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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

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