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A model containing both the Camassa-Holm and Degasperis-Procesi equations. (English) Zbl 1202.35231
Summary: A nonlinear dispersive partial differential equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases, is investigated. Although the $$H^{1}$$-norm of the solutions to the nonlinear model does not remain constants, the existence of its weak solutions in lower order Sobolev space $$H^s$$ with $$1<s \leqslant \frac 32$$ is established under the assumptions $$u_{0} \in H^s$$ and $$|| u_{0x}||_{L^{\infty }}< \infty$$. The local well-posedness of solutions for the equation in the Sobolev space $$H^s(R)$$ with $$s > \frac 32$$ is also developed.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35D30 Weak solutions to PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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##### References:
 [1] Bona, J.; Smith, R., The initial value problem for the Korteweg-de Vries equation, Philos. trans. R. soc. lond. ser. A, 278, 555-601, (1975) · Zbl 0306.35027 [2] Bressan, A.; Constantin, A., Global conservative solutions of the Camassa-Holm equation, Arch. ration. mech. anal., 183, 215-239, (2007) · Zbl 1105.76013 [3] Bressan, A.; Constantin, A., Global dissipative solutions of the Camassa-Holm equation, Anal. appl. (singap.), 5, 1, 1-27, (2007) · Zbl 1139.35378 [4] Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. rev. lett., 71, 1661-1664, (1993) · Zbl 0972.35521 [5] Coclite, G.M.; Holden, H.; Karlsen, K.H., Wellposedness for a parabolic-elliptic system, Discrete contin. dyn. syst., 13, 659-682, (2005) · Zbl 1082.35056 [6] Coclite, G.M.; Holden, H.; Karlsen, K.H., Well-posedness of higher-order Camassa-Holm equations, J. differential equations, 246, 929-963, (2009) · Zbl 1186.35003 [7] Coclite, G.M.; Holden, H.; Karlsen, K.H., Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. math. anal., 37, 1044-1069, (2005) · Zbl 1100.35106 [8] Constantin, A., The trajectories of particles in Stokes waves, Invent. math., 166, 523-535, (2006) · Zbl 1108.76013 [9] Constantin, A., Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. inst. Fourier, 50, 321-362, (2000) · Zbl 0944.35062 [10] Constantin, A., On the inverse spectral problem for the Camassa-Holm equation, J. funct. anal., 155, 352-363, (1998) · Zbl 0907.35009 [11] Constantin, A., On the scattering problem for the Camassa-Holm equation, Proc. R. soc. lond., 457, 953-970, (2001) · Zbl 0999.35065 [12] Constantin, A.; Lannes, D., The hydro-dynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. ration. mech. anal., 193, 165-186, (2009) · Zbl 1169.76010 [13] Constantin, A.; Ivanov, R., Poisson structure and action-angle variables for the Camassa-Holm equation, Lett. math. phys., 76, 93-108, (2006) · Zbl 1137.37032 [14] Constantin, A.; Gerdjikov, V.S.; Ivanov, R.I., Inverse scattering transform for the Camassa-Holm equation, Inverse problems, 22, 2197-2207, (2006) · Zbl 1105.37044 [15] Constantin, A.; McKean, H.P., A shallow water equation on the circle, Comm. pure appl. math., 52, 949-982, (1999) · Zbl 0940.35177 [16] Constantin, A.; Escher, J., Particle trajectories in solitary water waves, Bull. amer. math. soc., 44, 423-431, (2007) · Zbl 1126.76012 [17] 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 [18] Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta math., 181, 229-243, (1998) · Zbl 0923.76025 [19] Constantin, A.; Kolev, B., Geodesic flow on the diffeomorphism group of the circle, Comment. math. helv., 78, 787-804, (2003) · Zbl 1037.37032 [20] 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 [21] Constantin, A.; Strauss, W., Stability of peakons, Comm. pure appl. math., 53, 603-610, (2000) · Zbl 1049.35149 [22] Constantin, A.; Strauss, W., Stability of the Camassa-Holm solitons, J. nonlinear sci., 12, 415-422, (2002) · Zbl 1022.35053 [23] Christov, O.; Hakkaev, S., On the Cauchy problem for the periodic b-family of equations and of non-uniform continuity of Degasperis-Procesi equation, J. math. anal. appl., 360, 47-56, (2009) · Zbl 1178.35081 [24] Danchin, R., A note on well-posedness for Camassa-Holm equation, J. differential equations, 192, 429-444, (2003) · Zbl 1048.35076 [25] Degasperis, A.; Holm, D.; Hone, A., A new integral equation with peakon solutions, Theoret. and math. phys., 133, 1461-1472, (2002) [26] Degasperis, A.; Procesi, M., Asymptotic integrability, (), 23-37 · Zbl 0963.35167 [27] Fokas, A.; Fuchssteiner, B., Symplectic structures, their backlund transformation and hereditary symmetries, Phys. D, 4, 821-831, (1981) · Zbl 1194.37114 [28] Himonas, A.A.; Misioek, G.; Ponce, G.; Zhou, Y., Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. math. phys., 271, 511-522, (2007) · Zbl 1142.35078 [29] Johnson, R.S., Camassa-Holm, Korteweg de Vries and related models for water waves, J. fluid mech., 455, 63-82, (2002) · Zbl 1037.76006 [30] Johnson, R.S., On solutions of the Camassa-Holm equation, Proc. R. soc. lond., 459, 1687-1708, (2003) · Zbl 1039.76006 [31] Kato, T., Quasi-linear equations of evolution with applications to partial differential equations, (), 25-70 [32] Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. pure appl. math., 41, 891-907, (1988) · Zbl 0671.35066 [33] Kouranbaeva, S., The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. math. phys., 40, 857-868, (1999) · Zbl 0958.37060 [34] Lai, S.; Xu, Y., The compact and noncompact structures for two types of generalized Camassa-Holm-KP equations, Math. comput. modelling, 47, 1089-1098, (2008) · Zbl 1145.35448 [35] Lenells, J., Traveling wave solutions of the Camassa-Holm equation, J. differential equations, 217, 393-430, (2005) · Zbl 1082.35127 [36] Lenells, J., Conservation laws of the Camassa-Holm equation, J. phys. A, 38, 869-880, (2005) · Zbl 1076.35100 [37] Lenells, J., Infinite propagation speed of the Camassa-Holm equation, J. math. anal. appl., 325, 1468-1478, (2007) · Zbl 1160.35323 [38] Tian, L.; Gui, G.; Guo, B., The limit behavior of the solutions to a class of nonlinear dispersive wave equations, J. math. anal. appl., 341, 1311-1333, (2008) · Zbl 1137.35064 [39] 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 [40] Li, Y.; Zhang, J., The multiple-soliton solutions of the Camassa-Holm equation, Proc. R. soc. lond., 460, 2617-2627, (2004) · Zbl 1068.35109 [41] Li, L.C., Long time behaviour for a class of low-regularity solutions of the Camassa-Holm equation, Comm. math. phys., 285, 265-291, (2009) · Zbl 1228.35205 [42] Liu, Y.; Yin, Z.Y., Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. math. phys., 267, 801-820, (2006) · Zbl 1131.35074 [43] Matsuno, Y., Parametric representation for the multisoliton solution of the Camassa-Holm equation, J. phys. soc. Japan, 74, 1983-1987, (2005) · Zbl 1076.35102 [44] Matsuno, Y., The N-soliton solution of the Degasperis-Procesi equation, Inverse problems, 21, 2085-2101, (2005) · Zbl 1112.37072 [45] Mckean, H.P., Fredholm determinants and the Camassa-Holm hierarchy, Comm. pure appl. math., 56, 638-680, (2003) · Zbl 1047.37047 [46] Mclachlan, R.; Zhang, X.Y., Well-posedness of modified Camassa-Holm equations, J. differential equations, 246, 3241-3259, (2009) · Zbl 1203.35079 [47] 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 [48] Mustafa, O.G., Existence and uniqueness of low regularity solutions for the Dullin-Gottwald-Holm equation, Comm. math. phys., 265, 189-200, (2006) · Zbl 1106.35084 [49] Ohta, Y.; Maruno, K.I.; Feng, B.F., An integrable semi-discretization of Camassa-Holm equation and its determinant solution, J. phys. A: math. theoret., 41, 1-30, (2008) · Zbl 1180.76011 [50] Parker, A., On the Camassa-Holm equation and a direct method of solution: I. bilinear form and solitary waves, Proc. R. soc. lond., 460, 2929-2957, (2004) · Zbl 1068.35110 [51] Parker, A., On the Camassa-Holm equation and a direct method of solution: II. soliton solutions, Proc. R. soc. lond., 461, 3611-3632, (2005) · Zbl 1370.35236 [52] Parker, A., On the Camassa-Holm equation and a direct method of solution: III. N-soliton solutions, Proc. R. soc. lond., 461, 3893-3911, (2005) · Zbl 1370.35237 [53] Parkes, E.; Vakhnenko, V., Explicit solutions of the Camassa-Holm equation, Chaos solitons fractals, 26, 1309-1316, (2005) · Zbl 1072.35156 [54] Toland, J.F., Stokes waves, Topol. methods nonlinear anal., 7, 1-48, (1996) · Zbl 0897.35067 [55] Walter, W., Differential and integral inequalities, (1970), Springer-Verlag New York [56] Wu, S.; Yin, Z.Y., 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 [57] Wu, S.; Yin, Z.Y., Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation, J. differential equations, 246, 4309-4321, (2009) · Zbl 1195.35072 [58] Xin, Z.; Zhang, P., On the weak solutions to a shallow water equation, Comm. pure appl. math., 53, 1411-1433, (2000) · Zbl 1048.35092 [59] Xin, Z.; Zhang, P., On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. partial differential equations, 27, 1815-1844, (2002) · Zbl 1034.35115 [60] Xin, Z., Well-posedness, blowup, and global existence for an intergrable shallow water equation, Discrete contin. dyn. syst., 11, 393-411, (2004) [61] Yin, Z.Y., On the Cauchy problem for an intergrable equation with peakon solutions, Illinois J. math., 47, 649-666, (2003) · Zbl 1061.35142 [62] Yin, Z.Y., Global weak solutions for a new periodic integrable equation with peakon solutions, J. funct. anal., 212, 182-194, (2004) · Zbl 1059.35149
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