$$\begin{gathered}\partial_t \rho+\nabla\cdot(\rho u)=0\\ \partial_t(\rho u)+\nabla\cdot(\rho u\otimes u)+\nabla P(\rho)=\operatorname{div}\mathbb S(u)\end{gathered}$$

$$\partial_t\theta+R^\perp\theta\cdot\nabla\theta+\kappa\Lambda^\alpha\theta=f$$

$$\partial_t\omega+u\cdot\nabla\omega=0$$

$$\begin{gathered}\partial_t u-\Delta u+(u\cdot\nabla)u+\nabla p=0\\ \nabla \cdot u = 0\end{gathered}$$

$$\begin{gathered}\partial_t u-\Delta u+u\cdot\nabla u+\nabla p=(B\cdot\nabla)B\\ \partial_tB-\Delta B+u\cdot\nabla B-B\cdot\nabla u+\nabla\times((\nabla\times B)\times B)=0\\ \nabla\cdot u=\nabla\cdot B=0\end{gathered}$$

$$\begin{gathered}\partial_t v-\Delta_{\rm h}v+v\cdot\nabla v+\nabla p=N\theta e_3\\ \partial_t\theta-\Delta_{\rm h}\theta+v\cdot\nabla\theta=-Nv_3\\ \nabla\cdot v=0\end{gathered}$$

$$\begin{gathered}-\Delta U+(U\cdot\nabla)U+\nabla P=f\\ \nabla \cdot U = 0\end{gathered}$$

$$\begin{gathered}\partial_tu+\Omega e_3\times u-\Delta u+(u\cdot\nabla)u+\nabla p=0\\ \nabla\cdot u=0\end{gathered}$$

$$\begin{gathered}\partial_t \rho+\nabla\cdot(\rho u)=0\\ \varepsilon^2(\partial_tu+u\cdot\nabla u)+\nabla\rho=\varepsilon\nu\Delta u\end{gathered}$$

$$\begin{gathered}\partial_t u-\Delta_{\rm h}u+(u\cdot\nabla)u+\nabla p=0\\ \nabla\cdot u=0\end{gathered}$$

$$\kappa\Lambda^\alpha\Theta+R^\perp\Theta\cdot\nabla\Theta=f$$

$$\partial_t\omega+u\cdot\nabla\omega-\Delta\omega=\omega\cdot\nabla u$$

Mikihiro Fujii

Institution

Graduate School of Science, Nagoya City University

E-mail

fujii.mikihiro [at] nsc.nagoya-cu.ac.jp

Research

Mathematical analysis of nonlinear PDEs in fluid dynamics

My research field is the mathematical analysis of nonlinear partial differential equations. In particular, I mainly study partial differential equations arising in fluid dynamics, such as the Navier-Stokes equations. The methods I use are primarily harmonic analysis, and I am interested in well/ill-posedness in scaling critical spaces and quantitive properties via the asymptotic behavior or singular limit of solutions.