 10.4310/MRL.2019.v26.n4.a11.

[25]

N. N. Nekhorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk, 32 (1977), 5–66,287.

show all references

References:
[1]

P. BaldiM. Berti and R. Montalto, KAM for autonomous quasi-linear perturbations of KdV, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 33 (2016), 1589-1638.  doi: 10.1016/j.anihpc.2015.07.003.

[2]

D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations, Math. Z., 230 (1999), 345-387.  doi: 10.1007/PL00004696.

[3]

D. Bambusi and N. N. Nekhoroshev, Long time stability in perturbations of completely resonant PDE's, Acta Applicandae Mathematica, 70 (2002), 1-22.  doi: 10.1023/A:1013943111479.

[4]

M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Equations, 31 (2006), 959-985.  doi: 10.1080/03605300500358129.

[5]

L. BiascoL. Chierchia and D. Treschev, Stability of nearly integrable, degenerate Hamiltonian systems with two degrees of freedom, J. Nonlinear Sci., 16 (2006), 79-107.  doi: 10.1007/s00332-005-0692-7.

[6]

L. BiascoJ. E. Massetti and M. Procesi, An abstract birkhoff normal form theorem and exponential type stability of the 1d NLS, Communications in Mathematical Physics, 375 (2020), 2089-2153.  doi: 10.1007/s00220-019-03618-x.

[7]

A. Bounemoura, Generic perturbations of linear integrable hamiltonian systems, Regul. Chaotic Dyn., 21 (2016), 665-681.  doi: 10.1134/S1560354716060071.

[8]

A. Bounemoura and V. Kaloshin, Generic fast diffusion for a class of non-convex hamiltonians with two degrees of freedom, Mosc. Math. J., 14 (2014), 181-203.  doi: 10.17323/1609-4514-2014-14-2-181-203.

[9]

J. Bourgain, Aspects of long time behaviour of solutions of nonlinear hamiltonian evolution equations, Geom. Funct. Anal., 5 (1995), 105-140.  doi: 10.1007/BF01895664.

[10]

J. Bourgain, On the growth in time of higher sobolev norms of smooth solutions of hamiltonian PDE, Internat. Math. Res. Notices, (1996), 277–304. doi: 10.1155/S1073792896000207.

[11]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math., 181 (2010), 39-113.  doi: 10.1007/s00222-010-0242-2.

[12]

R. FeolaF. Giuliani and M. Procesi, Reducible kam tori for the Degasperis-Procesi equation, Communications in Mathematical Physics, 377 (2020), 1681-1759.  doi: 10.1007/s00220-020-03788-z.

[13]

P. Gérard and S. Grellier, The cubic Szegö equation, Ann. Sci. Éc. Norm. Supér, 43 (2010), 761-810.  doi: 10.24033/asens.2133.

[14]

P. Gérard and S. Grellier, Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDE, 5 (2012), 1139-1155.  doi: 10.2140/apde.2012.5.1139.

[15]

F. Giuliani, M. Guardia, P. Martin and S. Pasquali, Chaotic-like transfers of energy in hamiltonian pdes, Communications in Mathematical Physics, (2021).

[16]

B. Grébert and L. Thomann, Resonant dynamics for the quintic nonlinear Schrödinger equation, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 29 (2012), 455-477.  doi: 10.1016/j.anihpc.2012.01.005.

[17]

B. Grébert and C. Villegas-Blas, On the energy exchange between resonant modes in nonlinear Schrödinger equations, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 28 (2011), 127-134.  doi: 10.1016/j.anihpc.2010.11.004.

[18]

M. Guardia, E. Haus, Z. Hani, A. Maspero and M. Procesi, Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap tori for the 2d cubic nls equation, J. Eur. Math. Soc., (2020).

[19]

M. GuardiaE. Haus and M. Procesi, Growth of Sobolev norms for the analytic nls on $\mathbb{T}^2$, Adv. Math., 301 (2016), 615-692.  doi: 10.1016/j.aim.2016.06.018.

[20]

M. Guardia and V. Kaloshin, Growth of sobolev norms in the cubic defocusing nonlinear Schrödinger equation, J. Eur. Math. Soc., 17 (2015), 71-149.  doi: 10.4171/JEMS/499.

[21]

Z. Hani, Long-time instability and unbounded sobolev orbits for some periodic nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 211 (2014), 929-964.  doi: 10.1007/s00205-013-0689-6.

[22]

E. Haus and M. Procesi, Kam for beating solutions of the quintic NLS, Communications in Mathematical Physics, 354 (2017), 1101-1132.  doi: 10.1007/s00220-017-2925-7.

[23]

S. B. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations, Geom. Funct. Anal., 7 (1997), 338-363.  doi: 10.1007/PL00001622.

[24]

A. Maspero, Lower bounds on the growth of sobolev norms in some linear time dependent Schrödinger equations, Math. Res. Lett., 26 (2019), 1197-1215.  doi: 10.4310/MRL.2019.v26.n4.a11.

[25]

N. N. Nekhorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk, 32 (1977), 5–66,287.

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