AI 中文总结
研究二维凯勒-塞格尔型方程，通过在临界齐次特里贝尔-利佐金空间发展热流估计，引入时间精细化空间，建立多种估计，得到该方程在临界空间的局部和全局适定性，相关双分量系统也有类似结果。
AI 中文摘要
我们在临界齐次特里贝尔-利佐金空间中发展热流估计，并将其应用于二维凯勒-塞格尔型方程。对于\(q>2\)，证明了从\(\dot F^0_{1,q}(\mathbb R^2)\)到\(L^2(0,T;\dot F^1_{1,q}(\mathbb R^2))\)的热流无界，这促使引入新的时间精细化特里贝尔-利佐金空间。在此框架下，建立了一系列热平滑估计及相应的最大正则性。还证明了巴拿赫值皮特雷估计等。作为应用，得到二维抛物-椭圆凯勒-塞格尔方程在临界空间\(\dot F^0_{1,q}(\mathbb R^2)\)，\(2\le q<\infty\)中任意初值的局部适定性和小初值的全局适定性，相关双分量漂移-扩散系统也有类似结果。
英文摘要
We develop heat-flow estimates in critical homogeneous Triebel--Lizorkin spaces and apply them to two-dimensional Keller--Segel type equations. For $q>2$, we show that the heat-flow from $\dot F^0_{1,q}(\mathbb R^2)$ to $L^2(0,T;\dot F^1_{1,q}(\mathbb R^2))$ is unbounded. This motivates the introduction of new spaces, time-refined Triebel--Lizorkin spaces, in which the time norm is taken before the dyadic summation. In this framework, we establish a family of heat smoothing estimates together with the corresponding maximal regularity. We further prove Banach-valued Peetre estimates, Banach-valued Jawerth-type estimate, homogeneous Poisson estimate, and an endpoint bilinear estimate for the Keller--Segel drift. As an application, we obtain local well-posedness for arbitrary initial data and global well-posedness for sufficiently small initial data in the critical space $\dot F^0_{1,q}(\mathbb R^2)$, $2\le q<\infty$, for the two-dimensional parabolic--elliptic Keller--Segel equation. The same analytic framework also yields analogous well-posedness results for a related two-component drift--diffusion system.
