On the inviscid Proudman-Johnson equation

Constantin, Adrian; Wunsch, Marcus

doi:10.3792/pjaa.85.81

Cited by 16 publications

(23 citation statements)

References 12 publications

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“…(3) was first derived by Okamoto and Zhu [30] and further investigated in [27,6,14,35,36]. Remarkably, the parameter a interconnects several well-studied equations within the framework of the generalized Proudman-Johnson equation (3), (5).…”

Section: Significance Of the Generalized Proudman-johnson Equationmentioning

confidence: 97%

Global and singular solutions to the generalized Proudman–Johnson equation

Cho

Wunsch

2010

Journal of Differential Equations

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Dedicated to Professor Adrian Constantin on the occasion of his 40th birthday MSC: 35D05 74H35 35Q35 Keywords: The generalized Proudman-Johnson equation Global weak solutions Blow-up rate We show that there is a class of solutions to the generalized Proudman-Johnson equation which exist globally for all parameters a having the form − n+3 n+1 for n ∈ N, thereby extending a result of Bressan and Constantin (2005) [2]. Furthermore, we present new proofs of existence of solutions developing spontaneous singularities and compute the corresponding blow-up rates.

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Section: Significance Of the Generalized Proudman-johnson Equationmentioning

confidence: 97%

Global and singular solutions to the generalized Proudman–Johnson equation

Cho

Wunsch

2010

Journal of Differential Equations

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“…In [7], Constantin and the author proved the results of this section for the special case of the inviscid Proudman-Johnson equation corresponding to a = 1.…”

Section: Preserved Oddness and More: A Blow-up Criterionmentioning

confidence: 99%

The Generalized Proudman–Johnson Equation Revisited

Wunsch

2009

J. Math. Fluid Mech.

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We demonstrate the existence of solutions to the inviscid generalized Proudman-Johnson equation for parameters a lying in the open interval (−5, −1) which develop singularities in finite time; moreover, we show that there are solutions which exist for all times if a = −1. Finally, a simple blow-up criterion for solutions arising from a special class of initial data is given. Mathematics Subject Classification (2000). Primary 35B67; Secondary 35B35.

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“…(8) is known as the inviscid Proudman-Johnson equation [20]. In [23], a general solution formula for solutions of (8), along with blowup and global-in-time criteria, were established (see [3,4,24,19,21] for additional regularity results). Eq.…”

Section: Introductionmentioning

confidence: 99%

“…In (4), u denotes the two-dimensional fluid velocity, p the scalar pressure, e 2 the standard unit vector in the vertical direction, and θ represents either the temperature in the context of thermal convection, or the density in the modeling of geophysical fluids. Note that periodicity (3)i), ii) of f (t, x) results from periodicity (in x) of u(t, x, y), i.e.…”

Section: Introductionmentioning

confidence: 99%

Blowup in stagnation-point form solutions of the inviscid 2d Boussinesq equations

Sarria

2015

Journal of Differential Equations

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The 2d Boussinesq equations model large scale atmospheric and oceanic flows. Whether its solutions develop a singularity in finite-time remains a classical open problem in mathematical fluid dynamics. In this work, blowup from smooth nontrivial initial velocities in stagnation-point form solutions of this system is established. On an infinite strip = {(x, y) ∈ [0, 1] × R + }, we consider velocities of the form u = (f (t, x), −yf x (t, x)), with scalar temperature θ = yρ(t, x). Assuming f x (0, x) attains its global maximum only at points x * i located on the boundary of [0, 1], general criteria for finite-time blowup of the vorticity −yf xx (t, x * i ) and the time integral of f x (t, x * i ) are presented. Briefly, for blowup to occur it is sufficient that ρ(0, x) ≥ 0 and f (t, x * i ) = ρ(0, x * i ) = 0, while −yf xx (0, x * i ) = 0. To illustrate how vorticity may suppress blowup, we also construct a family of global exact solutions. A local-existence result and additional regularity criteria in terms of the time integral of f x (t, ·) L ∞ ([0,1]) are also provided.

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On the inviscid Proudman-Johnson equation

Abstract: We show that certain qualitative properties of classical solutions to the inviscid Proudman-Johnson equation are preserved as long as these solutions exist. This enables us to give a simple blow-up criterion.

Cited by 16 publications

References 12 publications

Global and singular solutions to the generalized Proudman–Johnson equation

Global and singular solutions to the generalized Proudman–Johnson equation

The Generalized Proudman–Johnson Equation Revisited

Blowup in stagnation-point form solutions of the inviscid 2d Boussinesq equations

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