Diffusion and Regularity

研究人员将研究呈现某种扩散行为的演化方程，这一行为在广义上被理解。最经典的扩散方程是热方程，它描述了一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“抛物线”的偏微分方程式中都可以观察到。一般而言，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。研究人员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，在具有积分扩散的方程中，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培训的机会。在过去的二十年里，积分-微分方程组的研究一直是一个非常活跃的研究领域。首席调查员在制定规律性结果方面发挥了中心作用。这些方程在很大程度上是由涉及跳跃过程的概率模型驱动的。最近，这些方法在确定性模型中也得到了应用。这位研究人员一直在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，描述了稀薄气体中粒子密度的演变。当假设粒子在小尺度上通过幂函数势相互排斥时，方程表现出一种微妙的正则化效应，由一个类似于分数阶椭圆型算子的非线性积分-微分扩散所驱动。在最奇异的情况下，玻耳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。主要研究人员还对研究更一般和抽象的完全非线性抛物型方程感兴趣，以了解它们的规律性和可能的奇性形成。即使在守恒律方程中，在没有任何显式扩散项的情况下，低阶Sobolev空间中也存在微妙的正则化效应，可以被认为是消失粘性近似的残余。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.