-
Previous Article
Phase transition layers for Fife-Greenlee problem on smooth bounded domain
- DCDS Home
- This Issue
-
Next Article
Traveling wave solutions of a highly nonlinear shallow water equation
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. |
[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] |
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 |
[4] |
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 |
[5] |
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 |
[6] |
David M. A. Stuart. Solitons on pseudo-Riemannian manifolds: stability and motion. Electronic Research Announcements, 2000, 6: 75-89. |
[7] |
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 |
[8] |
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 |
[9] |
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 |
[10] |
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 |
[11] |
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 |
[12] |
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 |
[13] |
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 |
[14] |
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 |
[15] |
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 |
[16] |
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 |
[17] |
Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511 |
[18] |
Michael Hochman. A note on universality in multidimensional symbolic dynamics. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 301-314. doi: 10.3934/dcdss.2009.2.301 |
[19] |
Andrei Agrachev, Ugo Boscain, Mario Sigalotti. A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 801-822. doi: 10.3934/dcds.2008.20.801 |
[20] |
Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia. The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2535-2560. doi: 10.3934/dcds.2014.34.2535 |
2016 Impact Factor: 1.099
Tools
Metrics
Other articles
by authors
[Back to Top]