zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations. (English) Zbl 1234.35222
Summary: Considered herein are the generalized Camassa-Holm and Degasperis-Procesi equations in the spatially periodic setting. The precise blow-up scenarios of strong solutions are derived for both of equations. Several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established. The exact blow-up rates are also determined. Finally, geometric descriptions of these two integrable equations from non-stretching invariant curve flows in centro-equiaffine geometries, pseudo-spherical surfaces and affine surfaces are given.
35Q53KdV-like (Korteweg-de Vries) equations
35Q35PDEs in connection with fluid mechanics
35B44Blow-up (PDE)
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35D35Strong solutions of PDE
[1]Arnold, V. I.: Sur la géométrie différentielle des groupes de Lie de dimenson infinite er ses application à l’hydrodynamique des fluids parfaits, Ann. inst. Fourier (Grenoble) 16, 319-361 (1966) · Zbl 0148.45301 · doi:10.5802/aif.233 · doi:numdam:AIF_1966__16_1_319_0
[2]Bressan, A.; Constantin, A.: Global solutions of the Hunter-Saxton equation, SIAM J. Math. anal. 37, 996-1026 (2005) · Zbl 1108.35024 · doi:10.1137/050623036
[3]Buttazo, G.; Giaquina, M.; Hildebrandt, S.: One-dimensional variational problems: an introduction, (1998)
[4]Camassa, R.; Holm, D. D.: An integrable shallow water equation with peaked solitons, Phys. rev. Lett. 71, 1661-1664 (1993) · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[5]Chern, S. S.; Tenenblat, K.: Pseudo-spherical surfaces and evolution equations, Stud. appl. Math. 74, 55-83 (1986) · Zbl 0605.35080
[6]Chern, S. S.; Terng, C. L.: An analogue of Bäcklund theorem in affine geometry, Rocky mountain J. Math. 10, 105-124 (1980) · Zbl 0407.53002 · doi:10.1216/RMJ-1980-10-1-105
[7]Chou, K. S.; Qu, C. Z.: Integrable equations arising from motions of plane curves I, Phys. D 162, 9-33 (2002) · Zbl 0987.35139 · doi:10.1016/S0167-2789(01)00364-5
[8]Chou, K. S.; Qu, C. Z.: Integrable equations arising from motions of plane curves II, J. nonlinear sci. 13, 487-517 (2003) · Zbl 1045.35063 · doi:10.1007/s00332-003-0570-0
[9]Coclite, G. M.; Karlsen, K. H.: On the well-posedness of the Degasperis-Procesi equation, J. funct. Anal. 233, 60-91 (2006) · Zbl 1090.35142 · doi:10.1016/j.jfa.2005.07.008
[10]Constantin, A.: On the Cauchy problem for the periodic Camassa-Holm equation, J. differential equations 141, 218-235 (1997) · Zbl 0889.35022 · doi:10.1006/jdeq.1997.3333
[11]Constantin, A.: On the blow-up of solutions of a periodic shallow water equation, J. nonlinear sci. 10, 391-399 (2000) · Zbl 0960.35083 · doi:10.1007/s003329910017
[12]Constantin, A.; Escher, J.: Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. pure appl. Math. 51, 475-504 (1998) · Zbl 0934.35153 · doi:10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
[13]Constantin, A.; Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations, Acta math. 181, 229-243 (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586
[14]Constantin, A.; Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z. 233, 75-91 (2000) · Zbl 0954.35136 · doi:10.1007/PL00004793
[15]Constantin, A.; Lannes, D.: The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. ration. Mech. anal. 192, 165-186 (2009) · Zbl 1169.76010 · doi:10.1007/s00205-008-0128-2
[16]Constantin, A.; Mckean, H. P.: A shallow water equation on the circle, Comm. pure appl. Math. 52, 949-982 (1999) · Zbl 0940.35177 · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[17]Degasperis, A.; Procesi, M.: Asymptotic integrability, , 23-37 (1999) · Zbl 0963.35167
[18]Degasperis, A.; Holm, D. D.; Hone, A. N. W.: Integrable and non-integrable equations with peakons, , 37-43 (2003) · Zbl 1053.37039
[19]J. Escher, B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, preprint, arXiv:0908.0508v1.
[20]J. Escher, M. Kohlmann, B. Kolev, Geometric aspects of the periodic μDP equation, preprint, arXiv:1004.0978v1.
[21]Escher, J.; Liu, Y.; Yin, Z.: Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. funct. Anal. 241, 457-485 (2006) · Zbl 1126.35053 · doi:10.1016/j.jfa.2006.03.022
[22]Escher, J.; Liu, Y.; Yin, Z.: Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana univ. Math. J. 56, 87-117 (2007) · Zbl 1124.35041 · doi:10.1512/iumj.2007.56.3040
[23]Fuchssteiner, B.; Fokas, A. S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4, 47-66 (1981/1982) · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[24]Goldstein, R. E.; Petrich, D. M.: The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane, Phys. rev. Lett. 67, 3203-3206 (1991) · Zbl 0990.37519 · doi:10.1103/PhysRevLett.67.3203
[25]Hasimoto, H.: A soliton on a vortex filament, J. fluid mech. 51, 477-485 (1972) · Zbl 0237.76010 · doi:10.1017/S0022112072002307
[26]Hunter, J. K.; Saxton, R.: Dynamics of director fields, SIAM J. Appl. math. 51, 1498-1521 (1991) · Zbl 0761.35063 · doi:10.1137/0151075
[27]Hunter, J. K.; Zheng, Y.: On a completely integrable hyperbolic variational equation, Phys. D 79, 361-386 (1994) · Zbl 0900.35387
[28]Kato, T.; Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations, Comm. pure appl. Math. 41, 891-907 (1988) · Zbl 0671.35066 · doi:10.1002/cpa.3160410704
[29]Khesin, B.; Lenells, J.; Misiolek, G.: Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. ann. 342, 617-656 (2008) · Zbl 1156.35082 · doi:10.1007/s00208-008-0250-3
[30]Khesin, B.; Misiolek, G.: Euler equations on homogeneous spaces and Virasoro orbits, Adv. math. 176, 116-144 (2003) · Zbl 1017.37039 · doi:10.1016/S0001-8708(02)00063-4
[31]Kouranbaeva, S.: The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. math. Phys. 40, 857-868 (1999) · Zbl 0958.37060 · doi:10.1063/1.532690
[32]Lenells, J.: The Hunter-Saxton equation describes the geodesic flow on a sphere, J. geom. Phys. 57, 2049-2064 (2007) · Zbl 1125.35085 · doi:10.1016/j.geomphys.2007.05.003
[33]Lenells, J.; Misiolek, G.; Tiğlay, F.: Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. math. Phys. 299, 129-161 (2010) · Zbl 1214.35059 · doi:10.1007/s00220-010-1069-9
[34]Liu, Y.; Yin, Z.: Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. math. Phys. 267, 801-820 (2006) · Zbl 1131.35074 · doi:10.1007/s00220-006-0082-5
[35]Lundmark, H.: Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. nonlinear sci. 17, 169-198 (2007) · Zbl 1185.35194 · doi:10.1007/s00332-006-0803-3
[36]Misiolek, G.: A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. geom. Phys. 24, 203-208 (1998) · Zbl 0901.58022 · doi:10.1016/S0393-0440(97)00010-7
[37]Misiolek, G.: Classical solutions of the periodic Camassa-Holm equation, Geom. funct. Anal. 12, 1080-1104 (2002) · Zbl 1158.37311 · doi:10.1007/PL00012648
[38]Reyes, E. G.: Geometric integrability of the Camassa-Holm equation, Lett. math. Phys. 59, 17-131 (2002) · Zbl 0997.35081 · doi:10.1023/A:1014933316169
[39]Whitham, G. B.: Linear and nonlinear waves, (1974) · Zbl 0373.76001
[40]Yin, Z.: On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math. 47, 649-666 (2003) · Zbl 1061.35142 · doi:http://www.math.uiuc.edu/~hildebr/ijm/fall03/final/yin.html
[41]Yin, Z.: On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. anal. 36, 272-283 (2004) · Zbl 1151.35321 · doi:10.1137/S0036141003425672