Partial Differential Equations
My research subject is mathematical analysis of partial differential equations which describe space-time evolution of waves with strong interactions, arising in various physical contexts, such as plasma, superfluid, and water waves. The goal is to derive from the equations universal properties of solutions independent of the physical backgrounds. Solutions of partial differential equations in general contain information of infinite dimensions. When the equation has nonlinearity in addition, it is almost always impossible to write down the solutions in explicit forms. It has become easier to calculate approximate solutions thanks to the progress of computers, yet they can obtain at most finite amount of information. Hence it is a role of mathematics to capture the whole possibility in infinite dimensions. The mathematical study of nonlinear wave equations has developed starting from the most basic problems such as unique existence of solutions, and mainly by analysis of individual specific solutions such as blow-up and solitons. Recently, it is gradually exploring new aspects of infinite dimensions, such as classification of entire set of solutions, relations between different solutions, and mechanism of dynamical transitions.
- 1999 Assistant Professor, Kobe University (Faculty of Sciences)
- 2001 Associate Professor, Nagoya University (Graduate School of Mathematics)
- 2005 Associate Professor, Kyoto University (Graduate School of Sciences)
- 2015 Professor, Osaka University (Graduate School of Information Science and Technology)
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