## Good-bye my friend… (In Memoriam Seiji Ukai)

Seiji Ukai passed away on November 7 in Kyoto at the age of 73. He was a great mathematician and a beautiful soul.

My first encounter with Seiji took place in a conference in Maui, back in 1997: there he had shown how to recast the celebrated Lanford validity theorem in terms of an abstract Cauchy-Kowalevskaya theorem. I was struck by his quiet voice and his habit to quote other scientists by first names within his talk — quite contrary to the Japanese standards of his generation. I asked him for a reprint, which later came to my office in beautiful, string-sealed envelope accompanied with marvelously fine writing. Years later in Osaka I had with him one of the best Japanese dinners in my life.

Seiji was expert in using clever analytic tools to handle equations from fluid mechanics. His study of the initial layer arising in the incompressible limit of the Euler equations [1], his breakthrough on the existence of solutions to the Kadomtsev-Petviashvili (KP) equations [2], for instance, have become classical works.

A particularly important contribution of his was the first proof of existence of solutions to the Boltzmann equation near equilibrium [3]. That celebrated work, published slightly after the 100th aniversary of the discovery of Boltzmann’s equation as we know it, was a milestone, combining careful a priori estimates on the collision operator with a tricky spectral analysis.

Seiji was known for his informal style, friendly nature, quiet temper and extremely modest manners. He was witty and pleasant, always looking carefully for the most beautiful proofs and paths, both in mathematics and in his daily life. He was beloved by everybody, colleagues and students.

Good-bye Seiji.

[1] The incompressible limit and the initial layer of the compressible Euler equation. *J. Math. Kyoto Univ. 26* (1986), no. 2, 323–331.

[2] Local solutions of the Kadomtsev-Petviashvili equation. *J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36* (1989), no. 2, 193–209.

[3] On the existence of global solutions of mixed problem for non-linear Boltzmann equation. *Proc. Japan Acad. 50* (1974), 179–184.