American Institute of Mathematical Sciences

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November  2016, 36(11): 6331-6377. doi: 10.3934/dcds.2016075

Groups of asymptotic diffeomorphisms

Robert McOwen 1, and  Peter Topalov 1,

1. 

Northeastern University, Boston, MA 02115, United States, United States

Received  October 2015 Revised  June 2016 Published  August 2016

We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at infinity; the remainder term lies in a weighted Sobolev space whose properties at infinity fit with the desired application. We show that two such classes of asymptotic diffeomorphisms form topological groups under composition. As such, they can be used in the study of fluid dynamics according to the approach of V. Arnold [1].
Keywords: Groups of diffeomorphisms, Camassa-Holm equation, Euler equation., asymptotic expansions, weighted Sobolev spaces.
Mathematics Subject Classification: Primary: 58D17, 35Q31; Secondary: 76N1.
Citation: Robert McOwen, Peter Topalov. Groups of asymptotic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6331-6377. doi: 10.3934/dcds.2016075
References:
[1]

V. Arnold, Sur la geometrié differentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluids parfaits,, Ann. Inst. Fourier, 16 (1966), 319.  doi: 10.5802/aif.233.  Google Scholar

[2]

R. Bartnik, The mass of an asymptotically flat manifold,, Comm. Pure Appl. Math, 39 (1986), 661.  doi: 10.1002/cpa.3160390505.  Google Scholar

[3]

I. Bondareva and M. Shubin, Uniqueness of the solution of the Cauchy problem for the Korteweg - de Vries equation in classes of increasing functions,, Moscow Univ. Math. Bulletin, 40 (1985), 53.   Google Scholar

[4]

I. Bondareva and M. Shubin, Equations of Korteweg-de Vries type in classes of increasing functions,, J. Soviet Math., 51 (1990), 2323.  doi: 10.1007/BF01094991.  Google Scholar

[5]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation,, J. Func. Anal., 15 (1974), 341.   Google Scholar

[6]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[7]

M. Cantor, Perfect fluid flows over $\mathbbR^n$ with asymptotic conditions,, J. Func. Anal., 18 (1975), 73.  doi: 10.1016/0022-1236(75)90030-0.  Google Scholar

[8]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Four. Grenoble, 50 (2000), 321.  doi: 10.5802/aif.1757.  Google Scholar

[9]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. Math., 92 (1970), 102.  doi: 10.2307/1970699.  Google Scholar

[10]

H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms,, Mem. Amer. Math. Soc., 226 (2013).  doi: 10.1090/S0065-9266-2013-00676-4.  Google Scholar

[11]

T. Kappeler, P. Perry, M. Shubin and P. Topalov, Solutions of mKdV in classes of functions unbounded at infinity,, J. Geom. Anal., 18 (2008), 443.  doi: 10.1007/s12220-008-9013-3.  Google Scholar

[12]

C. Kenig, G. Ponce and L. Vega, Global solutions for the KdV equation with unbounded data,, J. Diff. Equations, 139 (1997), 339.  doi: 10.1006/jdeq.1997.3297.  Google Scholar

[13]

R. McOwen, The behavior of the Laplacian on weighted Sobolev spaces,, Comm. Pure Appl. Math., 32 (1979), 783.  doi: 10.1002/cpa.3160320604.  Google Scholar

[14]

R. McOwen, Partial Differential Equations: Methods and Applications,, 2nd ed, (2003).   Google Scholar

[15]

R. McOwen and P. Topalov, Asymptotics in shallow water waves,, Discrete Contin. Dyn. Syst., 35 (2015), 3103.  doi: 10.3934/dcds.2015.35.3103.  Google Scholar

[16]

R. McOwen and P. Topalov, Spatial asymptotic expansions in the incompressible Euler equation,, arXiv:1606.08059., ().   Google Scholar

[17]

A. Menikoff, The existence of unbounded solutions of the Korteweg-de Vries equation,, Comm. Pure Appl. Math., 25 (1972), 407.  doi: 10.1002/cpa.3160250404.  Google Scholar

[18]

P. Michor and D. Mumford, A zoo of diffeomorphisms groups on $\mathbbR^n$,, Ann. Glob. Anal. Geom., 44 (): 529.  doi: 10.1007/s10455-013-9380-2.  Google Scholar

[19]

G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Visaro group,, J. Geom. Phys., 24 (1998), 203.  doi: 10.1016/S0393-0440(97)00010-7.  Google Scholar

[20]

D. Montgomery, On continuity in topological groups,, Bull. Amer. Math. Soc., 42 (1936), 879.  doi: 10.1090/S0002-9904-1936-06456-6.  Google Scholar

show all references

References:
[1]

V. Arnold, Sur la geometrié differentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluids parfaits,, Ann. Inst. Fourier, 16 (1966), 319.  doi: 10.5802/aif.233.  Google Scholar

[2]

R. Bartnik, The mass of an asymptotically flat manifold,, Comm. Pure Appl. Math, 39 (1986), 661.  doi: 10.1002/cpa.3160390505.  Google Scholar

[3]

I. Bondareva and M. Shubin, Uniqueness of the solution of the Cauchy problem for the Korteweg - de Vries equation in classes of increasing functions,, Moscow Univ. Math. Bulletin, 40 (1985), 53.   Google Scholar

[4]

I. Bondareva and M. Shubin, Equations of Korteweg-de Vries type in classes of increasing functions,, J. Soviet Math., 51 (1990), 2323.  doi: 10.1007/BF01094991.  Google Scholar

[5]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation,, J. Func. Anal., 15 (1974), 341.   Google Scholar

[6]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[7]

M. Cantor, Perfect fluid flows over $\mathbbR^n$ with asymptotic conditions,, J. Func. Anal., 18 (1975), 73.  doi: 10.1016/0022-1236(75)90030-0.  Google Scholar

[8]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Four. Grenoble, 50 (2000), 321.  doi: 10.5802/aif.1757.  Google Scholar

[9]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. Math., 92 (1970), 102.  doi: 10.2307/1970699.  Google Scholar

[10]

H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms,, Mem. Amer. Math. Soc., 226 (2013).  doi: 10.1090/S0065-9266-2013-00676-4.  Google Scholar

[11]

T. Kappeler, P. Perry, M. Shubin and P. Topalov, Solutions of mKdV in classes of functions unbounded at infinity,, J. Geom. Anal., 18 (2008), 443.  doi: 10.1007/s12220-008-9013-3.  Google Scholar

[12]

C. Kenig, G. Ponce and L. Vega, Global solutions for the KdV equation with unbounded data,, J. Diff. Equations, 139 (1997), 339.  doi: 10.1006/jdeq.1997.3297.  Google Scholar

[13]

R. McOwen, The behavior of the Laplacian on weighted Sobolev spaces,, Comm. Pure Appl. Math., 32 (1979), 783.  doi: 10.1002/cpa.3160320604.  Google Scholar

[14]

R. McOwen, Partial Differential Equations: Methods and Applications,, 2nd ed, (2003).   Google Scholar

[15]

R. McOwen and P. Topalov, Asymptotics in shallow water waves,, Discrete Contin. Dyn. Syst., 35 (2015), 3103.  doi: 10.3934/dcds.2015.35.3103.  Google Scholar

[16]

R. McOwen and P. Topalov, Spatial asymptotic expansions in the incompressible Euler equation,, arXiv:1606.08059., ().   Google Scholar

[17]

A. Menikoff, The existence of unbounded solutions of the Korteweg-de Vries equation,, Comm. Pure Appl. Math., 25 (1972), 407.  doi: 10.1002/cpa.3160250404.  Google Scholar

[18]

P. Michor and D. Mumford, A zoo of diffeomorphisms groups on $\mathbbR^n$,, Ann. Glob. Anal. Geom., 44 (): 529.  doi: 10.1007/s10455-013-9380-2.  Google Scholar

[19]

G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Visaro group,, J. Geom. Phys., 24 (1998), 203.  doi: 10.1016/S0393-0440(97)00010-7.  Google Scholar

[20]

D. Montgomery, On continuity in topological groups,, Bull. Amer. Math. Soc., 42 (1936), 879.  doi: 10.1090/S0002-9904-1936-06456-6.  Google Scholar

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