On the Cauchy problem of a weakly dissipative μ-Hunter–Saxton equation
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 267-279.

In this paper, we study the Cauchy problem of a weakly dissipative μ-Hunter–Saxton equation. We first establish the local well-posedness for the weakly dissipative μ-Hunter–Saxton equation by Kato's semigroup theory. Then, we derive the precise blow-up scenario for strong solutions to the equation. Moreover, we present some blow-up results for strong solutions to the equation. Finally, we give two global existence results to the equation.

DOI : 10.1016/j.anihpc.2013.02.008
Classification : 35Q35, 35G25, 58D05
Mots-clés : A weakly dissipative μ-Hunter–Saxton, Blow-up scenario, Blow-up, Strong solutions, Global existence 
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     title = {On the {Cauchy} problem of a weakly dissipative {\protect\emph{\ensuremath{\mu}}-Hunter{\textendash}Saxton} equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {267--279},
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Liu, Jingjing; Yin, Zhaoyang. On the Cauchy problem of a weakly dissipative μ-Hunter–Saxton equation. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 267-279. doi : 10.1016/j.anihpc.2013.02.008. http://www.numdam.org/articles/10.1016/j.anihpc.2013.02.008/

[1] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa–Holm equation, Nonlinear Anal. 46 (2001), 309-327 | MR | Zbl

[2] R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664 | MR | Zbl

[3] A. Constantin, On the inverse spectral problem for the Camassa–Holm equation, J. Funct. Anal. 155 (1998), 352-363 | MR | Zbl

[4] A. Constantin, On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci. 10 (2000), 391-399 | MR | Zbl

[5] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), 523-535 | MR | Zbl

[6] A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998), 229-243 | MR | Zbl

[7] A. Constantin, R.S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res. 40 (2008), 175-211 | MR | Zbl

[8] A. Constantin, B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A 35 (2002), R51-R79 | MR | Zbl

[9] A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2009), 165-186 | MR | Zbl

[10] J. Escher, M. Kohlmann, B. Kolev, Geometric aspects of the periodic μ-Degasperis–Procesi equation, Progr. Nonlinear Differential Equations Appl. 80 (2011), 193-209 | MR | Zbl

[11] J. Escher, S. Wu, Z. Yin, Global existence and blow-up phenomena for a weakly dissipative Degasperis–Procesi equation, Discrete Contin. Dyn. Syst. Ser. B 12 no. 3 (2009), 633-645 | MR | Zbl

[12] A. Fokas, B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981), 47-66 | MR | Zbl

[13] Y. Fu, Y. Liu, C. Qu, On the blow up structure for the generalized periodic Camassa–Holm and Degasperis–Procesi equations, J. Funct. Anal. 262 (2012), 3125-3158 | MR | Zbl

[14] J.M. Ghidaglia, Weakly damped forced Korteweg–de Vries equations behave as finite dimensional dynamical system in the long time, J. Differential Equations 74 (1988), 369-390 | MR | Zbl

[15] G.L. Gui, Y. Liu, M. Zhu, On the wave-breaking phenomena and global existence for the generalized periodic Camassa–Holm equation, Int. Math. Res. Not. IMRN 2012 (2012), 4858-4903, http://dx.doi.org/10.1093/imrn/rnr214 | Zbl

[16] J.K. Hunter, R. Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), 1498-1521 | MR | Zbl

[17] J.K. Hunter, Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation, Phys. D 79 (1994), 361-386 | MR | Zbl

[18] R.S. Johnson, Camassa–Holm, Korteweg–de Vries and related models for water waves, J. Fluid Mech. 455 (2002), 63-82 | MR | Zbl

[19] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral Theory and Differential Equations, Lecture Notes in Math. vol. 448, Springer-Verlag, Berlin (1975), 25-70 | MR

[20] T. Kato, G. Ponce, Commutator estimates and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988), 203-208 | MR

[21] B. Khesin, J. Lenells, G. Misiolek, Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann. 342 (2008), 617-656 | MR | Zbl

[22] M. Kohlmann, Global existence and blow-up for a weakly dissipative μDP equation, Nonlinear Anal. 74 (2011), 4746-4753 | MR | Zbl

[23] B. Kolev, Poisson brackets in hydrodynamics, Discrete Contin. Dyn. Syst. 19 (2007), 555-574 | MR | Zbl

[24] J. Lenells, The Hunter–Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys. 57 (2007), 2049-2064 | MR | Zbl

[25] J. Lenells, G. Misiolek, F. Tiğlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. Math. Phys. 299 (2010), 129-161 | MR | Zbl

[26] Y. Li, P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations 162 (2000), 27-63 | MR | Zbl

[27] P. Olver, P. Rosenau, Tri-Hamiltonian duality between solitons and solitary wave solutions having compact support, Phys. Rev. E (3) 53 (1996), 1900-1906 | MR

[28] E. Ott, R.N. Sudan, Damping of solitary waves, Phys. Fluids 13 (1970), 1432-1434

[29] X. Wei, Z. Yin, Global existence and blow-up phenomena for the periodic Hunter–Saxton equation with weak dissipation, J. Nonlinear Math. Phys. 18 (2011), 139 | MR | Zbl

[30] G.B. Whitham, Linear and Nonlinear Waves, Wiley–Interscience, New York, London, Sydney (1974) | MR | Zbl

[31] S. Wu, Z. Yin, Blow-up and decay of the solution of the weakly dissipative Degasperis–Procesi equation, SIAM J. Math. Anal. 40 no. 2 (2008), 475-490 | MR | Zbl

[32] S. Wu, Z. Yin, Blow-up phenomena and decay for the periodic Degasperis–Procesi equation with weak dissipation, J. Nonlinear Math. Phys. 15 (2008), 28-49 | MR

[33] S. Wu, Z. Yin, Global existence and blow-up phenomena for the weakly dissipative Camassa–Holm equation, J. Differential Equations 246 no. 11 (2009), 4309-4321 | MR | Zbl

[34] Z. Xin, P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math. 53 (2000), 1411-1433 | MR | Zbl

[35] Z. Yin, Well-posedness, global existence and blowup phenomena for an integrable shallow water equation, Discrete Contin. Dyn. Syst. 10 (2004), 393-411 | MR | Zbl

[36] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math. 47 (2003), 649-666 | MR | Zbl

[37] Z. Yin, On the structure of solutions to the periodic Hunter–Saxton equation, SIAM J. Math. Anal. 36 (2004), 272-283 | MR | Zbl

[38] Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl. 283 (2003), 129-139 | MR | Zbl

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