Groups of asymptotic diffeomorphisms

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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

Abstract
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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 and Continuous Dynamical Systems, 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), 319-361. doi: 10.5802/aif.233.
[2]	R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math, 39 (1986), 661-693. doi: 10.1002/cpa.3160390505.
[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), 53-57.
[4]	I. Bondareva and M. Shubin, Equations of Korteweg-de Vries type in classes of increasing functions, J. Soviet Math., 51 (1990), 2323-2332. doi: 10.1007/BF01094991.
[5]	J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Func. Anal., 15 (1974), 341-363.
[6]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[7]	M. Cantor, Perfect fluid flows over $\mathbb{R}^{N}$ with asymptotic conditions, J. Func. Anal., 18 (1975), 73-84. doi: 10.1016/0022-1236(75)90030-0.
[8]	A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Four. Grenoble, 50 (2000), 321-362. doi: 10.5802/aif.1757.
[9]	D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92 (1970), 102-163. doi: 10.2307/1970699.
[10]	H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms, Mem. Amer. Math. Soc., 226 (2013), no. 1062. doi: 10.1090/S0065-9266-2013-00676-4.
[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), 443-477. doi: 10.1007/s12220-008-9013-3.
[12]	C. Kenig, G. Ponce and L. Vega, Global solutions for the KdV equation with unbounded data, J. Diff. Equations, 139 (1997), 339-364. doi: 10.1006/jdeq.1997.3297.
[13]	R. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math., 32 (1979), 783-795. doi: 10.1002/cpa.3160320604.
[14]	R. McOwen, Partial Differential Equations: Methods and Applications, 2nd ed, Pearson, 2003.
[15]	R. McOwen and P. Topalov, Asymptotics in shallow water waves, Discrete Contin. Dyn. Syst., 35 (2015), 3103-3131. doi: 10.3934/dcds.2015.35.3103.
[16]	R. McOwen and P. Topalov, Spatial asymptotic expansions in the incompressible Euler equation, arXiv:1606.08059.
[17]	A. Menikoff, The existence of unbounded solutions of the Korteweg-de Vries equation, Comm. Pure Appl. Math., 25 (1972), 407-432. doi: 10.1002/cpa.3160250404.
[18]	P. Michor and D. Mumford, A zoo of diffeomorphisms groups on $\mathbb{R}^{N}$, Ann. Glob. Anal. Geom., 44, 529-540. doi: 10.1007/s10455-013-9380-2.
[19]	G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Visaro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7.
[20]	D. Montgomery, On continuity in topological groups, Bull. Amer. Math. Soc., 42 (1936), 879-882. doi: 10.1090/S0002-9904-1936-06456-6.

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), 319-361. doi: 10.5802/aif.233.
[2]	R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math, 39 (1986), 661-693. doi: 10.1002/cpa.3160390505.
[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), 53-57.
[4]	I. Bondareva and M. Shubin, Equations of Korteweg-de Vries type in classes of increasing functions, J. Soviet Math., 51 (1990), 2323-2332. doi: 10.1007/BF01094991.
[5]	J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Func. Anal., 15 (1974), 341-363.
[6]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[7]	M. Cantor, Perfect fluid flows over $\mathbb{R}^{N}$ with asymptotic conditions, J. Func. Anal., 18 (1975), 73-84. doi: 10.1016/0022-1236(75)90030-0.
[8]	A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Four. Grenoble, 50 (2000), 321-362. doi: 10.5802/aif.1757.
[9]	D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92 (1970), 102-163. doi: 10.2307/1970699.
[10]	H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms, Mem. Amer. Math. Soc., 226 (2013), no. 1062. doi: 10.1090/S0065-9266-2013-00676-4.
[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), 443-477. doi: 10.1007/s12220-008-9013-3.
[12]	C. Kenig, G. Ponce and L. Vega, Global solutions for the KdV equation with unbounded data, J. Diff. Equations, 139 (1997), 339-364. doi: 10.1006/jdeq.1997.3297.
[13]	R. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math., 32 (1979), 783-795. doi: 10.1002/cpa.3160320604.
[14]	R. McOwen, Partial Differential Equations: Methods and Applications, 2nd ed, Pearson, 2003.
[15]	R. McOwen and P. Topalov, Asymptotics in shallow water waves, Discrete Contin. Dyn. Syst., 35 (2015), 3103-3131. doi: 10.3934/dcds.2015.35.3103.
[16]	R. McOwen and P. Topalov, Spatial asymptotic expansions in the incompressible Euler equation, arXiv:1606.08059.
[17]	A. Menikoff, The existence of unbounded solutions of the Korteweg-de Vries equation, Comm. Pure Appl. Math., 25 (1972), 407-432. doi: 10.1002/cpa.3160250404.
[18]	P. Michor and D. Mumford, A zoo of diffeomorphisms groups on $\mathbb{R}^{N}$, Ann. Glob. Anal. Geom., 44, 529-540. doi: 10.1007/s10455-013-9380-2.
[19]	G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Visaro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7.
[20]	D. Montgomery, On continuity in topological groups, Bull. Amer. Math. Soc., 42 (1936), 879-882. doi: 10.1090/S0002-9904-1936-06456-6.

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