On the inviscid Proudman-Johnson equation

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On the inviscid Proudman-Johnson equation

Adrian Constantin and Marcus Wunsch

Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 85, Number 7 (2009), 81-83.

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.

First Page: Show

Primary Subjects: 35Q35

Secondary Subjects: 76B99

Keywords: Proudman-Johnson equation; blow-up

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.pja/1247849905
Digital Object Identifier: doi:10.3792/pjaa.85.81
Mathematical Reviews number (MathSciNet): MR2548017
Zentralblatt MATH identifier: 05651159

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References

K. Bardos and È. S. Titi, Uspekhi Mat. Nauk 62 (2007), no. 3(375), 5--46, translation in Russian Math. Surveys 62 (2007), no. 3, 409--451.

Mathematical Reviews (MathSciNet): MR2355417

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Comm. Math. Phys. 94 (1984), no. 1, 61--66.

Mathematical Reviews (MathSciNet): MR763762

Zentralblatt MATH: 0573.76029

Digital Object Identifier: doi:10.1007/BF01212349

Project Euclid: euclid.cmp/1103941230

X. Chen and H. Okamoto, Global existence of solutions to the Proudman-Johnson equation, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), no. 9, 149--152.

Mathematical Reviews (MathSciNet): MR1801677

Digital Object Identifier: doi:10.3792/pjaa.76.149

Project Euclid: euclid.pja/1148393431

S. Childress et al., Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form, J. Fluid Mech. 203 (1989), 1--22.

Mathematical Reviews (MathSciNet): MR1002875

Zentralblatt MATH: 0674.76013

Digital Object Identifier: doi:10.1017/S0022112089001357

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998), no. 2, 229--243.

Mathematical Reviews (MathSciNet): MR1668586

Zentralblatt MATH: 0923.76025

Digital Object Identifier: doi:10.1007/BF02392586

A. Constantin and J. Escher, Global solutions for quasilinear parabolic problems, J. Evol. Equ. 2 (2002), no. 1, 97--111.

Mathematical Reviews (MathSciNet): MR1890883

Zentralblatt MATH: 1004.35070

Digital Object Identifier: doi:10.1007/s00028-002-8081-2

P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 4, 603--621. (electronic).

Mathematical Reviews (MathSciNet): MR2338368

Digital Object Identifier: doi:10.1090/S0273-0979-07-01184-6

H. Kozono and Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000), no. 1, 191--200.

Mathematical Reviews (MathSciNet): MR1794270

Zentralblatt MATH: 0985.46015

Digital Object Identifier: doi:10.1007/s002200000267

A. Majda, and A. Bertozzi Vorticity and incompressible flow, Cambridge University Press, Cambridge, 2002.

Mathematical Reviews (MathSciNet): MR1867882

H. Okamoto, H. Well-posedness of the generalized Proudman-Johnson equation without viscosity, J. Math. Fluid Mech. 11 (2009), 46--59.

Mathematical Reviews (MathSciNet): MR2481858

Zentralblatt MATH: 1162.76311

Digital Object Identifier: doi:10.1007/s00021-007-0247-9

H. Okamoto H. and J. Zhu, Some similarity solutions of the Navier-Stokes equations and related topics, Proceedings of 1999 International Conference on Nonlinear Analysis (Taipei), Taiwanese J. Math. 4 (2000), no. 1, 65--103.

Mathematical Reviews (MathSciNet): MR1757984

Zentralblatt MATH: 0972.35090

I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point, J. Fluid Mech. 12 (1962), 161--168.

Mathematical Reviews (MathSciNet): MR136211

Zentralblatt MATH: 0113.19501

Digital Object Identifier: doi:10.1017/S0022112062000130

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