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Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity
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 |
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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