Geometric harmonic analysis for diffusive heat equations

调和分析和偏微分方程之间的相互作用有着悠久的历史，可以追溯到拉普拉斯方程的狄利克雷问题。如今，一个富有成效的地面的相互作用是研究的各个方面的偏微分方程。特别强调几何容量测量理论方法，本研究计划将通过线性-非线性热方程的柯西问题围绕上述两个核心区域之间的边界进行。特别要研究的主题包括：* 时空估计抛物Riesz型潜力和能力所产生的解决方案的非齐次热方程。非线性热方程的适定性，如半线性耗散、耗散准地转和广义欧拉方程和Navier-Stokes方程。Carleson型测度与Riesz型容量，通过对齐次热方程和Schroedinger方程的解的先验估计。与热半群算子相关的各种函数空间的函数论和/或算子论特征。尖锐的几何和/或解析不等式（如容性Brunn-Minkowski和对数Sobolev估计）和一些耗散方程的凸性。主要思想和技术将涉及发展函数空间及其算子上调和分析的一些几何特征，建立在复调和分析，算子势理论，和凸几何学在过去的几十年里被许多人所创造。期望本研究能为发展方程和凸几何分析领域提供有用的工具。

The interaction between harmonic analysis and partial differential equations (PDE) has a long history going back to the Dirichlet problem for the Laplace equation. Nowadays, a fruitful ground for the interplay is the study of various aspects of PDE. With special emphasis on the geometric capacity-measure theory approach, this proposed research program will be centered about the boundary between the above two core areas via the Cauchy problem for the linear-nonlinear heat equations. Particular topics to be investigated include:* Spacetime estimates for the parabolic Riesz type potentials and capacities generated by solutions to the inhomogeneous heat equations.* Wellposedness of the nonlinear heat equations such as the semilinear dissipative, dissipative quasi-geostrophic, and generalized Euler and Navier-Stokes equations.* Carleson type measures versus Riesz type capacities through a priori estimations of solutions to the homogeneous heat and Schroedinger equations in terms of the initial data.* Function-theoretic and/or operator-theoretic characterizations of various function spaces associated to the heat semigroup operators.* Sharp geometric and/or analytic inequalities (e.g. the capacitary Brunn-Minkowski and logarithmic Sobolev estimates) related to the convexity of potentials and some dissipative equations.The principle ideas and techniques will involve developing some geometric features of harmonic analysis over function spaces and their operators, building on many significant advances in complex-harmonic analysis, operator-potential theory, and convex geometry made in the last decades by a number of people. It is expected that this research will provide useful tools for the field of evolution equations and convex geometric analysis.