AI 中文总结
通过改进已有方法，建立三维Lipschitz区域Stokes算子的L^p预解估计，证明其生成一致有界解析半群，补充了插值相关定理，修正了大p下估计可能失效的传统认知。
AI 中文摘要
通过改进文献\uc810{AGH-2015, GS2026a}提出的方法，我们在三维欧氏空间中有界Lipschitz区域Ω中，对任意1<p≤∞，建立了Stokes算子的L^p预解估计。由此可得，Stokes算子在L^p_σ(Ω)中生成一个一致有界的解析半群。该结果尤为出人意料，因为长期以来人们认为三维Lipschitz区域中的L^p预解估计对大p可能不成立。在附录中，我们证明了关于L^2_σ(Ω)与L^∞_σ(Ω)之间插值的一个定理。
英文摘要
By refining the approach developed in \cite{AGH-2015, GS2026a}, we establish resolvent estimates in $L^p_σ(Ω)$ for the Stokes operator in a bounded Lipschitz domain $Ω$ in $\R^3$ for any $(3/2)-\e< p\le \infty$, where $\e>0$ depends on $Ω$. As a consequence, the Stokes operator generates a uniformly bounded analytic semigroup in $L^p_σ(Ω)$. The results are particularly surprising, as it is long believed that the $L^p$ resolvent estimates in three-dimensional Lipschitz domains may fail for large $p$. In the Appendices we prove a theorem on interpolation between $L^2_σ(Ω)$ and $L^\infty_σ(Ω)$. Given $1< p< \infty$ and a bounded Lipschitz domain $Ω$ in $\R^d, d\ge 2$, we show that the resolvent estimate cannot hold in $L^p(Ω; \C^d)$ unless the Helmholtz projection is bounded on $L^p(Ω; \C^d)$. This explains the counter-example constructed in \cite{Deuring-2001}.
Comments26 pages. Appendix B is added. Comments are welcome
