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Global and singular solutions to the generalized Proudman-Johnson equation. (English) Zbl 1204.35067
Summary: We show that there is a class of solutions to the generalized Proudman-Johnson equation which exist globally for all parameters $$a$$ having the form $$-\frac {n+3}{n+1}$$ for $$n \in \mathbb N$$, thereby extending a result of A. Bressan and A. Constantin [SIAM J. Math. Anal. 37, No. 3, 996–1026 (2005; Zbl 1108.35024)]. Furthermore, we present new proofs for the existence of solutions developing spontaneous singularities and compute the corresponding blow-up rates.

##### MSC:
 35D30 Weak solutions to PDEs 74H35 Singularities, blow-up, stress concentrations for dynamical problems in solid mechanics 35Q35 PDEs in connection with fluid mechanics 35B44 Blow-up in context of PDEs
blow-up rate
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##### References:
 [1] Bergh, J.; Löfström, J., Interpolation spaces (an introduction), Grundlehren math. wiss., vol. 223, (1976), Springer-Verlag · Zbl 0344.46071 [2] Bressan, A.; Constantin, A., Global solutions of the hunter – saxton equation, SIAM J. math. anal., 37, 3, 996-1026, (2005) · Zbl 1108.35024 [3] Burgers, J., A mathematical model illustrating the theory of turbulence, Adv. appl. mech., 1, 171-199, (1948) [4] Calogero, F., A solvable nonlinear wave equation, Stud. appl. math., 70, 189-199, (1984) · Zbl 0551.35056 [5] Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. rev. lett., 71, 11, 1661-1664, (1993) · Zbl 0972.35521 [6] Chen, X.; Okamoto, H., Global existence of solutions to the proudman – johnson equation, Proc. Japan acad., 76, 149-152, (2000) · Zbl 0966.35002 [7] Childress, S.; Ierley, G.R.; Spiegel, E.R.; Young, W.R., Blow-up of unsteady two-dimensional Euler and navier – stokes solutions having stagnation-point form, J. fluid mech., 203, 1-22, (1989) · Zbl 0674.76013 [8] Constantin, A., The trajectories of particles in Stokes waves, Invent. math., 166, 523-535, (2006) · Zbl 1108.76013 [9] Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta math., 181, 229-243, (1998) · Zbl 0923.76025 [10] Constantin, A.; Escher, J., Global solutions for quasilinear parabolic problems, J. evol. equ., 2, 97-111, (2002) · Zbl 1004.35070 [11] Constantin, A.; Escher, J., Particle trajectories in solitary water waves, Bull. amer. math. soc., 44, 423-431, (2007) · Zbl 1126.76012 [12] Constantin, A.; Kolev, B., On the geometric approach to the motion of inertial mechanical systems, J. phys. A, 35, R51-R79, (2002) · Zbl 1039.37068 [13] Constantin, A.; Lannes, D., The hydrodynamical relevance of the camassa – holm and the degasperis – procesi equations, Arch. ration. mech. anal., 192, 1, 165-186, (2009) · Zbl 1169.76010 [14] Constantin, A.; Wunsch, M., On the inviscid proudman – johnson equation, Proc. Japan acad. ser. A math. sci., 85, 7, 81-83, (2009) · Zbl 1179.35236 [15] Degasperis, A.; Procesi, M., Asymptotic integrability, (), 23-37 · Zbl 0963.35167 [16] Escher, J.; Yin, Zh., Initial boundary value problems for nonlinear dispersive equations, J. funct. anal., 256, 479-508, (2009) · Zbl 1193.35108 [17] Holm, D.D.; Staley, M.F., Wave structure and nonlinear balances in a family of evolutionary pdes, SIAM J. appl. dyn. syst., 2, 323-380, (2003) · Zbl 1088.76531 [18] Hunter, J.K.; Saxton, R., Dynamics of director fields, SIAM J. appl. math., 51, 1498-1521, (1991) · Zbl 0761.35063 [19] Johnson, R.S., Camassa – holm, korteweg – de Vries and related models for water waves, J. fluid mech., 457, 63-82, (2002) · Zbl 1037.76006 [20] Khesin, B.; Misiołek, G., Euler equations on homogeneous spaces and Virasoro orbits, Adv. math., 176, 116-144, (2003) · Zbl 1017.37039 [21] Kolev, B., Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations, Philos. trans. R. soc. A, 365, 1858, 2333-2357, (2007) · Zbl 1152.37344 [22] Korteweg, D.J.; de Vries, G., On the change of form of long waves advancing in a rectangular channel, and a new type of long stationary waves, Philos. mag. (5), 39, 422-443, (1895) · JFM 26.0881.02 [23] Lenells, J., The hunter – saxton equation describes the geodesic flow on a sphere, J. geom. phys., 57, 2049-2064, (2007) · Zbl 1125.35085 [24] Lenells, J., The hunter – saxton equation: a geometric approach, SIAM J. math. anal., 40, 266-277, (2008) · Zbl 1168.35424 [25] Misiołek, G., A shallow water equation as a geodesic flow on the bott – virasoro group and the KdV equation, Proc. amer. math. soc., 125, 203-208, (1998) · Zbl 0901.58022 [26] Morozov, O.I., Contact equivalence of the generalized hunter – saxton equation and the euler – poisson equation, preprint, available at [27] Okamoto, H., Well-posedness of the generalized proudman – johnson equation without viscosity, J. math. fluid mech., 11, 46-59, (2009) · Zbl 1162.76311 [28] Okamoto, H.; Ohkitani, K., On the role of the convection term in the equations of motion of incompressible fluid, J. phys. soc. Japan, 74, 2737-2742, (2005) · Zbl 1083.76007 [29] Okamoto, H.; Sakajo, T.; Wunsch, M., On a generalization of the constantin – lax – majda equation, Nonlinearity, 21, 2447-2461, (2008) · Zbl 1221.35300 [30] Okamoto, H.; Zhu, J., Some similarity solutions of the navier – stokes equations and related topics, Proceedings of 1999 international conference on nonlinear analysis (Taipei), Taiwanese J. math., 4, 65-103, (2000) · Zbl 0972.35090 [31] Pavlov, M.V., The Calogero equation and Liouville-type equations, Theoret. math. phys., 128, 927-932, (2001) · Zbl 0992.35098 [32] Proudman, I.; Johnson, K., Boundary-layer growth near a rear stagnation point, J. fluid mech., 12, 161-168, (1962) · Zbl 0113.19501 [33] Saxton, R.; Tığlay, F., Global existence of some infinite energy solutions for a perfect incompressible fluid, SIAM J. math. anal., 4, 1499-1515, (2008) · Zbl 1167.35436 [34] Tod, K.P., Einstein – weyl spaces and third-order differential equations, J. math. phys., 41, 5572-5581, (2000) · Zbl 0979.53050 [35] M. Wunsch, Asymptotics for nonlinear diffusion and fluid dynamics equations, Ph.D. thesis, University of Vienna, 2009 [36] Wunsch, M., The generalized proudman – johnson equation revisited, J. Math. Fluid Mech., available at · Zbl 1270.35372 [37] Yin, Zh., On the structure of solutions to the periodic hunter – saxton equation, SIAM J. math. anal., 36, 1, 272-283, (2004) · Zbl 1151.35321
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