• Previous Article
    On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients
  • DCDS Home
  • This Issue
  • Next Article
    Unique solvability of elliptic problems associated with two-phase incompressible flows in unbounded domains
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  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 & Continuous Dynamical Systems, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[1]

Annegret Glitzky. Energy estimates for electro-reaction-diffusion systems with partly fast kinetics. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 159-174. doi: 10.3934/dcds.2009.25.159

[2]

Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207

[3]

Shin-Ichiro Ei, Hirofumi Izuhara, Masayasu Mimura. Infinite dimensional relaxation oscillation in aggregation-growth systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1859-1887. doi: 10.3934/dcdsb.2012.17.1859

[4]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005

[5]

Xavier Cabré, Amadeu Delshams, Marian Gidea, Chongchun Zeng. Preface of Llavefest: A broad perspective on finite and infinite dimensional dynamical systems. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : i-iii. doi: 10.3934/dcds.201812i

[6]

María J. Garrido-Atienza, Oleksiy V. Kapustyan, José Valero. Preface to the special issue "Finite and infinite dimensional multivalued dynamical systems". Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : i-iv. doi: 10.3934/dcdsb.201705i

[7]

Kening Lu, Alexandra Neamţu, Björn Schmalfuss. On the Oseledets-splitting for infinite-dimensional random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1219-1242. doi: 10.3934/dcdsb.2018149

[8]

Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9

[9]

Vedran Sohinger. Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3733-3771. doi: 10.3934/dcds.2012.32.3733

[10]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281

[11]

Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3115-3138. doi: 10.3934/dcdsb.2018303

[12]

J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731

[13]

Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069

[14]

Carsten Burstedde. On the numerical evaluation of fractional Sobolev norms. Communications on Pure & Applied Analysis, 2007, 6 (3) : 587-605. doi: 10.3934/cpaa.2007.6.587

[15]

Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753

[16]

Matthieu Alfaro, Thomas Giletti. When fast diffusion and reactive growth both induce accelerating invasions. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3011-3034. doi: 10.3934/cpaa.2019135

[17]

Felipe Wallison Chaves-Silva, Sergio Guerrero, Jean Pierre Puel. Controllability of fast diffusion coupled parabolic systems. Mathematical Control & Related Fields, 2014, 4 (4) : 465-479. doi: 10.3934/mcrf.2014.4.465

[18]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405

[19]

Massimiliano Berti, Philippe Bolle. Fast Arnold diffusion in systems with three time scales. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 795-811. doi: 10.3934/dcds.2002.8.795

[20]

Marcel Freitag. The fast signal diffusion limit in nonlinear chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1109-1128. doi: 10.3934/dcdsb.2019211

2019 Impact Factor: 1.338

Article outline

Figures and Tables

[Back to Top]