American Institute of Mathematical Sciences

Advanced
March  2018, 38(3): 1553-1565. doi: 10.3934/dcds.2018064

On the universality of the incompressible Euler equation on compact manifolds

Terence Tao

UCLA Department of Mathematics, Los Angeles, CA 90095-1555, USA

Received  July 2017 Published  December 2017

The incompressible Euler equations on a compact Riemannian manifold
$(M,g)$
 take the form
$\partial_t u + \nabla_u u =- \mathrm{grad}_g p \\\mathrm{div}_g u =0.$
We show that any quadratic ODE
$\partial_t y =B(y,y)$
, where
$B \colon \mathbb{R}^n × \mathbb{R}^n \to \mathbb{R}^n$
 is a symmetric bilinear map, can be linearly embedded into the incompressible Euler equations for some manifold
$M$
 if and only if
$B$
 obeys the cancellation condition
$\langle B(y,y), y \rangle =0$
 for some positive definite inner product
$\langle,\rangle$
 on
$\mathbb{R}^n$
. This allows one to construct explicit solutions to the Euler equations with various dynamical features, such as quasiperiodic solutions, or solutions that transition from one steady state to another, and provides evidence for the "Turing universality" of such Euler flows.
Keywords: Euler equation, quadratic ODE, universality, embedding, Riemannian manifolds.
Mathematics Subject Classification: 35Q35, 37N10, 76B99.
Citation: Terence Tao. On the universality of the incompressible Euler equation on compact manifolds. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1553-1565. doi: 10.3934/dcds.2018064
References:
[1]

V. I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier, 16 (1966), 319-361.  doi: 10.5802/aif.233.  Google Scholar

[2]

M. S. Ashbaugh, C. C. Chicone and R. H. Cushman, The twisting tennis racket, J. Dyn. Diff. Eq., 3 (1991), 67-85.  doi: 10.1007/BF01049489.  Google Scholar

[3]

T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge University Press, 1998.  Google Scholar

[4]

S. Bromberg and A. Medina, Completeness of homogeneous quadratic vector fields, Qual. Theory Dyn. Syst., 6 (2005), 181-185.  doi: 10.1007/BF02972671.  Google Scholar

[5]

R. J. Dickson and L. M. Perko, Bounded quadratic systems in the plane, J. of Diff. Equs., 7 (1990), 251-273.  doi: 10.1016/0022-0396(70)90110-5.  Google Scholar

[6]

E. I. Dinaburg and Ya. G. Sinai, A quasilinear approximation for the three-dimensional Navier-Stokes system, Moscow Math. J., 1 (2001), 381-388.   Google Scholar

[7]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math.(2), 92 (1970), 102-163.  doi: 10.2307/1970699.  Google Scholar

[8]

S. Friedlander and N. Pavlovic, Blow-up in a three-dimensional vector model for the Euler equations, Comm. Pure Appl. Math., 57 (2004), 705-725.  doi: 10.1002/cpa.20017.  Google Scholar

[9]

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995.  Google Scholar

[10]

E. B. Gledzer, System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl., 18 (1973), 216-217.   Google Scholar

[11]

J. L. Kaplan and J. A. Yorke, Non associative real algebras and quadratic differential equations, Nonlinear Analysis, 3 (1979), 49-51.  doi: 10.1016/0362-546X(79)90033-6.  Google Scholar

[12]

N. H. Katz and N. Pavlović, Finite time blow-up for a dyadic model of the Euler equations, Trans. Amer. Math. Soc., 357 (2005), 695-708.  doi: 10.1090/S0002-9947-04-03532-9.  Google Scholar

[13]

K. Okhitani and M. Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully developed model of turbulence, Prog. Theor. Phys., 89 (1989), 329-341.  doi: 10.1143/PTP.81.329.  Google Scholar

[14]

T. Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation, J. Amer. Math. Soc., 29 (2016), 601-674.   Google Scholar

[15]

T. Tao, On the universality of potential well dynamics, Dynamics of Partial Differential Equations, 14 (2017), 219-238.  doi: 10.4310/DPDE.2017.v14.n3.a1.  Google Scholar

show all references

References:
[1]

V. I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier, 16 (1966), 319-361.  doi: 10.5802/aif.233.  Google Scholar

[2]

M. S. Ashbaugh, C. C. Chicone and R. H. Cushman, The twisting tennis racket, J. Dyn. Diff. Eq., 3 (1991), 67-85.  doi: 10.1007/BF01049489.  Google Scholar

[3]

T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge University Press, 1998.  Google Scholar

[4]

S. Bromberg and A. Medina, Completeness of homogeneous quadratic vector fields, Qual. Theory Dyn. Syst., 6 (2005), 181-185.  doi: 10.1007/BF02972671.  Google Scholar

[5]

R. J. Dickson and L. M. Perko, Bounded quadratic systems in the plane, J. of Diff. Equs., 7 (1990), 251-273.  doi: 10.1016/0022-0396(70)90110-5.  Google Scholar

[6]

E. I. Dinaburg and Ya. G. Sinai, A quasilinear approximation for the three-dimensional Navier-Stokes system, Moscow Math. J., 1 (2001), 381-388.   Google Scholar

[7]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math.(2), 92 (1970), 102-163.  doi: 10.2307/1970699.  Google Scholar

[8]

S. Friedlander and N. Pavlovic, Blow-up in a three-dimensional vector model for the Euler equations, Comm. Pure Appl. Math., 57 (2004), 705-725.  doi: 10.1002/cpa.20017.  Google Scholar

[9]

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995.  Google Scholar

[10]

E. B. Gledzer, System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl., 18 (1973), 216-217.   Google Scholar

[11]

J. L. Kaplan and J. A. Yorke, Non associative real algebras and quadratic differential equations, Nonlinear Analysis, 3 (1979), 49-51.  doi: 10.1016/0362-546X(79)90033-6.  Google Scholar

[12]

N. H. Katz and N. Pavlović, Finite time blow-up for a dyadic model of the Euler equations, Trans. Amer. Math. Soc., 357 (2005), 695-708.  doi: 10.1090/S0002-9947-04-03532-9.  Google Scholar

[13]

K. Okhitani and M. Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully developed model of turbulence, Prog. Theor. Phys., 89 (1989), 329-341.  doi: 10.1143/PTP.81.329.  Google Scholar

[14]

T. Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation, J. Amer. Math. Soc., 29 (2016), 601-674.   Google Scholar

[15]

T. Tao, On the universality of potential well dynamics, Dynamics of Partial Differential Equations, 14 (2017), 219-238.  doi: 10.4310/DPDE.2017.v14.n3.a1.  Google Scholar

[1]

Flavia Antonacci, Marco Degiovanni. On the Euler equation for minimal geodesics on Riemannian manifoldshaving discontinuous metrics. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 833-842. doi: 10.3934/dcds.2006.15.833
[2]

Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407
[3]

Andrea Natale, François-Xavier Vialard. Embedding Camassa-Holm equations in incompressible Euler. Journal of Geometric Mechanics, 2019, 11 (2) : 205-223. doi: 10.3934/jgm.2019011
[4]

YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure & Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1
[5]

Razvan C. Fetecau, Beril Zhang. Self-organization on Riemannian manifolds. Journal of Geometric Mechanics, 2019, 11 (3) : 397-426. doi: 10.3934/jgm.2019020
[6]

Rossella Bartolo. Periodic orbits on Riemannian manifolds with convex boundary. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 439-450. doi: 10.3934/dcds.1997.3.439
[7]

Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas. Stability of boundary distance representation and reconstruction of Riemannian manifolds. Inverse Problems & Imaging, 2007, 1 (1) : 135-157. doi: 10.3934/ipi.2007.1.135
[8]

David M. A. Stuart. Solitons on pseudo-Riemannian manifolds: stability and motion. Electronic Research Announcements, 2000, 6: 75-89.

[9]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097
[10]

Anna Maria Candela, J.L. Flores, M. Sánchez. A quadratic Bolza-type problem in a non-complete Riemannian manifold. Conference Publications, 2003, 2003 (Special) : 173-181. doi: 10.3934/proc.2003.2003.173
[11]

Keith Burns, Eugene Gutkin. Growth of the number of geodesics between points and insecurity for Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 403-413. doi: 10.3934/dcds.2008.21.403
[12]

Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701
[13]

Anthony M. Bloch, Rohit Gupta, Ilya V. Kolmanovsky. Neighboring extremal optimal control for mechanical systems on Riemannian manifolds. Journal of Geometric Mechanics, 2016, 8 (3) : 257-272. doi: 10.3934/jgm.2016007
[14]

Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013
[15]

Yuhua Sun. On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1743-1757. doi: 10.3934/cpaa.2015.14.1743
[16]

Simone Fiori. Synchronization of first-order autonomous oscillators on Riemannian manifolds. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1725-1741. doi: 10.3934/dcdsb.2018233
[17]

Marcelo Nogueira, Mahendra Panthee. On the Schrödinger-Debye system in compact Riemannian manifolds. Communications on Pure & Applied Analysis, 2020, 19 (1) : 425-453. doi: 10.3934/cpaa.2020022
[18]

Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002
[19]

David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319
[20]

Bernard Bonnard, Olivier Cots, Nataliya Shcherbakova. The Serret-Andoyer Riemannian metric and Euler-Poinsot rigid body motion. Mathematical Control & Related Fields, 2013, 3 (3) : 287-302. doi: 10.3934/mcrf.2013.3.287

2018 Impact Factor: 1.143

Tools

Article outline

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences