American Institute of Mathematical Sciences

March  2015, 20(2): 641-673. doi: 10.3934/dcdsb.2015.20.641

Global estimates and blow-up criteria for the generalized Hunter-Saxton system

 1 Department of Mathematics, University of Colorado, Boulder, CO 80309-0395

Received  July 2013 Revised  November 2014 Published  January 2015

The generalized, two-component Hunter-Saxton system comprises several well-known models of fluid dynamics and serves as a tool for the study of one-dimensional fluid convection and stretching. In this article a general representation formula for periodic solutions to the system, which is valid for arbitrary values of parameters $(\lambda,\kappa) \in \mathbb{R} \times \mathbb{R}$, is derived. This allows us to examine in great detail qualitative properties of blow-up as well as the asymptotic behaviour of solutions, including convergence to steady states in finite or infinite time.
Citation: Alejandro Sarria. Global estimates and blow-up criteria for the generalized Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 641-673. doi: 10.3934/dcdsb.2015.20.641
References:
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Titi, Global well-posedness of the three-dimensional stratified primitive equations with partial vertical mixing turbulence diffusion,, Comm. Math. Phys., 310 (2012), 537. doi: 10.1007/s00220-011-1409-4. Google Scholar [7] C. Cao, S. Ibrahim, K. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics,, preprint, (). Google Scholar [8] D. Chae, On the blow-up problem for the axisymmetric 3D Euler equations,, Nonlinearity, 21 (2008), 2053. doi: 10.1088/0951-7715/21/9/007. Google Scholar [9] X. Chen and H. Okamoto, Global existence of solutions to the Proudman-Johnson equation,, Proc.Japan Acad., 76 (2000), 149. doi: 10.3792/pjaa.76.149. Google Scholar [10] X. Chen and H. Okamoto, Global existence of solutions to the generalized Proudman-Johnson equation,, Proc.Japan Acad., 78 (2002), 136. doi: 10.3792/pjaa.78.136. Google Scholar [11] S. Childress, G. R. Ierley, E. A. Spiegel and W. R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes equations having stagnation-point form,, J. Fluid Mech., 203 (1989), 1. doi: 10.1017/S0022112089001357. Google Scholar [12] C. H. Cho and M. Wunsch, Global and singular solutions to the generalized Proudman-Johnson equation,, J. Diff. Eqns., 249 (2010), 392. doi: 10.1016/j.jde.2010.03.013. Google Scholar [13] C. H. Cho and M. Wunsch, Global weak solutions to the generalized Proudman-Johnson equation,, Commun. Pure Appl. Ana., 11 (2012), 1387. doi: 10.3934/cpaa.2012.11.1387. Google Scholar [14] A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Physics Letters A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar [15] A. Constantin and M. Wunsch, On the inviscid Proudman-Johnson equation,, Proc. Japan Acad. Ser. A Math. Sci., 85 (2009), 81. doi: 10.3792/pjaa.85.81. Google Scholar [16] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi Equations,, Arch. Rational Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar [17] P. Constantin, P. D. Lax and A. Majda, A simple one dimensional model for the three dimensional vorticity equation,, Commun. Pure Appl. Math., 38 (1985), 715. doi: 10.1002/cpa.3160380605. Google Scholar [18] C. M. Dafermos, Generalized characteristics and the Hunter-Saxton equation,, J. Hyperbol. Differ. Eq., 8 (2011), 159. doi: 10.1142/S0219891611002366. Google Scholar [19] H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,, Fluid Dyn. Res., 33 (2003), 73. doi: 10.1016/S0169-5983(03)00046-7. Google Scholar [20] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. I,, McGraw-Hill, (1981), 56. Google Scholar [21] V. P. Ermakov, Univ. Izv. Kiev 20,, 1 (1880)., (1880). Google Scholar [22] J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow up phenomena for the 2-component Camassa-Holm equations,, Discrete Contin. Dyn. Syst., 19 (2007), 493. doi: 10.3934/dcds.2007.19.493. Google Scholar [23] T. W. Gamelin, Complex Analysis,, Undergraduate Texts in Mathematics, (2001). doi: 10.1007/978-0-387-21607-2. Google Scholar [24] G. Gasper and M. Rahman, Basic Hypergeometric Series,, Encyclopedia of Mathematics and Its Applications, (2004). doi: 10.1017/CBO9780511526251. Google Scholar [25] A. E. Gill, Atmosphere-Ocean Dynamics,, Academic Press, (1982). Google Scholar [26] A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth,, J. Fluid Mech., 78 (1976), 237. doi: 10.1017/S0022112076002425. Google Scholar [27] C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system,, J. Differential Equations, 248 (2010), 2003. doi: 10.1016/j.jde.2009.08.002. Google Scholar [28] Z. Guo, Blow up and global solutions to a new integrable model with two components,, J. Math. Anal. Appl., 372 (2010), 316. doi: 10.1016/j.jmaa.2010.06.046. Google Scholar [29] Z. Guo and Y. Zhou, On Solutions to a two-component generalized Camassa-Holm equation,, Stud. Appl. Math., 124 (2010), 307. doi: 10.1111/j.1467-9590.2009.00472.x. Google Scholar [30] D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs,, SIAM J. Appl. Dyn. Syst., 2 (2003), 323. doi: 10.1137/S1111111102410943. Google Scholar [31] T. Y. Hou and C. Li, Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl,, Comm. Pure Appl. Math., 61 (2008), 661. doi: 10.1002/cpa.20212. Google Scholar [32] J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498. doi: 10.1137/0151075. Google Scholar [33] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid. Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224. Google Scholar [34] R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow,, Fluid Dyn. Res., 33 (2003), 97. doi: 10.1016/S0169-5983(03)00036-4. Google Scholar [35] P. Kevrekidis and Y. Drossinos, Nonlinearity from linearity: The Ermakov-Pinney equation revisited,, Mathematics and Computers in Simulation, 74 (2007), 196. doi: 10.1016/j.matcom.2006.10.005. Google Scholar [36] B. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits,, Adv. Math., 176 (2003), 116. doi: 10.1016/S0001-8708(02)00063-4. Google Scholar [37] B. Khesin, J. Lenells, G. Misiolek and S. C. Preston, Geometry of diffeomorphism groups, complete integrability and optimal transport,, Geom. and Funct. Anal., 23 (2013), 334. doi: 10.1007/s00039-013-0210-2. Google Scholar [38] J. Lenells, Weak geodesic flow and global solutions of the Hunter-Saxton equation,, Discrete Contin. Dyn. Syst., 18 (2007), 643. doi: 10.3934/dcds.2007.18.643. Google Scholar [39] J. Lenells and O. Lechtenfeld, On the $N=2$ supersymmetric Camassa-Holm and Hunter-Saxton equations,, J. Math. Phys., 50 (2009), 1. doi: 10.1063/1.3060125. Google Scholar [40] J. Lenells and M. Wunsch, The Hunter-Saxton system and the geodesics on a pseudosphere,, Commun. Part. Diff. Eq., 38 (2013), 860. doi: 10.1080/03605302.2013.771660. Google Scholar [41] J. Liu and Z. Yin, Blow-up phenomena and global existence for a periodic two-component Hunter-Saxton system,, preprint, (). Google Scholar [42] J. Liu and Z. Yin, Global weak solutions for a periodic two-component $\mu-$Hunter-Saxton system,, Monatshefte f$\dot u$r Mathematik, 168 (2012), 503. doi: 10.1007/s00605-011-0346-9. Google Scholar [43] A. J. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean,, Courant Lecture Notes in Mathematics, (2003). Google Scholar [44] G. Misiolek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080. doi: 10.1007/PL00012648. Google Scholar [45] K. Mohajer, A note on traveling wave solutions to the two-component Camassa-Holm equation,, J. Nonlinear Math. Phys., 16 (2009), 117. doi: 10.1142/S140292510900011X. Google Scholar [46] B. Moon and Y. Liu, Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system,, J. Differ. Equations, 253 (2012), 319. doi: 10.1016/j.jde.2012.02.011. Google Scholar [47] B. Moon, Solitary wave solutions of the generalized two-component Hunter-Saxton system,, Nonlinear Anal-Theor., 89 (2013), 242. doi: 10.1016/j.na.2013.05.004. Google Scholar [48] O. G. Mustafa, On smooth traveling waves of an integrable two-component Camassa-Holm equation,, Wave Motion, 46 (2009), 397. doi: 10.1016/j.wavemoti.2009.06.011. Google Scholar [49] H. Okamoto and K. Ohkitani, On the role of the convection term in the equations of motion of incompressible fluid,, J. Phys. Soc. Japan, 74 (2005), 2737. doi: 10.1143/JPSJ.74.2737. Google Scholar [50] H. Okamoto and J. Zhu, Some similarity solutions of the Navier-Stokes equations and related topics,, Taiwanese J. Math., 4 (2000), 65. Google Scholar [51] H. Okamoto, Well-posedness of the generalized Proudman-Johnson equation without viscosity,, J. Math. Fluid Mech., 11 (2009), 46. doi: 10.1007/s00021-007-0247-9. Google Scholar [52] H. Okamoto, T. Sakajo and M. Wunsch, On a generalization of the Constantin-Lax-Majda equation,, Nonlinearity, 21 (2008), 2447. doi: 10.1088/0951-7715/21/10/013. Google Scholar [53] M. V. Pavlov, The Gurevich-Zybin system,, J. Phys. A: Math. Gen., 38 (2005), 3823. doi: 10.1088/0305-4470/38/17/008. Google Scholar [54] J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987). Google Scholar [55] E. Pinney, The nonlinear differential equation $y''+p(x)y+cy^{-3}=0$,, Proc. Amer. Math. Soc., 1 (1950). Google Scholar [56] I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point,, J. Fluid Mech., 12 (1962), 161. doi: 10.1017/S0022112062000130. Google Scholar [57] A. Sarria and R. Saxton, Blow-up of solutions to the generalized inviscid Proudman-Johnson equation,, J. Math Fluid Mech., 15 (2013), 493. doi: 10.1007/s00021-012-0126-x. Google Scholar [58] A. Sarria and R. Saxton, The role of initial curvature in solutions to the generalized inviscid Proudman-Johnson equation,, Q. Appl. Math., (). Google Scholar [59] A. Sarria, Regularity of stagnation point-form solutions of the two-dimensional Euler equations,, Differential and Integral Equations, (). Google Scholar [60] A. Sarria and J. Wu, Blowup in stagnation-point form solutions of the inviscid 2d Boussinesq equations, submitted,, preprint, (). Google Scholar [61] R. Saxton and F. Tiglay, Global existence of some infinite energy solutions for a perfect incompressible fluid,, SIAM J. Math. Anal., 40 (2008), 1499. doi: 10.1137/080713768. Google Scholar [62] F. Tiglay, The periodic Cauchy problem of the modified Hunter-Saxton equation,, J. Evol. Eq., 5 (2005), 509. doi: 10.1007/s00028-005-0215-x. Google Scholar [63] H. Weyl, On the differential equations of the simplest boundary-layer problems,, Ann. Math., 43 (1942), 381. doi: 10.2307/1968875. Google Scholar [64] H. Wu and M. Wunsch, Global existence for the generalized two-component Hunter-Saxton system,, J. Math. Fluid Mech., 14 (2012), 455. doi: 10.1007/s00021-011-0075-9. Google Scholar [65] M. Wunsch, The generalized Proudman-Johnson equation revisited,, J. Math. Fluid Mech., 13 (2011), 147. doi: 10.1007/s00021-009-0004-3. Google Scholar [66] M. Wunsch, On the Hunter-Saxton system,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 647. doi: 10.3934/dcdsb.2009.12.647. Google Scholar [67] M. Wunsch, The generalized Hunter-Saxton system,, SIAM J. Math. Anal., 42 (2010), 1286. doi: 10.1137/090768576. Google Scholar [68] M. Wunsch, Weak geodesic flow on a semidirect product and global solutions to the periodic Hunter-Saxton system,, Nonlinear Anal-Theor., 74 (2011), 4951. doi: 10.1016/j.na.2011.04.041. Google Scholar [69] Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272. doi: 10.1137/S0036141003425672. Google Scholar [70] P. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,, Int. Math. Res. Notices, (2010), 1981. doi: 10.1093/imrn/rnp211. Google Scholar

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References:
 [1] E. W. Barnes, A New Development of the Theory of Hypergeometric Functions,, Proc. London Math. Soc.(2), 6 (1908), 141. doi: 10.1112/plms/s2-6.1.141. Google Scholar [2] R. Beals, D. H. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equation,, Appl. Anal., 78 (2001), 255. doi: 10.1080/00036810108840938. Google Scholar [3] A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation,, SIAM J. Math. Anal, 37 (2005), 996. doi: 10.1137/050623036. Google Scholar [4] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar [5] C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,, Ann. of Math., 166 (2007), 245. doi: 10.4007/annals.2007.166.245. Google Scholar [6] C. Cao and E. S. Titi, Global well-posedness of the three-dimensional stratified primitive equations with partial vertical mixing turbulence diffusion,, Comm. Math. Phys., 310 (2012), 537. doi: 10.1007/s00220-011-1409-4. Google Scholar [7] C. Cao, S. Ibrahim, K. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics,, preprint, (). Google Scholar [8] D. Chae, On the blow-up problem for the axisymmetric 3D Euler equations,, Nonlinearity, 21 (2008), 2053. doi: 10.1088/0951-7715/21/9/007. Google Scholar [9] X. Chen and H. Okamoto, Global existence of solutions to the Proudman-Johnson equation,, Proc.Japan Acad., 76 (2000), 149. doi: 10.3792/pjaa.76.149. Google Scholar [10] X. Chen and H. Okamoto, Global existence of solutions to the generalized Proudman-Johnson equation,, Proc.Japan Acad., 78 (2002), 136. doi: 10.3792/pjaa.78.136. Google Scholar [11] S. Childress, G. R. Ierley, E. A. Spiegel and W. R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes equations having stagnation-point form,, J. Fluid Mech., 203 (1989), 1. doi: 10.1017/S0022112089001357. Google Scholar [12] C. H. Cho and M. Wunsch, Global and singular solutions to the generalized Proudman-Johnson equation,, J. Diff. Eqns., 249 (2010), 392. doi: 10.1016/j.jde.2010.03.013. Google Scholar [13] C. H. Cho and M. Wunsch, Global weak solutions to the generalized Proudman-Johnson equation,, Commun. Pure Appl. Ana., 11 (2012), 1387. doi: 10.3934/cpaa.2012.11.1387. Google Scholar [14] A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Physics Letters A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar [15] A. Constantin and M. Wunsch, On the inviscid Proudman-Johnson equation,, Proc. Japan Acad. Ser. A Math. Sci., 85 (2009), 81. doi: 10.3792/pjaa.85.81. Google Scholar [16] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi Equations,, Arch. Rational Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar [17] P. Constantin, P. D. Lax and A. Majda, A simple one dimensional model for the three dimensional vorticity equation,, Commun. Pure Appl. Math., 38 (1985), 715. doi: 10.1002/cpa.3160380605. Google Scholar [18] C. M. Dafermos, Generalized characteristics and the Hunter-Saxton equation,, J. Hyperbol. Differ. Eq., 8 (2011), 159. doi: 10.1142/S0219891611002366. Google Scholar [19] H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,, Fluid Dyn. Res., 33 (2003), 73. doi: 10.1016/S0169-5983(03)00046-7. Google Scholar [20] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. I,, McGraw-Hill, (1981), 56. Google Scholar [21] V. P. Ermakov, Univ. Izv. Kiev 20,, 1 (1880)., (1880). Google Scholar [22] J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow up phenomena for the 2-component Camassa-Holm equations,, Discrete Contin. Dyn. Syst., 19 (2007), 493. doi: 10.3934/dcds.2007.19.493. Google Scholar [23] T. W. Gamelin, Complex Analysis,, Undergraduate Texts in Mathematics, (2001). doi: 10.1007/978-0-387-21607-2. Google Scholar [24] G. Gasper and M. Rahman, Basic Hypergeometric Series,, Encyclopedia of Mathematics and Its Applications, (2004). doi: 10.1017/CBO9780511526251. Google Scholar [25] A. E. Gill, Atmosphere-Ocean Dynamics,, Academic Press, (1982). Google Scholar [26] A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth,, J. Fluid Mech., 78 (1976), 237. doi: 10.1017/S0022112076002425. Google Scholar [27] C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system,, J. Differential Equations, 248 (2010), 2003. doi: 10.1016/j.jde.2009.08.002. Google Scholar [28] Z. Guo, Blow up and global solutions to a new integrable model with two components,, J. Math. Anal. Appl., 372 (2010), 316. doi: 10.1016/j.jmaa.2010.06.046. Google Scholar [29] Z. Guo and Y. Zhou, On Solutions to a two-component generalized Camassa-Holm equation,, Stud. Appl. Math., 124 (2010), 307. doi: 10.1111/j.1467-9590.2009.00472.x. Google Scholar [30] D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs,, SIAM J. Appl. Dyn. Syst., 2 (2003), 323. doi: 10.1137/S1111111102410943. Google Scholar [31] T. Y. Hou and C. Li, Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl,, Comm. Pure Appl. Math., 61 (2008), 661. doi: 10.1002/cpa.20212. Google Scholar [32] J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498. doi: 10.1137/0151075. Google Scholar [33] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid. Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224. Google Scholar [34] R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow,, Fluid Dyn. Res., 33 (2003), 97. doi: 10.1016/S0169-5983(03)00036-4. Google Scholar [35] P. Kevrekidis and Y. Drossinos, Nonlinearity from linearity: The Ermakov-Pinney equation revisited,, Mathematics and Computers in Simulation, 74 (2007), 196. doi: 10.1016/j.matcom.2006.10.005. Google Scholar [36] B. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits,, Adv. Math., 176 (2003), 116. doi: 10.1016/S0001-8708(02)00063-4. Google Scholar [37] B. Khesin, J. Lenells, G. Misiolek and S. C. Preston, Geometry of diffeomorphism groups, complete integrability and optimal transport,, Geom. and Funct. Anal., 23 (2013), 334. doi: 10.1007/s00039-013-0210-2. Google Scholar [38] J. Lenells, Weak geodesic flow and global solutions of the Hunter-Saxton equation,, Discrete Contin. Dyn. Syst., 18 (2007), 643. doi: 10.3934/dcds.2007.18.643. Google Scholar [39] J. Lenells and O. Lechtenfeld, On the $N=2$ supersymmetric Camassa-Holm and Hunter-Saxton equations,, J. Math. Phys., 50 (2009), 1. doi: 10.1063/1.3060125. Google Scholar [40] J. Lenells and M. Wunsch, The Hunter-Saxton system and the geodesics on a pseudosphere,, Commun. Part. Diff. Eq., 38 (2013), 860. doi: 10.1080/03605302.2013.771660. Google Scholar [41] J. Liu and Z. Yin, Blow-up phenomena and global existence for a periodic two-component Hunter-Saxton system,, preprint, (). Google Scholar [42] J. Liu and Z. Yin, Global weak solutions for a periodic two-component $\mu-$Hunter-Saxton system,, Monatshefte f$\dot u$r Mathematik, 168 (2012), 503. doi: 10.1007/s00605-011-0346-9. Google Scholar [43] A. J. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean,, Courant Lecture Notes in Mathematics, (2003). Google Scholar [44] G. Misiolek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080. doi: 10.1007/PL00012648. Google Scholar [45] K. Mohajer, A note on traveling wave solutions to the two-component Camassa-Holm equation,, J. Nonlinear Math. Phys., 16 (2009), 117. doi: 10.1142/S140292510900011X. Google Scholar [46] B. Moon and Y. Liu, Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system,, J. Differ. Equations, 253 (2012), 319. doi: 10.1016/j.jde.2012.02.011. Google Scholar [47] B. Moon, Solitary wave solutions of the generalized two-component Hunter-Saxton system,, Nonlinear Anal-Theor., 89 (2013), 242. doi: 10.1016/j.na.2013.05.004. Google Scholar [48] O. G. Mustafa, On smooth traveling waves of an integrable two-component Camassa-Holm equation,, Wave Motion, 46 (2009), 397. doi: 10.1016/j.wavemoti.2009.06.011. Google Scholar [49] H. Okamoto and K. Ohkitani, On the role of the convection term in the equations of motion of incompressible fluid,, J. Phys. Soc. Japan, 74 (2005), 2737. doi: 10.1143/JPSJ.74.2737. Google Scholar [50] H. Okamoto and J. Zhu, Some similarity solutions of the Navier-Stokes equations and related topics,, Taiwanese J. Math., 4 (2000), 65. Google Scholar [51] H. Okamoto, Well-posedness of the generalized Proudman-Johnson equation without viscosity,, J. Math. Fluid Mech., 11 (2009), 46. doi: 10.1007/s00021-007-0247-9. Google Scholar [52] H. Okamoto, T. Sakajo and M. Wunsch, On a generalization of the Constantin-Lax-Majda equation,, Nonlinearity, 21 (2008), 2447. doi: 10.1088/0951-7715/21/10/013. Google Scholar [53] M. V. Pavlov, The Gurevich-Zybin system,, J. Phys. A: Math. 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