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Global existence of solutions to the generalized Proudman-Johnson equation. (English) Zbl 1020.35002

Summary: We consider the equation \(f_{xxt}+ ff_{xxx}- af_x f_{xx}= \nu f_{xxxx}\), \(x\in (0,1)\), \(t> 0\), where \(a\in\mathbb{R}\) is a constant, with the periodic boundary condition. We show that any solution exists globally in time if \(-3\leq a\leq 1\).

MSC:

35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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