Quantum Friction, Resonance Problems and New Methods in Geometric Flows

量子摩擦问题涉及玻色爱因斯坦凝聚体中入侵的经典粒子的运动。PI考虑的是偏微分方程(PDE)模型上的问题。具体地说，PI使用耦合到经典粒子轨迹的非线性薛定谔方程，该方程是在对玻色爱因斯坦凝聚体取平均场极限后获得的。PI将证明以下三个结果：(1)如果经典粒子的初始速度高于玻色-爱因斯坦凝聚体中的音速，即超音速，则粒子将由于摩擦而减速，直到速度达到音速；(2)如果粒子的初始速度是亚音速，则粒子将随着时间的推移以弹道方式运动；(3)整个系统将收敛到某个惯性模式。从技术上讲，PI必须更好地理解积分-微分方程，以及非线性偏微分方程组中的其他技术，例如费米黄金法则。第二个问题是共振问题。PI希望在线性和非线性薛定谔方程的背景下考虑该问题。通常，微扰展开中的简并或共振的存在使问题变得困难。PI希望对范式变换有更好的理解，然后解决薛定谔方程中的一些问题。第三个问题是发展一种新的方法来计算几何流，特别是平均曲率流和Ricci流。不同于以前的工作，例如惠斯肯，PI主要使用调制方程和谱分析来执行几乎精确的估计，而不是最大值原理和熵估计。希望PI能在这里解决一些悬而未决的问题。例如，在平均曲率流的背景下，演化的曲面将在有限时间内坍塌，并在塌陷点周围形成一个圆柱体。然而，圆柱体是否独一无二是一个悬而未决的问题。PI希望解决这个问题，在Ricci流中也有类似的问题。量子摩擦在当今有很多应用。一个例子是测试粒子的速度，例如中微子，通过将粒子发射到某种介质，例如Ar。这种现象被称为切伦科夫辐射(诺贝尔奖1958)。尽管切伦科夫辐射很重要，但人们对它的数学理解并不令人满意。在更广泛的背景下，这个问题属于非平衡统计力学和量子流体，这两个领域目前很流行。第二个问题，共振问题，将加深对规范形变换的理解，并有助于解决其他问题，如动力学系统(特别是KAM理论)和量子力学中的自旋模型。对于第三个问题，近年来，人们利用平均曲率流对不同表面的拓扑结构进行分类，并估计了广义相对论中的质量。PI的技术提供了关于曲面演变的更精确的信息。希望PI的方法能在那里找到应用。

The problem of quantum friction concerns the motion of an invading classical particle in the Bose Einstein condensate. The PI considers the problem on a partial differential equation (PDE) model. Specifically the PI uses a nonlinear Schrodinger equation coupled to the trajectory of a classical particle, obtained after taking the mean field limit on the Bose Einstein condensate. The PI will prove the following three results: (1) if the initial speed of the classical particle is higher than the speed of sound in the Bose Einstein condensate, i.e. supersonic, then the particle will decelerate due to the friction until the speed reaches the speed of sound, (2) if the initial speed of the particle is subsonic, then the particle will travel ballistically as the time goes to infinity, (3) the whole system will converge to some inertial mode. Technically, the PI has to develop a better understanding of integro-differential equations, together with other techniques in nonlinear PDEs, for example Fermi Golden rules. The second problem is the resonance problem. The PI wants to consider the problem in the context of linear and nonlinear Schrodinger equations. Very often the presence of degeneracy, or resonance, in perturbation expansions makes the problem hard. The PI hopes to develop a better understanding of normal form transformations, and then to tackle some of the problems in Schrodinger equations. The third problem is to develop a new method for geometric flows, specifically mean curvature flow and Ricci flow. Different from the previous works, by Huisken for example, the PI mainly uses modulation equations and spectral analysis to perform almost precise estimates, instead of the maximum principle and entropy estimates. Hopefully the PI can solve some open problems here. For example, in the context of mean curvature flow, the evolving surface will collapse in finite time and form a cylinder around the collapsing point. However, whether the cylinder is unique is an open problem. The PI hopes to solve it, also a similar problem in Ricci flow.Quantum friction has many applications nowadays. One example is to test the speed of particles, for example neutrinos, by shooting the particle to some medium, for example argon. This phenomenon is known as Cerenkov radiation (Noble prize 1958). Despite its importance, the mathematical understanding of Cerenkov radiation is not satisfactory. In a broader context, the problem is in the class of non-equilibrium statistical mechanics and quantum fluid, which are popular at the moment. The second problem, the resonance problem, will deepen the understanding of normal form transformations, and help to tackle other problems, for example, in dynamical system (specifically KAM theory), and spin model in quantum mechanics. For the third problem, in the recent years, people have applied mean curvature flow to classify topological structures of different surfaces, and have estimated the amount of mass in general relativity. The PI's techniques provide more precise information on the evolution of the surfaces. Hopefully the PI's method will find applications there.