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)
Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation. (English) Zbl 1195.35072
This paper investigates the Cauchy problem on the real line for the weakly dissipative Camassa-Holm (wdCH) equation $$u_t+3uu_x-u_{xxt}+\lambda(u-u_{xx})=2u_xu_{xx}+uu_{xxx}, \tag1$$ where $\lambda>0$ is a constant. The non-dissipative limiting case $\lambda\rightarrow0$ of (1) recovers the celebrated Camassa-Holm (CH) equation. As is noted therein, the wdCH equation admits solutions whose properties are markedly different to those of the CH equation. Significantly, and as a consequence of the extra dissipative term, (1) has no travelling-wave solutions. The main achievement of this work is to obtain a global existence result for (strong) solutions of the wCH equation having certain profiles. The authors demonstrate that, following an approach due to {\it A. Constantin} [Ann. Inst. Fourier 50, No. 2, 321--362 (2000; Zbl 0944.35062)], solutions of (1) either exist globally in time or blow up in finite time for a wide class of initial profiles. The work begins by recapping some known results on the well-posedness and blow-up scenarios for the wdCH equation. By first deriving a useful $L^\infty$-norm for strong solutions of (1), the authors proceed to establish a novel global existence theorem for these solutions. To conclude the study, they also prove a new blow-up result (for strong solutions) and determine a breaking point where the gradient of the solution becomes infinite. The results of this study extend and improve on those previously reported for the wdCH equation. This study will be of interest to those working on nonlinear wave equations -- especially shallow-water wave equations like the CH equation -- and who are concerned with the analytic and dynamic character of their solutions.

35B44Blow-up (PDE)
35G25Initial value problems for nonlinear higher-order PDE
35Q35PDEs in connection with fluid mechanics
Full Text: DOI
[1] Beals, R.; Sattinger, D.; Szmigielski, J.: Acoustic scattering and the extended Korteweg -- de Vries hierarchy, Adv. math. 140, 190-206 (1998) · Zbl 0919.35118 · doi:10.1006/aima.1998.1768
[2] Beals, R.; Sattinger, D.; Szmigielski, J.: Multipeakons and a theorem of Stieltjes, Inverse problems 15, 1-4 (1999) · Zbl 0923.35154 · doi:10.1088/0266-5611/15/1/001
[3] Bressan, A.; Constantin, A.: Global conservative solutions of the Camassa -- Holm equation, Arch. ration. Mech. anal. 183, 215-239 (2007) · Zbl 1105.76013 · doi:10.1007/s00205-006-0010-z
[4] Bressan, A.; Constantin, A.: Global dissipative solutions of the Camassa -- Holm equation, Anal. appl. (Singap.) 5, 1-27 (2007) · Zbl 1139.35378 · doi:10.1142/S0219530507000857
[5] Camassa, R.; Holm, 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
[6] Camassa, R.; Holm, D.; Hyman, J.: A new integrable shallow water equation, Adv. appl. Mech. 31, 1-33 (1994) · Zbl 0808.76011
[7] Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. inst. Fourier (Grenoble) 50, 321-362 (2000) · Zbl 0944.35062 · doi:10.5802/aif.1757 · numdam:AIF_2000__50_2_321_0
[8] Constantin, A.: On the scattering problem for the Camassa -- Holm equation, Proc. R. Soc. lond. Ser. A 457, 953-970 (2001) · Zbl 0999.35065 · doi:10.1098/rspa.2000.0701
[9] Constantin, A.: The trajectories of particles in Stokes waves, Invent. math. 166, 523-535 (2006) · Zbl 1108.76013 · doi:10.1007/s00222-006-0002-5
[10] Constantin, A.; Escher, J.: Global existence and blow-up for a shallow water equation, Ann. sc. Norm. super. Pisa cl. Sci. (4) 26, 303-328 (1998) · Zbl 0918.35005 · numdam:ASNSP_1998_4_26_2_303_0
[11] Constantin, A.; Escher, J.: Wave breaking for nonlinear nonlocal shallow water equation, Acta math. 181, 229-243 (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586
[12] Constantin, A.; Escher, J.: Well-posedness, global existence, and blowup 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.: 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
[14] Constantin, A.; Escher, J.: Particle trajectories in solitary water waves, Bull. amer. Math. soc. 44, 423-431 (2007) · Zbl 1126.76012 · doi:10.1090/S0273-0979-07-01159-7
[15] Constantin, A.; Gerdjikov, V.; Ivanov, R.: Inverse scattering transform for the Camassa -- Holm equation, Inverse problems 22, 2197-2207 (2006) · Zbl 1105.37044 · doi:10.1088/0266-5611/22/6/017
[16] Constantin, A.; Kappeler, T.; Kolev, B.; Topalov, P.: On geodesic exponential maps of the Virasoro group, Ann. global anal. Geom. 31, 155-180 (2007) · Zbl 1121.35111 · doi:10.1007/s10455-006-9042-8
[17] Constantin, A.; Kolev, B.: Geodesic flow on the diffeomorphism group of the circle, Comment. math. Helv. 78, 787-804 (2003) · Zbl 1037.37032 · doi:10.1007/s00014-003-0785-6
[18] Constantin, A.; Molinet, L.: Global weak solutions for a shallow water equation, Comm. math. Phys. 211, 45-61 (2000) · Zbl 1002.35101 · doi:10.1007/s002200050801
[19] Constantin, A.; Strauss, W. A.: Stability of peakons, Comm. pure appl. Math. 53, 603-610 (2000) · Zbl 1049.35149 · doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
[20] Constantin, A.; Strauss, W. A.: Stability of the Camassa -- Holm solitons, J. nonlinear sci. 12, 415-422 (2002) · Zbl 1022.35053 · doi:10.1007/s00332-002-0517-x
[21] Escher, J.; Yin, Z.: Initial boundary value problems of the Camassa -- Holm equation, Comm. partial differential equations 33, 377-395 (2008) · Zbl 1145.35031 · doi:10.1080/03605300701318872
[22] Escher, J.; Yin, Z.: Initial boundary value problems for nonlinear dispersive wave equations, J. funct. Anal. 256, 479-508 (2009) · Zbl 1193.35108 · doi:10.1016/j.jfa.2008.07.010
[23] Fokas, A.; Fuchssteiner, B.: Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D 4, 47-66 (1981) · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[24] Fuchssteiner, B.: Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa -- Holm equation, Phys. D 95, 229-243 (1996) · Zbl 0900.35345 · doi:10.1016/0167-2789(96)00048-6
[25] Ghidaglia, J. M.: Weakly damped forced Korteweg -- de Vries equations behave as a finite dimensional dynamical system in the long time, J. differential equations 74, 369-390 (1988) · Zbl 0668.35084 · doi:10.1016/0022-0396(88)90010-1
[26] Ionescu-Kruse, D.: Variational derivation of the Camassa -- Holm shallow water equation, J. nonlinear math. Phys. 14, 303-312 (2007) · Zbl 1157.76005 · doi:10.2991/jnmp.2007.14.3.1
[27] Ivanov, R. I.: Water waves and integrability, Philos. trans. R. soc. Lond. ser. A 365, 2267-2280 (2007) · Zbl 1152.76322 · doi:10.1098/rsta.2007.2007
[28] Johnson, R. S.: Camassa -- Holm, Korteweg -- de Vries and related models for water waves, J. fluid mech. 455, 63-82 (2002) · Zbl 1037.76006 · doi:10.1017/S0022112001007224
[29] Li, Y.; Olver, P.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. differential equations 162, 27-63 (2000) · Zbl 0958.35119 · doi:10.1006/jdeq.1999.3683
[30] Ott, E.; Sudan, R. N.: Damping of solitary waves, Phys. fluids 13, 1432-1434 (1970)
[31] Rodriguez-Blanco, G.: On the Cauchy problem for the Camassa -- Holm equation, Nonlinear anal. 46, 309-327 (2001) · Zbl 0980.35150 · doi:10.1016/S0362-546X(01)00791-X
[32] Wu, S.; Yin, Z.: Blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa -- Holm equation, J. math. Phys. 47, 1-12 (2006) · Zbl 1111.35067 · doi:10.1063/1.2158437
[33] Wu, S.; Yin, Z.: Blowup and decay of solution to the weakly dissipative Camassa -- Holm equation, Acta math. Appl. sin. 30, 996-1003 (2007) · Zbl 1164.35482
[34] Xin, Z.; Zhang, P.: On the weak solutions to a shallow water equation, Comm. pure appl. Math. 53, 1411-1433 (2000) · Zbl 1048.35092 · doi:10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5
[35] Yin, Z.: Well-posedness, blowup and global existence for an integrable shallow water equation, Discrete contin. Dyn. syst. 11, 393-411 (2004) · Zbl 1061.35123 · doi:10.3934/dcds.2004.11.393
[36] Yin, Z.: Well-posedness, global solutions and blowup phenomena for a nonlinearly dispersive wave equation, J. evol. Equ. 4, 391-419 (2004) · Zbl 1073.35064 · doi:10.1007/s00028-004-0166-7