-
Previous Article
Traveling wave solutions of a highly nonlinear shallow water equation
- DCDS Home
- This Issue
-
Next Article
Phase transition layers for Fife-Greenlee problem on smooth bounded domain
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. 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. |
[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. 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. |
[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. |
[1] |
Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031 |
[2] |
Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021008 |
[3] |
Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017 |
[4] |
Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020374 |
[5] |
Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001 |
[6] |
Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 |
[7] |
Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605 |
[8] |
Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012 |
[9] |
Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020169 |
[10] |
Wei-Chieh Chen, Bogdan Kazmierczak. Traveling waves in quadratic autocatalytic systems with complexing agent. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020364 |
[11] |
Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170 |
[12] |
Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229 |
[13] |
Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020 doi: 10.3934/jgm.2020031 |
[14] |
Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415 |
[15] |
Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003 |
[16] |
Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015 |
[17] |
Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020108 |
[18] |
Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020400 |
[19] |
Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390 |
[20] |
Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011 |
2019 Impact Factor: 1.338
Tools
Article outline
[Back to Top]