## Journal of the Mathematical Society of Japan

### Blow-up problems in the strained vorticity dynamics and critical exponents

Hisashi OKAMOTO

#### Abstract

Two partial differential equations are studied from the view-point of critical exponents. They are equations for a scalar unknown of one spatial variable, and produce self-similar solutions of the Navier-Stokes equations. Global existence and blow-up are examined for them, and the critical exponent separating them is determined.

#### Article information

Source
J. Math. Soc. Japan, Volume 65, Number 4 (2013), 1079-1099.

Dates
First available in Project Euclid: 24 October 2013

https://projecteuclid.org/euclid.jmsj/1382620186

Digital Object Identifier
doi:10.2969/jmsj/06541079

Mathematical Reviews number (MathSciNet)
MR3127817

Zentralblatt MATH identifier
1282.35096

Subjects
Primary: 35B44: Blow-up
Secondary: 35B33: Critical exponents 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

#### Citation

OKAMOTO, Hisashi. Blow-up problems in the strained vorticity dynamics and critical exponents. J. Math. Soc. Japan 65 (2013), no. 4, 1079--1099. doi:10.2969/jmsj/06541079. https://projecteuclid.org/euclid.jmsj/1382620186

#### References

• 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), 61–66.
• J. M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. in Appl. Mech., 1 (1948), 171–199.
• P. Constantin, P. D. Lax and A. J. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math., 38 (1985), 715–724.
• C. R. Doering and J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts Appl. Math., Cambridge University Press, 1995.
• H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t =\Delta u + u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109–124.
• Th. Gallay and C. E. Wayne, Global stability of vortex solutions of the two dimensional Navier-Stokes equation, Comm. Math. Phys., 255 (2005), 97–129.
• J. D. Gibbon, The three-dimensional Euler equations: Where do we stand?, Phys. D, 237 (2008), 1894–1904.
• Y. Giga and M.-H. Giga, Nonlinear Partial Differential Equations, Kyoritsu Shuppan, 1999 (in Japanese). English translation by M.-H. Giga, Y. Giga and J. Saal, Progr. Nonlinear Differential Equations Appl., 79, Springer-Verlag, 2010.
• Y. Giga and T. Kambe, Large time behavior of the vorticity of two-dimensional viscous flow and its application to vortex formation, Comm. Math. Phys., 117 (1988), 549–568.
• R. E. Grundy and R. McLaughlin, Three-dimensional blow-up solutions of the Navier-Stokes equations, IMA J. Appl. Math., 63 (1999), 287–306.
• T. Y. Hou and Z. Lei, On the stabilizing effect of convection in three-dimensional incompressible flows, Comm. Pure. Appl. Math., 62 (2009), 501–564.
• T. Kambe, A class of exact solutions of two-dimensional viscous flow, J. Phys. Soc. Japan, 52 (1983), 834–841.
• T. Kambe, A class of exact solutions of the Navier-Stokes equations, Fluid Dynam. Res., 1 (1986), 21–31.
• S. Kida and K. Ohkitani, Spatiotemporal intermittency and instability of a forced turbulence, Phys. Fluids A, 4 (1992), 1018–1027.
• H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev., 32, (1990), 262–288.
• T. S. Lundgren, Strained spiral vortex model for turbulent fine structure, Phys. Fluids, 25 (1982), 2193–2203
• Y. Maekawa, On the existence of Burgers vortices for high Reynolds numbers, J. Math. Anal. Appl., 349 (2009), 181–200.
• Y. Maekawa, Existence of asymmetric Burgers vortices and their asymptotic behavior at large circulations, Math. Models Methods Appl. Sci., 19 (2009), 669–705.
• A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math., 39 (1986), S187–S220.
• H. K. Moffatt, The interaction of skewed vortex pairs: a model for blow-up of the Navier-Stokes equations, J. Fluid Mech., 409 (2000), 51–68.
• H. K. Moffatt, S. Kida and K. Ohkitani, Stretched vortices –- the sinews of turbulence; large-Reynolds-number asymptotics, J. Fluid Mech., 259 (1994), 241–264.
• K.-I. Nakamura, H. Okamoto and H. Yagisita, Blow-up solutions appearing in the vorticity dynamics with linear strain, J. Math. Fluid Mech., 6 (2004), 157–168.
• K. Ohkitani and J. D. Gibbon, Numerical study of singularity formation in a class of Euler and Navier-Stokes flows, Phys. Fluid, 12 (2000), 3181–3194.
• K. Ohkitani and H. Okamoto, Blow-up problems modeled from the strain-vorticity dynamics, In: Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics, Sapporo, 2001, (eds. H. Fujita, S. T. Kuroda and H. Okamoto), Sûrikaisekikenkyûsyo Kôkyûroku, 1234 (2001), 240–250. Downloadable at http://www.kurims.kyoto-u.ac.jp/\~ kyodo/kokyuroku/contents/pdf/1234-20.pdf
• H. Okamoto, Exact solutions of the Navier-Stokes equations via Leray's scheme, Japan J. Indust. Appl. Math., 14 (1997), 169–197.
• H. Okamoto and K. Ohkitani, On the role of the convection term in the equations of motion of incompressible fluid, J. Phys. Soc. Japan, 74 (2005), 2737–2742.
• A. C. Robinson and P. G. Saffman, Stability and structure of stretched vortices, Stud. Appl. Math., 70 (1984), 163–181.
• P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser, Basel, 2007.
• E. Yanagida, Blow-up of solutions of nonlinear heat equations, In: Blow-up and Aggregation, University of Tokyo Press, 2006, 1–50. (in Japanese)