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.

Classification: 35Q35, 35G25, 58D05

Keywords: A weakly dissipative μ-Hunter–Saxton, Blow-up scenario, Blow-up, Strong solutions, Global existence

@article{AIHPC_2014__31_2_267_0, author = {Liu, Jingjing and Yin, Zhaoyang}, title = {On the Cauchy problem of a weakly dissipative $\mu$-Hunter--Saxton equation}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, publisher = {Elsevier}, volume = {31}, number = {2}, year = {2014}, pages = {267-279}, doi = {10.1016/j.anihpc.2013.02.008}, zbl = {1302.35320}, mrnumber = {3181669}, language = {en}, url = {http://www.numdam.org/item/AIHPC_2014__31_2_267_0} }

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, Volume 31 (2014) no. 2, pp. 267-279. doi : 10.1016/j.anihpc.2013.02.008. http://www.numdam.org/item/AIHPC_2014__31_2_267_0/

[1] On the Cauchy problem for the Camassa–Holm equation, Nonlinear Anal. 46 (2001), 309-327 | MR 1851854 | Zbl 0980.35150

,[2] An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664 | MR 1234453 | Zbl 0972.35521

, ,[3] On the inverse spectral problem for the Camassa–Holm equation, J. Funct. Anal. 155 (1998), 352-363 | MR 1624553 | Zbl 0907.35009

,[4] On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci. 10 (2000), 391-399 | MR 1752603 | Zbl 0960.35083

,[5] The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), 523-535 | MR 2257390 | Zbl 1108.76013

,[6] Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998), 229-243 | MR 1668586 | Zbl 0923.76025

, ,[7] Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res. 40 (2008), 175-211 | MR 2369543 | Zbl 1135.76007

, ,[8] On the geometric approach to the motion of inertial mechanical systems, J. Phys. A 35 (2002), R51-R79 | MR 1930889 | Zbl 1039.37068

, ,[9] The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2009), 165-186 | MR 2481064 | Zbl 1169.76010

, ,[10] Geometric aspects of the periodic μ-Degasperis–Procesi equation, Progr. Nonlinear Differential Equations Appl. 80 (2011), 193-209 | MR 3052578 | Zbl 1250.58005

, , ,[11] 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 2525161 | Zbl 1225.35035

, , ,[12] Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981), 47-66 | MR 636470 | Zbl 1194.37114

, ,[13] On the blow up structure for the generalized periodic Camassa–Holm and Degasperis–Procesi equations, J. Funct. Anal. 262 (2012), 3125-3158 | MR 2885950 | Zbl 1234.35222

, , ,[14] 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 952903 | Zbl 0668.35084

,[15] 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 1252.35240

, , ,[16] Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), 1498-1521 | MR 1135995 | Zbl 0761.35063

, ,[17] On a completely integrable nonlinear hyperbolic variational equation, Phys. D 79 (1994), 361-386 | MR 1306466 | Zbl 0900.35387

, ,[18] Camassa–Holm, Korteweg–de Vries and related models for water waves, J. Fluid Mech. 455 (2002), 63-82 | MR 1894796 | Zbl 1037.76006

,[19] 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 407477

,[20] Commutator estimates and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988), 203-208 | MR 951744

, ,[21] Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann. 342 (2008), 617-656 | MR 2430993 | Zbl 1156.35082

, , ,[22] Global existence and blow-up for a weakly dissipative μDP equation, Nonlinear Anal. 74 (2011), 4746-4753 | MR 2810714 | Zbl 1220.35152

,[23] Poisson brackets in hydrodynamics, Discrete Contin. Dyn. Syst. 19 (2007), 555-574 | MR 2335765 | Zbl 1139.53040

,[24] The Hunter–Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys. 57 (2007), 2049-2064 | MR 2348278 | Zbl 1125.35085

,[25] Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. Math. Phys. 299 (2010), 129-161 | MR 2672800 | Zbl 1214.35059

, , ,[26] Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations 162 (2000), 27-63 | MR 1741872 | Zbl 0958.35119

, ,[27] Tri-Hamiltonian duality between solitons and solitary wave solutions having compact support, Phys. Rev. E (3) 53 (1996), 1900-1906 | MR 1401317

, ,[28] Damping of solitary waves, Phys. Fluids 13 (1970), 1432-1434

, ,[29] Global existence and blow-up phenomena for the periodic Hunter–Saxton equation with weak dissipation, J. Nonlinear Math. Phys. 18 (2011), 139 | MR 2786940 | Zbl 1215.35101

, ,[30] Linear and Nonlinear Waves, Wiley–Interscience, New York, London, Sydney (1974) | MR 483954 | Zbl 0373.76001

,[31] 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 2438773 | Zbl 1216.35126

, ,[32] Blow-up phenomena and decay for the periodic Degasperis–Procesi equation with weak dissipation, J. Nonlinear Math. Phys. 15 (2008), 28-49 | MR 2434723

, ,[33] Global existence and blow-up phenomena for the weakly dissipative Camassa–Holm equation, J. Differential Equations 246 no. 11 (2009), 4309-4321 | MR 2517772 | Zbl 1195.35072

, ,[34] On the weak solutions to a shallow water equation, Comm. Pure Appl. Math. 53 (2000), 1411-1433 | MR 1773414 | Zbl 1048.35092

, ,[35] Well-posedness, global existence and blowup phenomena for an integrable shallow water equation, Discrete Contin. Dyn. Syst. 10 (2004), 393-411 | MR 2083424 | Zbl 1061.35123

,[36] On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math. 47 (2003), 649-666 | MR 2007229 | Zbl 1061.35142

,[37] On the structure of solutions to the periodic Hunter–Saxton equation, SIAM J. Math. Anal. 36 (2004), 272-283 | MR 2083862 | Zbl 1151.35321

,[38] Global existence for a new periodic integrable equation, J. Math. Anal. Appl. 283 (2003), 129-139 | MR 1994177 | Zbl 1033.35121

,