June  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

* Corresponding author: François-Xavier Vialard

Received  April 2018 Revised  March 2019 Published  May 2019

Fund Project: The first author was supported by the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n. PCOFUND-GA-2013-609102, through the PRESTIGE programme coordinated by Campus France

Recently, Gallouët and Vialard [11] showed that the CH equation can be embedded in the incompressible Euler equation on a non compact Riemannian manifold. After surveying this result from a geometric point of view, we extend it to a broader class of PDEs, namely the so-called CH2 equations and the Holm-Staley $b$-family of equations. A salient feature of these embeddings is the cone singularity of the Riemannian manifold on which the incompressible Euler equation is considered.

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
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

L. P. Eisenhart, Dynamical trajectories and geodesics, Annals of Mathematics, 30 (1928), 591-606. doi: 10.2307/1968307. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

D. D. HolmJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

B. KhesinJ. LenellsG. 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. Google Scholar

[16]

B. KhesinG. Misiolek and K. Modin, Geometric hydrodynamics via Madelung transform, Proc. Natl. Acad. Sci. USA, 115 (2018), 6165-6170. doi: 10.1073/pnas.1719346115. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

L. P. Eisenhart, Dynamical trajectories and geodesics, Annals of Mathematics, 30 (1928), 591-606. doi: 10.2307/1968307. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

D. D. HolmJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

B. KhesinJ. LenellsG. 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. Google Scholar

[16]

B. KhesinG. Misiolek and K. Modin, Geometric hydrodynamics via Madelung transform, Proc. Natl. Acad. Sci. USA, 115 (2018), 6165-6170. doi: 10.1073/pnas.1719346115. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

Figure 1.  A diagram illustrating the key relations leading of the embedding of $ H^{{\rm{div}}} $ geodesic equations into incompressible Euler.
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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