## Research Theme

Research on nonlinear partial differential equations

## Overview of Research

##### Mathematical analysis of fluid from a macro-perspective

- By deepening and developing the methods that our group have worked on thus far, functional analysis, real analysis, Fourier restriction, computer-assisted authentication, variational method, and viscous analysis, etc., as well as by introducing new approaches for nonlinear partial differential equations, we will conduct research on the existence of stationary solutions for Navier-Stokes equations and nonlinear Schrodinger equations, and on the behavior of solutions in infinite distances.
- Establish stability theories for stationary solutions. In particular, we aim to develop intrinsically new methods for spectral analysis of variable coefficient linear partial differential operators, and contribute to new developments in theories of stabilities, which have remained unresolved thus far. We will also work together with the department of mechanical science and aeronautics to tackle unresolved issues such as aerodynamic noise pollution.
- Carry out mathematical analysis of fluid motions with free boundaries. We will employ methods developed from the principle of maximum regularity. As with point 2. above, an analysis of asymptotical behavior of solutions in infinite time and distances through spectral analysis. The relationship with application lies in the mathematical analysis of multiphase flows, and the first step toward clarifying complex flows in multiphase flows is to conduct joint research of benchmark phenomena and their mathematical analyses, together with the department of mechanical science and aeronautics.
- Carry out research on nonlinear Schrodinger equations that describe the dynamics of low-temperature superfluid liquid helium layers and Bose-Einstein condensates. In particular, mathematical analyses based on functional analysis, computer-assisted authentication, and viscous analysis methods will be carried out.

##### Mathematical analysis of fluid from a meso-perspective

- Investigate the motion of water molecules from the standpoint of kinetic theory, and establish the fundamental similarities and differences between the microscopic picture of atoms and molecules, and the macroscopic continuum picture.
- Establish mathematically the overall space of molecular motion, introduce an invariant probability measure, define the energy functional, and derive mathematical equations for fluids using the variational method.
- Take up the challenge of conducting joint research with the department of mechanical science and aeronautics on unresolved fluid mechanic issues such as the occurrence of bubbles in water and cavitation.
- Conduct joint research with the applied chemistry departments on the construction of mathematical models, such as for chemical reactions in nanofluids.