, Volume 11, Issue 1, pp 46-59

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

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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.

Dedicated to Professor Giovanni Paolo Galdi on the occasion of his 60th birthday
Communicated by the Editors
Partly supported by the Grant-in-Aid for Scientific Research from JSPS No. 14204007.