# American Institute of Mathematical Sciences

July  2012, 11(4): 1387-1396. doi: 10.3934/cpaa.2012.11.1387

## Global weak solutions to the generalized Proudman-Johnson equation

 1 Department of Mathematics, National Chung Cheng University, Min-Hsiung, Chia-Yi 621, Taiwan 2 Swiss Federal Institute of Technology Zurich, Department of Mathematics, 8092 Zurich, Switzerland

Received  October 2010 Revised  May 2011 Published  January 2012

We consider the generalized Proudman-Johnson equation, in which an artificial parameter $a$ controlling the impact of convection was introduced to the Proudman-Johnson equation ([33]). In the present paper, we are going to show that there are global weak solutions to the generalized Proudman-Johnson equation for certain parameter $a$'s.
Citation: Chien-Hong Cho, Marcus Wunsch. Global weak solutions to the generalized Proudman-Johnson equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1387-1396. doi: 10.3934/cpaa.2012.11.1387
##### References:
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Fluid Mech., 203 (1989), 1-22. doi: 10.1017/S0022112089001357.  Google Scholar [7] C.-H. Cho and M. Wunsch, Global and singular solutions to the generalized Proudman-Johnson equation, J. Differential Equations, 249 (2010), 392-413. doi: 10.1016/j.jde.2010.03.013.  Google Scholar [8] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757.  Google Scholar [9] A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506. doi: 10.1063/1.1845603.  Google Scholar [10] A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar [11] A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.  Google Scholar [12] A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A: Math. Gen., 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201.  Google Scholar [13] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar [14] A. Constantin and M. Wunsch, On the inviscid Proudman-Johnson equation, Proc. Japan Acad. Ser. A Math. Sci., 85 (2009), 81-83. doi: 10.3792/pjaa.85.81.  Google Scholar [15] J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153 doi: 10.1007/s00209-010-0778-2.  Google Scholar [16] J. Escher, M. Kohlmann and B. Kolev, Geometric aspects of the periodic $\mu$DP equation, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 193-209. Google Scholar [17] J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, preprint, ().   Google Scholar [18] L. C. Evans, "Partial Differential Equations,'' AMS Graduate Studies in Mathematics, 1998.  Google Scholar [19] Y. Fu, Y. Liu and C. Qu, On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations,, preprint, ().   Google Scholar [20] D. Henry, Infinite propagation speed for the Degasperis-Procesi equation, J. Math. Anal. Appl., 311 (2005), 755-759. doi: 10.1016/j.jmaa.2005.03.001.  Google Scholar [21] J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075.  Google Scholar [22] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224.  Google Scholar [23] B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4.  Google Scholar [24] S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690.  Google Scholar [25] J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, Journal of Geometry and Physics, 57 (2007), 2049-2064. doi: 10.1016/j.geomphys.2007.05.003.  Google Scholar [26] J. Lenells, The Hunter-Saxton equation: a geometric approach, SIAM J. Math. Anal., 40 (2008), 266-277. doi: 10.1137/050647451.  Google Scholar [27] J. Lenells, Weak geodesic flow and global solutions of the Hunter-Saxton equation, Discrete Contin. Dyn. Syst., 18 (2007), 643-656. doi: 10.3934/dcds.2007.18.643.  Google Scholar [28] J. Lenells, G. Misiołek and F. Tığlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. Math. Phys., 299 (2010), 129-161. doi: 10.1007/s00220-010-1069-9.  Google Scholar [29] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198. doi: 10.1007/s00332-006-0803-3.  Google Scholar [30] G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group and the KdV equation, Proc. Amer. Math. Soc., 125 (1998), 203-208.  Google Scholar [31] O. G. Mustafa, A note on the Degasperis-Procesi equation, J. Nonlinear Math. Phys., 12 (2005), 10-14. doi: 10.2991/jnmp.2005.12.1.2.  Google Scholar [32] H. Okamoto, Well-posedness of the generalized Proudman-Johnson equation without viscosity, J. Math. Fluid Mech., 11 (2009), 46-59. doi: 10.1007/s00021-007-0247-9.  Google Scholar [33] H. Okamoto and J. Zhu, Some similarity solutions of the Navier-Stokes equations and related topics, Taiwanese J. Math., 4 (2000), 65-103.  Google Scholar [34] M. V. Pavlov, The Calogero equation and Liouville-type equations, Theoretical and Mathematical Physics, 128 (2001), 927-932. doi: 10.1023/A:1010454217405.  Google Scholar [35] I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point, J. Fluid Mech., 12 (1962), 161-168. doi: 10.1017/S0022112062000130.  Google Scholar [36] R. Saxton and F. Tığlay, Global existence of some infinite energy solutions for a perfect incompressible fluid, SIAM J. Math. Anal., 4 (2008), 1499-1515. doi: 10.1137/080713768.  Google Scholar [37] M. Wunsch, "Asymptotics for Nonlinear Diffusion and Fluid Dynamics Equations,'' Ph.D.-Thesis at the University of Vienna, 2009. Google Scholar [38] M. Wunsch, The generalized Proudman-Johnson equation revisited, J. Math. Fluid Mech., 13 (2009), 147-154. doi: 10.1007/s00021-009-0004-3.  Google Scholar [39] Z.-Y. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283. doi: 10.1137/S0036141003425672.  Google Scholar

show all references

##### References:
 [1] A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026. doi: 10.1137/050623036.  Google Scholar [2] J. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1 (1948), 171-99. doi: 10.1016/S0065-2156(08)70100-5.  Google Scholar [3] F. Calogero, A solvable nonlinear wave equation, Stud. Appl. Math., 70 (1984), 189-199.  Google Scholar [4] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar [5] X. Chen and H. Okamoto, Global existence of solutions to the Proudman-Johnson equation, Proc. Japan Acad., 76 (2000), 149-152. doi: 10.3792/pjaa.76.149.  Google Scholar [6] S. Childress, G. R. Ierley, E. R. Spiegel and W. R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form, J. Fluid Mech., 203 (1989), 1-22. doi: 10.1017/S0022112089001357.  Google Scholar [7] C.-H. Cho and M. Wunsch, Global and singular solutions to the generalized Proudman-Johnson equation, J. Differential Equations, 249 (2010), 392-413. doi: 10.1016/j.jde.2010.03.013.  Google Scholar [8] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757.  Google Scholar [9] A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506. doi: 10.1063/1.1845603.  Google Scholar [10] A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar [11] A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.  Google Scholar [12] A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A: Math. Gen., 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201.  Google Scholar [13] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar [14] A. Constantin and M. Wunsch, On the inviscid Proudman-Johnson equation, Proc. Japan Acad. Ser. A Math. Sci., 85 (2009), 81-83. doi: 10.3792/pjaa.85.81.  Google Scholar [15] J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153 doi: 10.1007/s00209-010-0778-2.  Google Scholar [16] J. Escher, M. Kohlmann and B. Kolev, Geometric aspects of the periodic $\mu$DP equation, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 193-209. Google Scholar [17] J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, preprint, ().   Google Scholar [18] L. C. Evans, "Partial Differential Equations,'' AMS Graduate Studies in Mathematics, 1998.  Google Scholar [19] Y. Fu, Y. Liu and C. Qu, On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations,, preprint, ().   Google Scholar [20] D. Henry, Infinite propagation speed for the Degasperis-Procesi equation, J. Math. Anal. Appl., 311 (2005), 755-759. doi: 10.1016/j.jmaa.2005.03.001.  Google Scholar [21] J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075.  Google Scholar [22] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224.  Google Scholar [23] B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4.  Google Scholar [24] S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690.  Google Scholar [25] J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, Journal of Geometry and Physics, 57 (2007), 2049-2064. doi: 10.1016/j.geomphys.2007.05.003.  Google Scholar [26] J. Lenells, The Hunter-Saxton equation: a geometric approach, SIAM J. Math. Anal., 40 (2008), 266-277. doi: 10.1137/050647451.  Google Scholar [27] J. Lenells, Weak geodesic flow and global solutions of the Hunter-Saxton equation, Discrete Contin. Dyn. Syst., 18 (2007), 643-656. doi: 10.3934/dcds.2007.18.643.  Google Scholar [28] J. Lenells, G. Misiołek and F. Tığlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. Math. Phys., 299 (2010), 129-161. doi: 10.1007/s00220-010-1069-9.  Google Scholar [29] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198. doi: 10.1007/s00332-006-0803-3.  Google Scholar [30] G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group and the KdV equation, Proc. Amer. Math. Soc., 125 (1998), 203-208.  Google Scholar [31] O. G. Mustafa, A note on the Degasperis-Procesi equation, J. Nonlinear Math. Phys., 12 (2005), 10-14. doi: 10.2991/jnmp.2005.12.1.2.  Google Scholar [32] H. Okamoto, Well-posedness of the generalized Proudman-Johnson equation without viscosity, J. Math. Fluid Mech., 11 (2009), 46-59. doi: 10.1007/s00021-007-0247-9.  Google Scholar [33] H. Okamoto and J. Zhu, Some similarity solutions of the Navier-Stokes equations and related topics, Taiwanese J. Math., 4 (2000), 65-103.  Google Scholar [34] M. V. Pavlov, The Calogero equation and Liouville-type equations, Theoretical and Mathematical Physics, 128 (2001), 927-932. doi: 10.1023/A:1010454217405.  Google Scholar [35] I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point, J. Fluid Mech., 12 (1962), 161-168. doi: 10.1017/S0022112062000130.  Google Scholar [36] R. Saxton and F. Tığlay, Global existence of some infinite energy solutions for a perfect incompressible fluid, SIAM J. Math. Anal., 4 (2008), 1499-1515. doi: 10.1137/080713768.  Google Scholar [37] M. Wunsch, "Asymptotics for Nonlinear Diffusion and Fluid Dynamics Equations,'' Ph.D.-Thesis at the University of Vienna, 2009. Google Scholar [38] M. Wunsch, The generalized Proudman-Johnson equation revisited, J. Math. Fluid Mech., 13 (2009), 147-154. doi: 10.1007/s00021-009-0004-3.  Google Scholar [39] Z.-Y. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283. doi: 10.1137/S0036141003425672.  Google Scholar
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