Hiroshi Fujita

His most widely cited paper, published in 1966,{{cite journal |first=Hiroshi |last=Fujita|title=On the blowing up of solutions of the Cauchy problem for {{math|u_t {{=}} Δu + u^1+α }} |journal=Journal of the Faculty of Science, University of Tokyo. Sect. 1|volume=13 |year=1966 |issue=2 |pages=109–124 |doi=10.15083/00039873 |url=https://repository.dl.itc.u-tokyo.ac.jp/record/39882/files/jfs130201.pdf }} studied the partial differential equation

:$\backslash frac\{\backslash partial\; u\}\{\backslash partial\; t\}=\backslash frac\{\backslash partial^2u\}\{\backslash partial\; x\_1^2\}+\backslash cdots+\backslash frac\{\backslash partial^2u\}\{\backslash partial\; x\_n^2\}+u^p,$

and showed that there is a "threshold" value {{math|p₀ > 1}} for which {{math|p > p₀}} implies the existence of nonconstant solutions which exist for all positive {{mvar|t}} and all real values of the {{math|x}} variables. By contrast, if {{mvar|p}} is between {{math|1}} and {{math|p₀}} then such solutions cannot exist. This paper initiated the study of similar and analogous phenomena for various parabolic and hyperbolic partial differential equations. The impact of Fujita's paper is described by the well-known survey articles of Levine (1990){{cite journal |first=Howard A. |last=Levine |title=The role of critical exponents in blowup theorems |journal=SIAM Rev. |volume=32 |year=1990 |issue=2 |pages=262–288 |doi=10.1137/1032046 |url=https://lib.dr.iastate.edu/math_pubs/44 }} and Deng & Levine (2000).{{cite journal |first1=Keng |last1=Deng |first2=Howard A. |last2=Levine |title=The role of critical exponents in blow-up theorems: the sequel |journal=J. Math. Anal. Appl. |volume=243 |year=2000 |issue=1 |pages=85–126 |doi=10.1006/jmaa.1999.6663 |doi-access=free }}

In collaboration with Kato, Fujita applied the semigroup approach in evolutionary partial differential equations to the Navier–Stokes equations of fluid mechanics. They found the existence of unique locally defined strong solutions under certain fractional derivative-based assumptions on the initial velocity. Their approach has been adopted by other influential works, such as Giga & Miyakawa (1985), to allow for different assumptions on the initial velocity.{{cite journal |first1=Yoshikazu |last1=Giga |first2=Tetsuro |last2=Miyakawa |title=Solutions in {{math|L_r}} of the Navier–Stokes initial value problem |journal=Arch. Rational Mech. Anal. |volume=89 |year=1985 |issue=3 |pages=267–281 |doi=10.1007/BF00276875 |bibcode=1985ArRMA..89..267G |s2cid=122196523 }} The full understanding of the smoothness and maximal extension of such solutions is currently considered as a major problem of partial differential equations and mathematical physics.

• Tosio Kato and Hiroshi Fujita. On the nonstationary Navier-Stokes system. Rend. Sem. Mat. Univ. Padova 32 (1962), 243–260.
• Hiroshi Fujita and Tosio Kato. On the Navier-Stokes initial value problem. I. Arch. Rational Mech. Anal. 16 (1964), 269–315.
• Hiroshi Fujita. [https://repository.dl.itc.u-tokyo.ac.jp/record/39882/files/jfs130201.pdf On the blowing up of solutions of the Cauchy problem for {{math|u_t {{=}} Δu + u^1+α}}. J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124.]
• Mathematical theory of sedimentation analysis (book)
• Functional-Analytic Methods for Partial Differential Equations (1990, Springer), Proceedings of a Conference and a Symposium held in Tokyo, Japan, July 3–9, 1989. Edited by Hiroshi Fujita, Teruo Ikebe and Shige T. Kuroda.
• Proceedings of the Ninth International Congress on Mathematical Education, Edited by Hiroshi Fujita et al.

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Category:Japanese mathematicians

Category:1928 births

Category:People from Osaka

Category:Scientists from Osaka

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Category:University of Tokyo alumni

Category:Living people