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On the universality of the incompressible Euler equation on compact manifolds
UCLA Department of Mathematics, Los Angeles, CA 90095-1555, USA |
$(M,g)$ |
$\partial_t u + \nabla_u u =- \mathrm{grad}_g p \\\mathrm{div}_g u =0.$ |
$\partial_t y =B(y,y)$ |
$B \colon \mathbb{R}^n × \mathbb{R}^n \to \mathbb{R}^n$ |
$M$ |
$B$ |
$\langle B(y,y), y \rangle =0$ |
$\langle,\rangle$ |
$\mathbb{R}^n$ |
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. |
[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. |
[3] |
T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge University Press, 1998. |
[4] |
S. Bromberg and A. Medina,
Completeness of homogeneous quadratic vector fields, Qual. Theory Dyn. Syst., 6 (2005), 181-185.
doi: 10.1007/BF02972671. |
[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. |
[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.
|
[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. |
[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. |
[9] |
U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995. |
[10] |
E. B. Gledzer,
System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl., 18 (1973), 216-217.
|
[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. |
[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. |
[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. |
[14] |
T. Tao,
Finite time blowup for an averaged three-dimensional Navier-Stokes equation, J. Amer. Math. Soc., 29 (2016), 601-674.
|
[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. |
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. |
[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. |
[3] |
T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge University Press, 1998. |
[4] |
S. Bromberg and A. Medina,
Completeness of homogeneous quadratic vector fields, Qual. Theory Dyn. Syst., 6 (2005), 181-185.
doi: 10.1007/BF02972671. |
[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. |
[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.
|
[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. |
[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. |
[9] |
U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995. |
[10] |
E. B. Gledzer,
System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl., 18 (1973), 216-217.
|
[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. |
[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. |
[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. |
[14] |
T. Tao,
Finite time blowup for an averaged three-dimensional Navier-Stokes equation, J. Amer. Math. Soc., 29 (2016), 601-674.
|
[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. |
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