Some similarity solutions of the Navier-Stokes equations and related topics

Taiwanese Journal of Mathematics

Volumn 4, Issue 1, 2000, Pages 65-103

  Okamoto, Hisashi ^a   Zhu, Jinghui ^a  

^a KYOTO UNIVERSITY   (Japan)

Author keywords

Blow up; Navier Stokes equation; Nonlocal convection; Similar transform

Indexed keywords

EID: 0002628126     PISSN: 10275487     EISSN: None     Source Type: Journal    
DOI: 10.11650/twjm/1500407199     Document Type: Article

References (36)

1. G. R. Baker, X. Li and A. C. Morlet, Analytic structure of two 1D-transport equations with nonlocal fluxes, Phys. D 91 (1996), 349-375.
2. J. F. Brady and A. Acrivos, Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier-Stokes equations with reverse flow, J. Fluid Mech. 112 (1981), 127-150.
3. C. Budd, B. Dold and A. Stuart, Blowup in a partial differential equation with conserved first integral, SIAM J. Appl. Math. 53 (1993), 718-742.
4. C. Budd, B. Dold and A. Stuart, Blowup in a system of partial differential equations with conserved first integral. Part II: Problems with convection, SIAM J. Appl. Math. 54 (1994), 610-640.
5. S. Childress, G. R. Ierley, E. A. Spiegel and W.R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point, J. Fluid Mech. 203 (1989), 1-22.
6. P. Constantin, Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations, Comm. Math. Phys. 104 (1986), 311-326.
7. P. Constantin, A. J. Majda and E. G. Tabak, Singular front formation in a model for quasigeostrophic flow, Phys. Fluid 6 (1994), 9-11.
8. _, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity 7 (1994), 1495-1533.
9. 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.
10. S. M. Cox, Two-dimensional flow of a viscous fluid in a channel with porous walls, J. Fluid Mech. 227 (1991), 1-33.
11. K. Deng and H. A. Levine, The role of critical exponents in blowup theorems: the sequel, preprint.
12. W. E and B. Engquist, Blowup of solutions of the unsteady Prandtl's equation, Comm. Pure Appl. Math. 50 (1997), 1287-1293.
13. 1+α, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109-124.
14. A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425-447.
15. Y. Giga and R. V. Kohn, Asymptotic self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297-319.
16. Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40.
17. R. E. Grundy and R. McLaughlin, Global blow-up of separable solutions of the vorticity equation, IMA J. Appl. Math. 59 (1997), 287-307.
18. R. E. Grundy and R. McLaughlin, Three-dimensional blow-up solutions of the Navier-Stokes equations, IMA J. Appl. Math., to appear.
19. K. Hiemenz, Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszlynder, Dinglers J. 326 (1911), 321-324, 344-348, 357-362, 372-376, 391-393, 407-410.
20. T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Func. Anal. 56 (1984), 15-28.
21. t = -A(t)u + F(u), Arch. Rational Mech. Anal. 51 (1973), 371-386.
22. H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), 262-288.
23. 2, Appl. Math. Optim. 2 (1976), 337-350.
24. H. Okamoto, Localization of singularities in inviscid limit - numerical examples, in: Navier-Stokes Equations: Theory and Numerical Methods, R. Salvi, ed., Longman, 1998, pp. 220-236.
25. H. Okamoto and M. Shoji, A spectral method for unsteady two-dimensional Navier-Stokes equations, in: Proc. 3-rd China-Japan Seminar on Numerical Mathematics, Z.-C. Shi and M. Mori, eds., Science Press, Beijing, 1998, pp. 253-260.
26. H. Okamoto and M. Shoji, Boundary layer in unsteady two-dimensional Navier-Stokes equation, in: Recent Developments in Domain Decomposition Methods and Flow Problems, H. Fujita et al., eds., GAKUTO International Series, Mathematical Science and Applications, Vol.11, 1998, pp. 171-180.
27. I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point, J. Fluid Mech. 12 (1962), 161-168.
28. D. Riabouchinsky, Quelques considérations sur les mouvements plans rotationnels d'un liquide, C. R. Acad. Sci. Paris Sér. I Math. 179 (1924), 1133-1136.
29. S. Schochet, Explicit solutions of the viscous model vorticity equation, Comm. Pure Appl. Math. 39 (1986), 531-537.
30. K.-G. Shi, On the existence of solutions of an equation arising in the theory of laminar flow in a uniformly porous channel with injection, SIAM J. Appl. Math. (1987), 526-533.
31. F. M. Skalak and C.-Y. Wang, On the nonunique solutions of laminar flow through a porous tube or channel, SIAM J. Appl. Math. (1978), 535-544.
32. M. Tabata, A finite difference approach to the number of peaks of solutions for semilinear parabolic problems, J. Math. Soc. Japan 32 (1980), 171-191.
33. H. Weyl, On the differential equations of the simplest boundary-layer problems, Ann. Math. 43 (1942), 381-407.
34. P. J. Zandbergen and D. Dijkstra, Von Kármán swirling flows, Ann. Rev. Fluid Mech. 19 (1987), 465-491.
35. M. B. Zaturska, P. G. Drazin and W. H. H. Banks, On the flow of a viscous fluid driven along a channel by suction at porous walls, Fluid Dynam. Res. 4 (1988), 151-178.
36. J. Zhu, Numerical study of stagnation-point flows of incompressible fluid, Japan J. Indust. Appl. Math., to appear.