分野別セミナー

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 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 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).