Blowup for the inviscid Proudman-Johnson equation with the homogeneous three-point boundary condition

The Proudman-Johnson equation is the reduced equation of the 2-dimensional incompressible Euler or Navier-Stokes equations in fluid dynamics. In this article, we show the blowup phenomenon of the third-order inviscid Proudman-Johnson equation with the homogeneous three-point boundary condition fx(0, t) = f(1, t) = 0, (1) and the initial condition H₀ = ∫¹₀ fx(x, 0)dx > 0. (2) In detail, if the above initial-boundary condition is satisfied, we apply the integration method to show that the corresponding C³ solutions of the inviscid Proudman-Johnson equation blow up on or before the finite time 1/(2H₀). Our blowup phenomenon for the inviscid Proudman-Johnson equation contrasts with Chen and Okamoto’s global existence result for the viscous equation with the homogeneous four-point boundary condition in “X. F. Chen and H. Okamoto, Global Existence of Solutions to the Proudman-Johnson Equation, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 149–152”. Copyright © 2013 Manwai Yuen.