American Institute of Mathematical Sciences

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

Groups of asymptotic diffeomorphisms

 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].
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
 [1] Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065 [2] Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047 [3] Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883 [4] Milena Stanislavova, Atanas Stefanov. Attractors for the viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 159-186. doi: 10.3934/dcds.2007.18.159 [5] Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871 [6] Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067 [7] 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 [8] Stephen C. Anco, Elena Recio, María L. Gandarias, María S. Bruzón. A nonlinear generalization of the Camassa-Holm equation with peakon solutions. Conference Publications, 2015, 2015 (special) : 29-37. doi: 10.3934/proc.2015.0029 [9] Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230 [10] Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305 [11] Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713 [12] Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026 [13] Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181 [14] Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483 [15] Priscila Leal da Silva, Igor Leite Freire. An equation unifying both Camassa-Holm and Novikov equations. Conference Publications, 2015, 2015 (special) : 304-311. doi: 10.3934/proc.2015.0304 [16] Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194 [17] David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629 [18] Shaoyong Lai, Qichang Xie, Yunxi Guo, YongHong Wu. The existence of weak solutions for a generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 45-57. doi: 10.3934/cpaa.2011.10.45 [19] Alberto Bressan, Geng Chen, Qingtian Zhang. Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 25-42. doi: 10.3934/dcds.2015.35.25 [20] Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139

2018 Impact Factor: 1.143