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This special issue is dedicated to Professor Hideo Kozono, Waseda University, on the occasion of his 60th birthday.

Hideo Kozono was born in Yokohama, Japan on November 29, 1958. In 1987, he earned his doctorate degree from Hokkaido University under the supervision of Shozo Koshi. At that time, he was also visiting Tokyo Institute of Technology and studying under Atsushi Inoue. In that same year, he was appointed to an assistant professorship at Nagoya University. In 1991, he was promoted to an associate professor at Kyushu University. In 1993, he returned to Nagoya University and remained there until 1999. In 1999, he was promoted to a full professor at Tohoku University, where he then spent the next 13 years. In 2012, he moved to Waseda University, where he is currently a professor. He has also held a professorship at Tohoku Univerisity since 2018.

From September 1988 to July 1990 and from March 1996 to July 1996, he received a grant from the Alexander von Humboldt Foundation and carried out research visits to University of Paderborn. Since those visits, he has maintained close friendships and research collaborations with many German researchers, especially with Reinhard Farwig and Hermann Sohr.

He published more than 140 research papers and made numerous and important contributions to the functional analytic research on equations in fluid mechanics and to harmonic analysis. He started his research with the analysis of 2D incompressible Euler equations in a time dependent domain [1] and the MHD equations for an incompressible fluid [2, 3]. Then, he studied the asymptotic behavior of the solutions to the incompressible Navier–Stokes equations in bounded or unbounded domains [4,5,6,7,8,9,10,11,12] jointly with Tohru Ozawa, Takayoshi Ogawa, and Hermann Sohr. The Navier–Stokes system has been one of his main subjects up to the present. Jointly with Hermann Sohr and Masao Yamazaki, he found the relation between the net force exerted on the obstacle and the \(L^3\) integrability of stationary solutions to the 3D Navier–Stokes equations and Stokes equations in exterior domains [13, 14]. Later, he also found the relation between the net force and asymptotic decay rate with respect to the time variable of non-stationary solutions to the 3D Navier–Stokes equations in exterior domains [15, 16]. These results showed that the weak \(L^3\) space is more important than the \(L^3\) space for the research of solutions to the 3D Navier–Stokes equations in exterior domains.

In the mid-1990s, Kozono succeeded in proving the uniqueness of the Leray–Hopf weak solutions in the endpoint space of Serrin’s class \(L^{\infty }(0,T;L^n)\) jointly with Hermann Sohr [17]. Also, with Sohr, he proved the regularity of the Leray–Hopf weak solutions in \(L^{\infty }(0,T;L^n)\) under some additional conditions [18].

Jointly with Mitsuhiro Nakao, in [19], he introduced a new approach for the study of time-periodic solutions to the Navier–Stokes equations in unbounded domains by the use of an integral equation defined on an infinite half-line. Their approach enables time-periodic solutions to be investigated by modern functional analytic methods and operator theory.

At the same time, Kozono started to apply harmonic analysis techniques to the study of nonlinear partial differential equations, mainly for the Navier–Stokes equations and obtained many important results. Jointly with Masao Yamazaki, in [20, 21], he introduced new function spaces of Besov-Morrey type and proved the well-posedness of the Navier–Stokes equations in those spaces, which we now call Kozono-Yamazaki spaces. In the mid-1990s, the Kozono-Yamazaki space \(N^{n/p-1}_{p,q,\infty }\) was the largest class for the global well-posedness of the Navier–Stokes equations. Also with Yamazaki, he established the \(L^p_w\) (weak \(L^p\)) theory for the exterior Navier–Stokes problem, and obtained its well-posedness, as well as stability results [22,23,24,25,26,27].

He also established new bilinear estimates and critical Sobolev inequalities in BMO, Besov, and Triebel-Lizorkin spaces [28,29,30,31] jointly with Takayoshi Ogawa, Yasushi Taniuchi, and Yukihiro Shimada. As applications, he obtained new regularity and extension criteria for the Navier–Stokes and Euler equations.

From 2005, Kozono published an outstanding series of joint works with Reinhard Farwig and Hermann Sohr [32,33,34,35,36,37]. In those papers, he established the so-called \(\tilde{L}^q\) theory on the Navier–Stokes equations in very general unbounded domains that may have unbounded boundaries. Among these papers, he showed the existence of weak solutions which satisfy several regularity properties. This result was published in Acta Mathematica, and now regarded as one of the fundamental theories on the Navier–Stokes equations.

From 2008, jointly with Yoshie Sugiyama, Kozono also obtained important results on the Keller–Segel system in the \(L^p\) setting [38,39,40,41,42,43]. Together with Masanori Miura, they also showed the wel-posedness of the Keller–Segel system coupled with Navier–Stokes fluids [44, 45].

From 2009, jointly with Taku Yanagisawa, he has been working on the structure of vector fields in multi-connected domains. They generalized the div-curl lemma and identified a relation between the solvability of the stationary Navier–Stokes equations and the topological structure of the domains [46,47,48,49,50,51]. They went on to generalize these results to exterior domains with Anton Seyfert, Matthias Hieber and Senjo Shimizu [52,53,54,55].

Recently, jointly with Yutaka Terasawa and Yuta Wakasugi, he proved Liouville-type theorems for the stationary Navier–Stokes equations [56, 57]. It is notable that their Liouville-type theorems are applicable to some 2D unbounded domains including exterior domains, while many known Liouville-type theorems for 2D Navier–Stokes equations were proven only for the whole plane \(\mathbb R^2\).

Jointly with Senjo Shimizu, he also proved the well-posedness of non-stationary Navier–Stokes equations with external forces in Lorentz spaces or Besov spaces with scaling invariant norms by establishing the maximal Lorentz regularity (in the time variable ) on the Stokes equations in Besov spaces [58,59,60,61,62]. Their existence theorem enables us to handle such singular data as the Dirac measure and the single layer potential supported on the sphere.

Very recently, jointly with Reinhard Farwig and David Wegmann, he started to investigate the incompressible Navier–Stokes equations in domains with moving boundaries [63]. He also proved the maximal regularity on the Stokes equations with a moving boundary. Together with Kazuyuki Tsuda, they succeeded in constructing time-periodic solutions in \(L^q\) spaces to the Navier–Stokes equations in a bounded domain with a moving boundary [64]. As for other important papers of Kozono, we refer the readers to [66,67,68,69,70,71,72,73,74,75].

Kozono has received several prizes for his remarkable achievements in mathematics. In 2002, he received the Siebold prize from the Federal Republic of Germany. In 2014, he was awarded the Autumn Prize of the Mathematical Society of Japan for his contributions on “Harmonic analytic research on stationary and nonstationary problems for the incompressible Navier–Stokes equations”. In 2016, he also received the “Commendation for Science and Technology" from the Japanese Minister of Education, Culture, Sports, Science and Technology.

Kozono has also contributed to the mathematical community and been a member of several committees. Most significantly, he was the president of the Mathematical Society of Japan from May 2017 to May 2019. From April 2017, he has been one of coordinators of the JSPS Japanese-German Graduate Externship “Mathematical Fluid Dynamics”, jointly operated by Waseda University, The University of Tokyo, and TU Darmstadt, which is founded by JSPS and DFG. Since 2021, he is the chairman of Japan’s National Committee for the International Mathematical Union. He is now recognized as one of the leading mathematicians in the field of partial differential equations.

This volume is inspired by the international conference “Mathematical Fluid Mechanics and Related Topics,” held on September 5–7, 2018 at Tokyo Institute of Technology, on the occasion of Professor Kozono’s 60th birthday, which was organized by Hideyuki Miura, Takahiro Okabe, Tomoyuki Suzuki, Ryo Takada, Yasushi Taniuchi, Erika Ushikoshi, and Hidemitsu Wadade, all his former doctoral students. We express our gratitude to the organizers and participants of that conference and especially to all the authors who contributed to this volume during this difficult time of the COVID-19 pandemic.

To end this preface, we would like to thank the referees for their critical reviews and useful comments on the original papers. We would especially like to thank Professor Hideo Kozono for an everlasting friendship and important mathematics.

  • Guest editors:

  • Kazuhiro Ishige

  • Tohru Ozawa

  • Senjo Shimizu

  • Yasushi Taniuchi

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Japanese “Kagami-Biraki” ceremony at the banquet of Conference dedicated to Professor Kozono’s 60th birthday (2018). From left to right, Reinhard Farwig, Takaaki Nishida, Hideo Kozono, Shigetoshi Kuroda, Yukio Kaneda, and Hisashi Okamoto.