Journal of Mathematical Fluid Mechanics

, Volume 11, Issue 1, pp 46–59

Well-Posedness of the Generalized Proudman–Johnson Equation Without Viscosity


    • Research Institute for Mathematical SciencesKyoto University

DOI: 10.1007/s00021-007-0247-9

Cite this article as:
Okamoto, H. J. math. fluid mech. (2009) 11: 46. doi:10.1007/s00021-007-0247-9


The generalized Proudman–Johnson equation, which was derived from the Navier–Stokes equations by Jinghui Zhu and the author, are considered in the case where the viscosity is neglected and the periodic boundary condition is imposed. The equation possesses two nonlinear terms: the convection and stretching terms. We prove that the solution exists globally in time if the stretching term is weak in the sense to be specified below. We also discuss on blow-up solutions when the stretching term is strong.

Mathematics Subject Classification (2000).

Primary 76B45Secondary 34B15


Generalized Proudman–Johnson equationblow-up

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