Geometric harmonic analysis for diffusive heat equations

扩散热方程的几何调和分析

• 批准号: 261100-2007
• 负责人: Xiao, Jie
• 金额: $ 0.87万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=364958
• 关键词: Geometric; harmonic; analysis; diffusive; heat

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.

调和分析与偏微分方程的相互作用由来已久，最早可以追溯到拉普拉斯方程的狄里克莱特问题。如今，对偏微分方程各个方面的研究是相互作用的一个富有成效的基础。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，以上述两个核心区的边界为中心。要研究的特殊课题包括：*由非齐次热方程解产生的抛物型Riesz型势和容量的时空估计。*非线性热方程的适定性，如半线性耗散、耗散准地转、和推广的Euler和Navier-Stokes方程。*Carleson型度量与Riesz型容量通过根据初始数据对齐次热方程和薛定谔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。*与势的凸性有关的尖锐的几何和/或分析不等式(例如，电容Brunn-Minkowski和对数Sobolev估计)和一些耗散方程。基本思想和技术将涉及发展函数空间及其算子上的调和分析的一些几何特征，建立在复调和分析、算子势理论、算子势理论的许多重大进展的基础上以及最近几十年由许多人制作的凸几何学。期望本文的研究将为演化方程和凸几何分析领域提供有用的工具。