TAIWANESE JOURNAL OF MATHEMATICS
Vol. 4, No. 1, pp. 65-103, March 2000
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SOME SIMILARITY SOLUTIONS OF THE NAVIER-STOKES EQUATIONS AND RELATED TOPICS |
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Hisashi Okamoto* and Jinghui Zhu** |
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| Abstract. |
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| We consider a semilinear equation arising from the Navier-Stokes equations - the governing equations of viscous fluid motion - and related model equations. The solutions of the semilinear equation represent a certain class of exact solutions of the Navier-Stokes equations. Both the equation and our models have nonlocal terms. We will show that the nonlocality will play an intriguing role for the blow-up and/or global existence of the solutions and that the convection term, which is often neglected in the study of the blow-up problems, plays a very decisive role. In addition to our new contributions, open problems and known facts are surveyed. |
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Received December 15, 1999.
Communicated by P. Y. Wu.
2000 Mathematics Subject Classification: 35K55, 35Q30, 76D05.
Key words and phrases:
Navier-Stokes equation, similar transform, blow-up, nonlocal convection.* Supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan.
** On leave from Dept. of Mathematics, Xiamen University, China. Supported by the scholarship awarded by the Ministry of Education, Science, Sports and Culture of Japan.