Global weak solutions to the generalized Proudman-Johnson equation

Previous Article
Introduction to the special issue on hydrodynamic model equations
CPAA Home
This Issue
Next Article
Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity

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
Related Papers

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.

[1]	Changchun Liu, Jingxue Yin, Juan Zhou. Existence of weak solutions for a generalized thin film equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 465-480. doi: 10.3934/cpaa.2007.6.465
[2]	Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883
[3]	Shaoyong Lai, Qichang Xie, Yunxi Guo, YongHong Wu. The existence of weak solutions for a generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 45-57. doi: 10.3934/cpaa.2011.10.45
[4]	Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643
[5]	Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026
[6]	Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713
[7]	Kazuhiro Ishige, Tatsuki Kawakami, Kanako Kobayashi. Global solutions for a nonlinear integral equation with a generalized heat kernel. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 767-783. doi: 10.3934/dcdss.2014.7.767
[8]	Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[9]	Xueke Pu, Boling Guo, Jingjun Zhang. Global weak solutions to the 1-D fractional Landau-Lifshitz equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 199-207. doi: 10.3934/dcdsb.2010.14.199
[10]	Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215
[11]	Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681
[12]	Miao Liu, Weike Wang. Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1203-1222. doi: 10.3934/cpaa.2014.13.1203
[13]	Zijuan Wen, Meng Fan, Asim M. Asiri, Ebraheem O. Alzahrani, Mohamed M. El-Dessoky, Yang Kuang. Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 407-420. doi: 10.3934/mbe.2017025
[14]	Mahmut Çalik, Marcel Oliver. Weak solutions for generalized large-scale semigeostrophic equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 939-955. doi: 10.3934/cpaa.2013.12.939
[15]	Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867
[16]	Bo Su and Martin Burger. Global weak solutions of non-isothermal front propagation problem. Electronic Research Announcements, 2007, 13: 46-52.
[17]	Zhen Lei, Yi Zhou. BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 575-583. doi: 10.3934/dcds.2009.25.575
[18]	Bernard Ducomet, Eduard Feireisl, Hana Petzeltová, Ivan Straškraba. Global in time weak solutions for compressible barotropic self-gravitating fluids. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 113-130. doi: 10.3934/dcds.2004.11.113
[19]	Liu Rui. The explicit nonlinear wave solutions of the generalized $b$-equation. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1029-1047. doi: 10.3934/cpaa.2013.12.1029
[20]	Ezzeddine Zahrouni. On the Lyapunov functions for the solutions of the generalized Burgers equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 391-410. doi: 10.3934/cpaa.2003.2.391

2016 Impact Factor: 0.801

American Institute of Mathematical Sciences