Diffusion and Regularity

扩散性和规律性

• 批准号: 2054888
• 负责人: Luis Silvestre
• 金额: $ 27.51万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054888&HistoricalAwards=false
• 关键词: Diffusion; Regularity

The investigator will study evolutionary equations which exhibit some sort of diffusive behavior, understood in a broad sense. The most classical diffusion equation is the heat equation, describing the evolution of the distribution of temperature in some homogeneous media. One remarkable characteristic of the heat equation is that, even for very irregular initial data, the solutions become immediately smooth. This behavior is observed in many other partial differential equations that are classified as "parabolic". In general, the regularization effect of the equation plays a key role in our understanding of the nature of solutions and in obtaining a priori bounds for them. The investigator seeks to understand the regularization effect in equations whose diffusion appears in a nonstandard form. For example, in equations with integral diffusion, like the Boltzmann equation from statistical mechanics, in fully nonlinear parabolic equations, or in entropy solutions of inviscid conservation laws. The project provides research training opportunities for graduate students.The study of integro-differential equations has been a very active area of research in the last twenty years. The principal investigator played a central role in the development of regularity results. The equations are motivated, for the most part, from probabilistic models involving jump processes. More recently, the methods are also finding applications in deterministic models. The investigator has been studying the Boltzmann equation from this perspective. It is a model from statistical mechanics describing the evolution of the density of particles in a dilute gas. When the particles are assumed to repel each other in small scales by a power law potential, the equation exhibits a subtle regularization effect, driven by a nonlinear integro-differential diffusion that acts like an elliptic operator of fractional order. In the most singular scenario, the Boltzmann equation turns into the Landau equation, which is a kinetic equation with a more classical second order diffusion. The principal investigator is also interested in studying more general and abstract fully nonlinear parabolic equations, in order to understand their regularity and possible singularity formation. Even in conservation law equations, without any explicit diffusion term, there is a subtle regularization effect in low order Sobolev spaces that can be arguably understood as a remnant of the vanishing viscosity approximation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

研究者将研究表现出某种扩散行为的演化方程，从广义上理解。最经典的扩散方程是热方程，它描述的是某些均匀介质中温度分布的演化。热方程的一个显著特点是，即使对于非常不规则的初始数据，解也会立即变得光滑。这种行为在许多其他被归类为“抛物型”的偏微分方程中也可以观察到。一般来说，方程的正则化效应在我们理解解的性质和获得解的先验界方面起着关键作用。研究者试图理解扩散以非标准形式出现的方程中的正则化效应。例如，在积分扩散方程中，如统计力学中的玻尔兹曼方程，在完全非线性抛物方程中，或在无粘守恒定律的熵解中。该项目为研究生提供了研究训练的机会。积分微分方程的研究在过去的二十年里一直是一个非常活跃的研究领域。主要研究者在规律性结果的开发中发挥了核心作用。方程的动机，在大多数情况下，从概率模型涉及跳跃过程。最近，这些方法也在确定性模型中找到了应用。研究者一直从这个角度研究玻尔兹曼方程。这是一个来自统计力学的模型，描述了稀气体中粒子密度的演变。当粒子被假设为在小尺度上通过幂律势相互排斥时，方程表现出微妙的正则化效应，由非线性积分微分扩散驱动，其作用类似于分数阶椭圆算子。在最奇异的情况下，玻尔兹曼方程变成朗道方程，这是一个具有更经典的二阶扩散的动力学方程。主要研究者也有兴趣研究更一般和抽象的完全非线性抛物方程，以了解其规律性和可能的奇异性形成。即使在守恒定律方程中，没有任何显式扩散项，低阶Sobolev空间中也存在一种微妙的正则化效应，这可以被理解为消失粘性近似的残余。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

项目成果

Singular solutions to parabolic equations in nondivergence form

非散度形式的抛物型方程的奇异解

• DOI: 10.2422/2036-2145.202011_110
• 发表时间: 2022
• 期刊: ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
• 作者: Silvestre, Luis

Entropy dissipation estimates for the Boltzmann equation without cut-off

无截止的玻尔兹曼方程的熵耗散估计

• DOI: 10.3934/krm.2023006
• 发表时间: 2023
• 期刊: Kinetic and Related Models
• 影响因子: 1
• 作者: Chaker, Jamil;Silvestre, Luis
• 通讯作者: Silvestre, Luis

Holder estimates for kinetic Fokker-Planck equations up to the boundary

动态 Fokker-Planck 方程达到边界的 Holder 估计

• 发表时间: 2022
• 期刊: Ars inveniendi analytica
• 作者: Luis Silvestre

Global regularity estimates for the Boltzmann equation without cut-off

• DOI: 10.1090/jams/986
• 发表时间: 2019-09
• 期刊: Journal of the American Mathematical Society
• 影响因子: 3.9
• 作者: C. Imbert;L. Silvestre

Solutions to the non-cutoff Boltzmann equation uniformly near a Maxwellian

• DOI: 10.3934/mine.2023034
• 发表时间: 2021-06
• 期刊: Mathematics in Engineering
• 影响因子: 1
• 作者: L. Silvestre;Stanley Snelson