题目📃：New Approach to the Free Boundary Problem of the Navier-Stokes Equations

报告人👀：Prof. Keiichi Watanabe （Suwa University of Science）

时间🍍：2024年3月5日（星期二） 15:30-16:30

地点：宁静楼110室

摘要：In this talk, we consider the maximal regularity theorem of the Stokes equations with free boundary conditions in the half-space within $L_1$-in-time and $\mathcal{B}^s_{q,1}$-in-space framework with $(q,s)$ satisfying $1 < q < \infty$ and $-1 + 1/q < s < 1/q$, where $\mathcal{B}^s_{q,1}$ stands for either homogeneous or inhomogeneous Beosv spaces. The proof is based on density, duality, and interpolation techniques since the operator-valued Fourier multiplier theorem due to Weis (2001) cannot be used in our functional framework. To be precise, the maximal $L_1$-regularity theorem is proved by estimating the Fourier-Laplace inverse transform of the solution to the generalized Stokes resolvent problem with inhomogeneous boundary conditions, and thus our theory can be regarded as an extension of a classical $C_0$-analytic semigroup theory. As an application, we show the unique existence of a local strong solution to the Navier-Stokes equations with free boundary conditions for arbitrary initial data in $B^s_{q,1} (\mathbb{R}^d_+)$, where $q$ and $s$ satisfy $d−1 < q \le d$ and $−1+d/q < s < 1/q$, respectively. If we assume that the initial data a are small in $\dot B^{-1+d/q}_{q,1} (\mathbb{R}^d_+)^d$, $d−1 < q < 2d$, then the unique existence of a global strong solution to the system is proved. This talk is based on a joint work with Yoshihiro Shibata (Waseda University).

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