Figure 1.
A diagram illustrating the key relations leading of the embedding of $ H^{{\rm{div}}} $ geodesic equations into incompressible Euler.
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2019, 11(2): 205-223.
doi: 10.3934/jgm.2019011
Embedding Camassa-Holm equations in incompressible Euler
| 1. | INRIA, Paris, 2 Rue Simone Iff, 75012 Paris, France |
| 2. | Université Paris-Est Marne-la-Vallée, 5 Boulevard Descartes, 77420 Champs-sur-Marne, France |
Recently, Gallouët and Vialard [
Mathematics Subject Classification:
Primary: 58F15, 58F17; Secondary: 53C35.
Citation:
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
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References:
| [1] |
V. Arnold,
Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361.
doi: 10.5802/aif.233. |
| [2] |
R. Bishop and B. O'Neill,
Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1-49.
doi: 10.1090/S0002-9947-1969-0251664-4. |
| [3] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
| [4] |
A. Constantin and B. Kolev,
Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.
doi: 10.1007/s00014-003-0785-6. |
| [5] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
| [6] |
A. Constantin and D. Lannes,
The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations, Archive for Rational Mechanics and Analysis, 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
| [7] |
H. H. Dai,
Model equations for nonlinear dispersive waves in a compressible mooney-rivlin rod, Acta Mechanica, 127 (1998), 193-207.
doi: 10.1007/BF01170373. |
| [8] |
D. G. 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. |
| [9] |
L. P. Eisenhart,
Dynamical trajectories and geodesics, Annals of Mathematics, 30 (1928), 591-606.
doi: 10.2307/1968307. |
| [10] |
M. Fisher and J. Schiff,
The Camassa Holm equation: Conserved quantities and the initial value problem, Physics Letters A, 259 (1999), 371-376.
doi: 10.1016/S0375-9601(99)00466-1. |
| [11] |
T. Gallouët and F.-X. Vialard, The Camassa–Holm equation as an incompressible Euler equation: A geometric point of view, Journal of Differential Equations, 264 (2018), 4199–4234, URL http://www.sciencedirect.com/science/article/pii/S0022039617306435.
doi: 10.1016/j.jde.2017.12.008. |
| [12] |
D. D. Holm, J. E. Marsden and T. S. Ratiu,
The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.
doi: 10.1006/aima.1998.1721. |
| [13] |
D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation, in The Breadth of Symplectic and Poisson Geometry, Springer, 232 (2005), 203–235.
doi: 10.1007/0-8176-4419-9_8. |
| [14] |
D. D. Holm and M. F. Staley,
Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE, Physics Letters A, 308 (2003), 437-444.
doi: 10.1016/S0375-9601(03)00114-2. |
| [15] |
B. Khesin, J. Lenells, G. Misiolek and S. C. Preston,
Geometry of diffeomorphism groups, complete integrability and geometric statistics, Geom. Funct. Anal., 23 (2013), 334-366.
doi: 10.1007/s00039-013-0210-2. |
| [16] |
B. Khesin, G. Misiolek and K. Modin,
Geometric hydrodynamics via Madelung transform, Proc. Natl. Acad. Sci. USA, 115 (2018), 6165-6170.
doi: 10.1073/pnas.1719346115. |
| [17] |
M. Kohlmann, The two-dimensional periodic b-equation on the diffeomorphism group of the torus, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 465205, 17pp.
doi: 10.1088/1751-8113/44/46/465205. |
| [18] |
S. Kouranbaeva,
The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868.
doi: 10.1063/1.532690. |
| [19] |
P. W. Michor and D. Mumford,
Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245.
|
| [20] |
P. W. Michor and D. Mumford,
On Euler's equation and 'EPDiff', J. Geom. Mech., 5 (2013), 319-344.
doi: 10.3934/jgm.2013.5.319. |
| [21] |
L. Molinet,
On well-posedness results for Camassa-Holm equation on the line: A survey, Journal of Nonlinear Mathematical Physics, 11 (2004), 521-533.
doi: 10.2991/jnmp.2004.11.4.8. |
| [22] |
S. C. Preston,
The geometry of barotropic flow, Journal of Mathematical Fluid Mechanics, 15 (2013), 807-821.
doi: 10.1007/s00021-013-0142-5. |
| [23] |
T. Tao,
On the universality of the incompressible Euler equation on compact manifolds, Discrete Contin. Dyn. Syst., 38 (2018), 1553-1565.
doi: 10.3934/dcds.2018064. |
| [24] |
T. Tao, Embedding the Boussinesq equations in the incompressible Euler equations on a manifold, https://terrytao.wordpress.com/2017/12/14/embedding-the-boussinesq-equations-in-the-incompressible-euler-equations-on-a-manifold/, 2017. |
show all references
References:
| [1] |
V. Arnold,
Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361.
doi: 10.5802/aif.233. |
| [2] |
R. Bishop and B. O'Neill,
Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1-49.
doi: 10.1090/S0002-9947-1969-0251664-4. |
| [3] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
| [4] |
A. Constantin and B. Kolev,
Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.
doi: 10.1007/s00014-003-0785-6. |
| [5] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
| [6] |
A. Constantin and D. Lannes,
The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations, Archive for Rational Mechanics and Analysis, 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
| [7] |
H. H. Dai,
Model equations for nonlinear dispersive waves in a compressible mooney-rivlin rod, Acta Mechanica, 127 (1998), 193-207.
doi: 10.1007/BF01170373. |
| [8] |
D. G. 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. |
| [9] |
L. P. Eisenhart,
Dynamical trajectories and geodesics, Annals of Mathematics, 30 (1928), 591-606.
doi: 10.2307/1968307. |
| [10] |
M. Fisher and J. Schiff,
The Camassa Holm equation: Conserved quantities and the initial value problem, Physics Letters A, 259 (1999), 371-376.
doi: 10.1016/S0375-9601(99)00466-1. |
| [11] |
T. Gallouët and F.-X. Vialard, The Camassa–Holm equation as an incompressible Euler equation: A geometric point of view, Journal of Differential Equations, 264 (2018), 4199–4234, URL http://www.sciencedirect.com/science/article/pii/S0022039617306435.
doi: 10.1016/j.jde.2017.12.008. |
| [12] |
D. D. Holm, J. E. Marsden and T. S. Ratiu,
The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.
doi: 10.1006/aima.1998.1721. |
| [13] |
D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation, in The Breadth of Symplectic and Poisson Geometry, Springer, 232 (2005), 203–235.
doi: 10.1007/0-8176-4419-9_8. |
| [14] |
D. D. Holm and M. F. Staley,
Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE, Physics Letters A, 308 (2003), 437-444.
doi: 10.1016/S0375-9601(03)00114-2. |
| [15] |
B. Khesin, J. Lenells, G. Misiolek and S. C. Preston,
Geometry of diffeomorphism groups, complete integrability and geometric statistics, Geom. Funct. Anal., 23 (2013), 334-366.
doi: 10.1007/s00039-013-0210-2. |
| [16] |
B. Khesin, G. Misiolek and K. Modin,
Geometric hydrodynamics via Madelung transform, Proc. Natl. Acad. Sci. USA, 115 (2018), 6165-6170.
doi: 10.1073/pnas.1719346115. |
| [17] |
M. Kohlmann, The two-dimensional periodic b-equation on the diffeomorphism group of the torus, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 465205, 17pp.
doi: 10.1088/1751-8113/44/46/465205. |
| [18] |
S. Kouranbaeva,
The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868.
doi: 10.1063/1.532690. |
| [19] |
P. W. Michor and D. Mumford,
Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245.
|
| [20] |
P. W. Michor and D. Mumford,
On Euler's equation and 'EPDiff', J. Geom. Mech., 5 (2013), 319-344.
doi: 10.3934/jgm.2013.5.319. |
| [21] |
L. Molinet,
On well-posedness results for Camassa-Holm equation on the line: A survey, Journal of Nonlinear Mathematical Physics, 11 (2004), 521-533.
doi: 10.2991/jnmp.2004.11.4.8. |
| [22] |
S. C. Preston,
The geometry of barotropic flow, Journal of Mathematical Fluid Mechanics, 15 (2013), 807-821.
doi: 10.1007/s00021-013-0142-5. |
| [23] |
T. Tao,
On the universality of the incompressible Euler equation on compact manifolds, Discrete Contin. Dyn. Syst., 38 (2018), 1553-1565.
doi: 10.3934/dcds.2018064. |
| [24] |
T. Tao, Embedding the Boussinesq equations in the incompressible Euler equations on a manifold, https://terrytao.wordpress.com/2017/12/14/embedding-the-boussinesq-equations-in-the-incompressible-euler-equations-on-a-manifold/, 2017. |
Figure 2.
A peakons-antipeakons collision represented in the new polar coordinates variables at different time points. The two curves in blue and green are scaled versions of each others: they represent the image under the map $ \Psi(\theta,r) = r \sqrt{\partial_\theta \varphi(\theta)} e^{i\varphi(\theta)} $ of two circles on the complex plane with different radii.
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