# American Institute of Mathematical Sciences

August  2014, 34(8): 3155-3170. doi: 10.3934/dcds.2014.34.3155

## Steady-states and traveling-wave solutions of the generalized Constantin--Lax--Majda equation

 1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502 2 Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan 3 Departement Mathematik, ETH Zürich, 8092 Zurich, Switzerland

Received  May 2013 Published  January 2014

Steady-states and traveling-waves of the generalized Constantin--Lax--Majda equation are computed and their asymptotic behavior is described. Their relation with possible blow-up and the Benjamin--Ono equation is discussed.
Citation: Hisashi Okamoto, Takashi Sakajo, Marcus Wunsch. Steady-states and traveling-wave solutions of the generalized Constantin--Lax--Majda equation. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3155-3170. doi: 10.3934/dcds.2014.34.3155
##### References:
 [1] H. Brezis, Blow-up for $u_t - \Delta u = g(u)$ revisited, Adv. Diff. Eqns., 1 (1996), 73-90.  Google Scholar [2] A. Castro and D. Cordoba, Infinite energy solutions of the surface quasi-geostrophic equation, Adv. Math., 225 (2010), 1820-1829. doi: 10.1016/j.aim.2010.04.018.  Google Scholar [3] A. Córdoba, D. Córdoba and M. A. Fontelos, Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. Math. Pure Appl., 86 (2006), 529-540. doi: 10.1016/j.matpur.2006.08.002.  Google Scholar [4] A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. Math., 162 (2005), 1-13. doi: 10.4007/annals.2005.162.1377.  Google Scholar [5] P. Constantin, P. D. Lax and A. J. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math., 38 (1985), 715-724. doi: 10.1002/cpa.3160380605.  Google Scholar [6] S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Stat. Phys., 59 (1990), 1251-1263. doi: 10.1007/BF01334750.  Google Scholar [7] S. De Gregorio, A partial differential equation arising in a 1D model for the 3D vorticity equation, Math. Meth. Appl. Sci., 19 (1996), 1233-1255. doi: 10.1002/(SICI)1099-1476(199610)19:15<1233::AID-MMA828>3.0.CO;2-W.  Google Scholar [8] J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation, Comm. Pure Appl. Anal., 11 (2012), 1407-1419. doi: 10.3934/cpaa.2012.11.1407.  Google Scholar [9] M. Fila and H. Matano, Blow-up in nonlinear heat equations from the dynamical systems point of view, in Handbook of Dynamical Systems, 2, North-Holland, Amsterdam, 2002, 723-758. doi: 10.1016/S1874-575X(02)80035-2.  Google Scholar [10] S. Hamada, Numerical solutions of Serrin's equations by double exponential transformation, Publ. RIMS, 43 (2007), 795-817. doi: 10.2977/prims/1201012042.  Google Scholar [11] T. Hou, C. Li, Z. Shi, S. Wang and X. Yu, On singularity formation of a nonlinear nonlocal system, Arch. Rational Mech. Anal., 199 (2011), 117-144. doi: 10.1007/s00205-010-0319-5.  Google Scholar [12] Y. Katznelson, An Introduction to Harmonic Analysis, 3rd Ed., Camb. Univ. Press, 2004.  Google Scholar [13] K. Kobayashi, H. Okamoto and J. Zhu, Numerical computation of water and solitary waves by the double exponential transform, J. Comp. Appl. Math., 152 (2003), 229-241. doi: 10.1016/S0377-0427(02)00708-2.  Google Scholar [14] H. Kozono and Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Commun. Math. Phys., 214 (2000), 191-200. doi: 10.1007/s002200000267.  Google Scholar [15] Yu. P. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, 1964. Google Scholar [16] Y. Matsuno, Bilinear Transformation Method, Academic Press, 1984.  Google Scholar [17] M. Mori, A. Nurmuhammad and M. Muhammad, DE-sinc method for second order singularly perturbed boundary value problems, Japan J. Indust. Appl. Math., 26 (2009), 41-63. doi: 10.1007/BF03167545.  Google Scholar [18] M. Nagayama, H. Okamoto and J. Zhu, On the blow-up of some similarity solutions of the Navier-Stokes equations, Quader. di Mat., 10 (2003), 137-162.  Google Scholar [19] K. Ohkitani, The Fefferman-Stein decomposition for the Constantin-Lax-Majda equation: Regularity criteria for inviscid fluid dynamics revisited, J. Math. Phys., 53 (2012), 115607. doi: 10.1063/1.4738639.  Google Scholar [20] 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), 2737-2742. doi: 10.1143/JPSJ.74.2737.  Google Scholar [21] H. Okamoto, T. Sakajo and M. Wunsch, On a generalization of the Constantin-Lax-Majda equation, Nonlinearity, 21 (2008), 2447-2461. doi: 10.1088/0951-7715/21/10/013.  Google Scholar [22] T. Okayama, T. Matsuo and M. Sugihara, Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind, J. Comp. Appl. Math., 234 (2010), 1211-1227. doi: 10.1016/j.cam.2009.07.049.  Google Scholar [23] H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. doi: 10.1143/JPSJ.39.1082.  Google Scholar [24] E. Yanagida, Blow-up of Solutions of the Nonlinear Heat Equations, in Blow-up and Aggregation, (ed. by E. Yanagida), University of Tokyo Press 2006, 1-50 (in Japanese). Google Scholar [25] M. Wunsch, The generalized Constantin-Lax-Majda equation, Comm. Math. Sci., 9 (2011), 929-936. doi: 10.4310/CMS.2011.v9.n3.a12.  Google Scholar [26] M. Wunsch, On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric, J. Nonlinear Math. Phys., 17 (2010), 7-11. doi: 10.1142/S1402925110000544.  Google Scholar [27] M. Wunsch, The generalized Constantin-Lax-Majda equation revisited, Comm. Math. Sci., 9 (2011), 929-936. doi: 10.4310/CMS.2011.v9.n3.a12.  Google Scholar

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##### References:
 [1] H. Brezis, Blow-up for $u_t - \Delta u = g(u)$ revisited, Adv. Diff. Eqns., 1 (1996), 73-90.  Google Scholar [2] A. Castro and D. Cordoba, Infinite energy solutions of the surface quasi-geostrophic equation, Adv. Math., 225 (2010), 1820-1829. doi: 10.1016/j.aim.2010.04.018.  Google Scholar [3] A. Córdoba, D. Córdoba and M. A. Fontelos, Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. Math. Pure Appl., 86 (2006), 529-540. doi: 10.1016/j.matpur.2006.08.002.  Google Scholar [4] A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. Math., 162 (2005), 1-13. doi: 10.4007/annals.2005.162.1377.  Google Scholar [5] P. Constantin, P. D. Lax and A. J. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math., 38 (1985), 715-724. doi: 10.1002/cpa.3160380605.  Google Scholar [6] S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Stat. Phys., 59 (1990), 1251-1263. doi: 10.1007/BF01334750.  Google Scholar [7] S. De Gregorio, A partial differential equation arising in a 1D model for the 3D vorticity equation, Math. Meth. Appl. Sci., 19 (1996), 1233-1255. doi: 10.1002/(SICI)1099-1476(199610)19:15<1233::AID-MMA828>3.0.CO;2-W.  Google Scholar [8] J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation, Comm. Pure Appl. Anal., 11 (2012), 1407-1419. doi: 10.3934/cpaa.2012.11.1407.  Google Scholar [9] M. Fila and H. Matano, Blow-up in nonlinear heat equations from the dynamical systems point of view, in Handbook of Dynamical Systems, 2, North-Holland, Amsterdam, 2002, 723-758. doi: 10.1016/S1874-575X(02)80035-2.  Google Scholar [10] S. Hamada, Numerical solutions of Serrin's equations by double exponential transformation, Publ. RIMS, 43 (2007), 795-817. doi: 10.2977/prims/1201012042.  Google Scholar [11] T. Hou, C. Li, Z. Shi, S. Wang and X. Yu, On singularity formation of a nonlinear nonlocal system, Arch. Rational Mech. Anal., 199 (2011), 117-144. doi: 10.1007/s00205-010-0319-5.  Google Scholar [12] Y. Katznelson, An Introduction to Harmonic Analysis, 3rd Ed., Camb. Univ. Press, 2004.  Google Scholar [13] K. Kobayashi, H. Okamoto and J. Zhu, Numerical computation of water and solitary waves by the double exponential transform, J. Comp. Appl. Math., 152 (2003), 229-241. doi: 10.1016/S0377-0427(02)00708-2.  Google Scholar [14] H. Kozono and Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Commun. Math. Phys., 214 (2000), 191-200. doi: 10.1007/s002200000267.  Google Scholar [15] Yu. P. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, 1964. Google Scholar [16] Y. Matsuno, Bilinear Transformation Method, Academic Press, 1984.  Google Scholar [17] M. Mori, A. Nurmuhammad and M. Muhammad, DE-sinc method for second order singularly perturbed boundary value problems, Japan J. Indust. Appl. Math., 26 (2009), 41-63. doi: 10.1007/BF03167545.  Google Scholar [18] M. Nagayama, H. Okamoto and J. Zhu, On the blow-up of some similarity solutions of the Navier-Stokes equations, Quader. di Mat., 10 (2003), 137-162.  Google Scholar [19] K. Ohkitani, The Fefferman-Stein decomposition for the Constantin-Lax-Majda equation: Regularity criteria for inviscid fluid dynamics revisited, J. Math. Phys., 53 (2012), 115607. doi: 10.1063/1.4738639.  Google Scholar [20] 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), 2737-2742. doi: 10.1143/JPSJ.74.2737.  Google Scholar [21] H. Okamoto, T. Sakajo and M. Wunsch, On a generalization of the Constantin-Lax-Majda equation, Nonlinearity, 21 (2008), 2447-2461. doi: 10.1088/0951-7715/21/10/013.  Google Scholar [22] T. Okayama, T. Matsuo and M. Sugihara, Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind, J. Comp. Appl. Math., 234 (2010), 1211-1227. doi: 10.1016/j.cam.2009.07.049.  Google Scholar [23] H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. doi: 10.1143/JPSJ.39.1082.  Google Scholar [24] E. Yanagida, Blow-up of Solutions of the Nonlinear Heat Equations, in Blow-up and Aggregation, (ed. by E. Yanagida), University of Tokyo Press 2006, 1-50 (in Japanese). Google Scholar [25] M. Wunsch, The generalized Constantin-Lax-Majda equation, Comm. Math. Sci., 9 (2011), 929-936. doi: 10.4310/CMS.2011.v9.n3.a12.  Google Scholar [26] M. Wunsch, On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric, J. Nonlinear Math. Phys., 17 (2010), 7-11. doi: 10.1142/S1402925110000544.  Google Scholar [27] M. Wunsch, The generalized Constantin-Lax-Majda equation revisited, Comm. Math. Sci., 9 (2011), 929-936. doi: 10.4310/CMS.2011.v9.n3.a12.  Google Scholar
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