Article

Journal of Mathematical Fluid Mechanics

, Volume 11, Issue 1, pp 46-59

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

  • Hisashi OkamotoAffiliated withResearch Institute for Mathematical Sciences, Kyoto University Email author 

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Abstract.

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 76B45 Secondary 34B15

Keywords.

Generalized Proudman–Johnson equation blow-up