Global weak solutions to the generalized Proudman-Johnson equation

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

Global weak solutions to the generalized Proudman-Johnson equation

Chien-Hong Cho ^1, and Marcus Wunsch ^2,

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

Abstract
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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.

Keywords: global weak solutions., The generalized Proudman-Johnson equation.

Mathematics Subject Classification: 35D05, 74H35, 35Q3.

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:

[1]	A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation,, SIAM J. Math. Anal., 37 (2005), 996. doi: 10.1137/050623036.
[2]	J. Burgers, A mathematical model illustrating the theory of turbulence,, Adv. Appl. Mech., 1 (1948), 171. doi: 10.1016/S0065-2156(08)70100-5.
[3]	F. Calogero, A solvable nonlinear wave equation,, Stud. Appl. Math., 70 (1984), 189.
[4]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661.
[5]	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.
[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. doi: 10.1017/S0022112089001357.
[7]	C.-H. Cho and M. Wunsch, Global and singular solutions to the generalized Proudman-Johnson equation,, J. Differential Equations, 249 (2010), 392. doi: 10.1016/j.jde.2010.03.013.
[8]	A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757.
[9]	A. Constantin, Finite propagation speed for the Camassa-Holm equation,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1845603.
[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. doi: 10.1007/PL00004793.
[11]	A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Comment. Math. Helv., 78 (2003), 787. doi: 10.1007/s00014-003-0785-6.
[12]	A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems,, J. Phys. A: Math. Gen., 35 (2002). doi: 10.1088/0305-4470/35/32/201.
[13]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2.
[14]	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.
[15]	J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation,, Math. Z., 269 (2011), 1137. doi: 10.1007/s00209-010-0778-2.
[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.
[17]	J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, preprint, ().
[18]	L. C. Evans, "Partial Differential Equations,'', AMS Graduate Studies in Mathematics, (1998).
[19]	Y. Fu, Y. Liu and C. Qu, On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations,, preprint, ().
[20]	D. Henry, Infinite propagation speed for the Degasperis-Procesi equation,, J. Math. Anal. Appl., 311 (2005), 755. doi: 10.1016/j.jmaa.2005.03.001.
[21]	J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498. doi: 10.1137/0151075.
[22]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224.
[23]	B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits,, Adv. Math., 176 (2003), 116. doi: 10.1016/S0001-8708(02)00063-4.
[24]	S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857. doi: 10.1063/1.532690.
[25]	J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere,, Journal of Geometry and Physics, 57 (2007), 2049. doi: 10.1016/j.geomphys.2007.05.003.
[26]	J. Lenells, The Hunter-Saxton equation: a geometric approach,, SIAM J. Math. Anal., 40 (2008), 266. doi: 10.1137/050647451.
[27]	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.
[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. doi: 10.1007/s00220-010-1069-9.
[29]	H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation,, J. Nonlinear Sci., 17 (2007), 169. doi: 10.1007/s00332-006-0803-3.
[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.
[31]	O. G. Mustafa, A note on the Degasperis-Procesi equation,, J. Nonlinear Math. Phys., 12 (2005), 10. doi: 10.2991/jnmp.2005.12.1.2.
[32]	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.
[33]	H. Okamoto and J. Zhu, Some similarity solutions of the Navier-Stokes equations and related topics,, Taiwanese J. Math., 4 (2000), 65.
[34]	M. V. Pavlov, The Calogero equation and Liouville-type equations,, Theoretical and Mathematical Physics, 128 (2001), 927. doi: 10.1023/A:1010454217405.
[35]	I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point,, J. Fluid Mech., 12 (1962), 161. doi: 10.1017/S0022112062000130.
[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. doi: 10.1137/080713768.
[37]	M. Wunsch, "Asymptotics for Nonlinear Diffusion and Fluid Dynamics Equations,'', Ph.D.-Thesis at the University of Vienna, (2009).
[38]	M. Wunsch, The generalized Proudman-Johnson equation revisited,, J. Math. Fluid Mech., 13 (2009), 147. doi: 10.1007/s00021-009-0004-3.
[39]	Z.-Y. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272. doi: 10.1137/S0036141003425672.

show all references

References:

[1]	A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation,, SIAM J. Math. Anal., 37 (2005), 996. doi: 10.1137/050623036.
[2]	J. Burgers, A mathematical model illustrating the theory of turbulence,, Adv. Appl. Mech., 1 (1948), 171. doi: 10.1016/S0065-2156(08)70100-5.
[3]	F. Calogero, A solvable nonlinear wave equation,, Stud. Appl. Math., 70 (1984), 189.
[4]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661.
[5]	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.
[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. doi: 10.1017/S0022112089001357.
[7]	C.-H. Cho and M. Wunsch, Global and singular solutions to the generalized Proudman-Johnson equation,, J. Differential Equations, 249 (2010), 392. doi: 10.1016/j.jde.2010.03.013.
[8]	A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757.
[9]	A. Constantin, Finite propagation speed for the Camassa-Holm equation,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1845603.
[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. doi: 10.1007/PL00004793.
[11]	A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Comment. Math. Helv., 78 (2003), 787. doi: 10.1007/s00014-003-0785-6.
[12]	A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems,, J. Phys. A: Math. Gen., 35 (2002). doi: 10.1088/0305-4470/35/32/201.
[13]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2.
[14]	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.
[15]	J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation,, Math. Z., 269 (2011), 1137. doi: 10.1007/s00209-010-0778-2.
[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.
[17]	J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, preprint, ().
[18]	L. C. Evans, "Partial Differential Equations,'', AMS Graduate Studies in Mathematics, (1998).
[19]	Y. Fu, Y. Liu and C. Qu, On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations,, preprint, ().
[20]	D. Henry, Infinite propagation speed for the Degasperis-Procesi equation,, J. Math. Anal. Appl., 311 (2005), 755. doi: 10.1016/j.jmaa.2005.03.001.
[21]	J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498. doi: 10.1137/0151075.
[22]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224.
[23]	B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits,, Adv. Math., 176 (2003), 116. doi: 10.1016/S0001-8708(02)00063-4.
[24]	S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857. doi: 10.1063/1.532690.
[25]	J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere,, Journal of Geometry and Physics, 57 (2007), 2049. doi: 10.1016/j.geomphys.2007.05.003.
[26]	J. Lenells, The Hunter-Saxton equation: a geometric approach,, SIAM J. Math. Anal., 40 (2008), 266. doi: 10.1137/050647451.
[27]	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.
[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. doi: 10.1007/s00220-010-1069-9.
[29]	H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation,, J. Nonlinear Sci., 17 (2007), 169. doi: 10.1007/s00332-006-0803-3.
[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.
[31]	O. G. Mustafa, A note on the Degasperis-Procesi equation,, J. Nonlinear Math. Phys., 12 (2005), 10. doi: 10.2991/jnmp.2005.12.1.2.
[32]	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.
[33]	H. Okamoto and J. Zhu, Some similarity solutions of the Navier-Stokes equations and related topics,, Taiwanese J. Math., 4 (2000), 65.
[34]	M. V. Pavlov, The Calogero equation and Liouville-type equations,, Theoretical and Mathematical Physics, 128 (2001), 927. doi: 10.1023/A:1010454217405.
[35]	I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point,, J. Fluid Mech., 12 (1962), 161. doi: 10.1017/S0022112062000130.
[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. doi: 10.1137/080713768.
[37]	M. Wunsch, "Asymptotics for Nonlinear Diffusion and Fluid Dynamics Equations,'', Ph.D.-Thesis at the University of Vienna, (2009).
[38]	M. Wunsch, The generalized Proudman-Johnson equation revisited,, J. Math. Fluid Mech., 13 (2009), 147. doi: 10.1007/s00021-009-0004-3.
[39]	Z.-Y. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272. doi: 10.1137/S0036141003425672.

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