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带自由边界条件的Navier-Stokes方程的最大$L_1$-正则性：基于广义半群理论

Maximal $L_1$-regularity of the Navier-Stokes equations with free boundary conditions via a generalized semigroup theory

Yoshihiro Shibata, Keiichi Watanabe

备注：
Minor version. Accepted for publication in Journal of Differential Equations
发表信息：
Journal of Differential Equations 426 (2025), 495-605
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AI总结：

本文通过广义半群理论，在时间$L_1$和空间$\mathcal B^s_{q,1}$框架下证明了带自由边界条件的Stokes方程的最大$L_1$-正则性，并应用于Navier-Stokes方程，得到了局部和全局强解的唯一存在性。

摘要：

本文发展了一种新方法，用于证明半空间$\mathbb R^d_+$（$d \ge 2$）中带自由边界条件的Stokes方程在时间$L_1$和空间$\mathcal B^s_{q, 1}$框架下的最大正则性定理，其中$(q, s)$满足$1 < q < \infty$和$1 + 1 / q < s < 1 / q$，这里$\mathcal B^s_{q, 1}$表示齐次或非齐次Besov空间。特别地，我们在时间$L_1$和空间$\mathcal B^s_{q,1}$框架下建立了一个广义半群理论，该理论将经典的$C_0$-解析半群理论推广到非齐次边界条件的情形。最大$L_1$-正则性定理通过估计带非齐次边界条件的广义Stokes预解问题解的Fourier--Laplace逆变换来证明，其中使用了密度和内插论证。该最大$L_1$-正则性定理被应用于证明：对于任意初始数据$\boldsymbol a$属于$B^s_{q, 1} (\mathbb R^d_+)^d$，其中$q$和$s$分别满足$d-1 < q \le d$和$-1+d/q < s < 1/q$，带自由边界条件的Navier--Stokes方程存在唯一的局部强解。若进一步假设初始数据$\boldsymbol a$在$\dot B^{1 + d / q}_{q, 1} (\mathbb R^d_+)^d$中很小，其中$d 1 < q < 2 d$，则证明了该系统存在唯一的全局强解。

英文摘要：

This paper develops a new approach to show the maximal regularity theorem of the Stokes equations with free boundary conditions in the half-space $\mathbb R^d_+$, $d \ge 2$, within the $L_1$-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 Besov spaces. In particular, we establish a generalized semigroup theory within an $L_1$-in-time and $\mathcal B^s_{q,1}$-in-space framework, which extends a classical $C_0$-analytic semigroup theory to the case of inhomogeneous boundary conditions. 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, where density and interpolation arguments are used. The maximal $L_1$-regularity theorem is applied to show the unique existence of a local strong solution to the Navier--Stokes equations with free boundary conditions for arbitrary initial data $\boldsymbol a$ in $B^s_{q, 1} (\mathbb R^d_+)^d$, 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 $\boldsymbol a$ are small in $\dot B^{1 + d / q}_{q, 1} (\mathbb R^d_+)^d$, $d 1 < q < 2 d$, then the unique existence of a global strong solution to the system is proved.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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