Scaling limit in dispersive equations

本项目旨在发展物理学中三个基本方程的研究。首先，考虑了等离子体物理学基本方程之一的二流体欧拉-麦克斯韦方程的稳定性问题。目的是证明，在一定条件下，平衡的小扰动不会产生激波，而且，实际上，等离子体会回到平衡状态，即使在没有耗散的情况下。这样的结果将具有重大的物理和数学意义，因为已知在没有自洽电磁场的情况下，可压缩欧拉方程是不成立的。第二部分研究了弯曲背景下的薛定谔方程。在这种情况下，欧几里得理论中的许多经典工具失效了，由于背景几何的影响，人们期望出现许多新的现象。特别地，将研究体积的增长对能量临界方程解的整体存在性和整体行为的影响。最后，在第三部分中，利用之前发展的思想将一些关于四阶齐次方程的已知结果升级到更物理的非齐次方程。了解流体如何在没有任何摩擦的情况下通过自一致的电磁场稳定，代表了纯数学、应用数学和物理学之间的跨学科合作，并在流体工程中有应用。更具体地说，证明静止的等离子体在小扰动下是稳定的将是一个重大的物理发现，而且肯定会大大增强我们控制等离子体的方法。这是特别合适的，因为等离子体稳定性是限制托卡马克性能的主要因素之一。最后，在工业应用之外，等离子体代表了宇宙中99%以上的物质的状态，任何能更好地理解这种状态的工作都是非常重要的。色散方程和曲线空间（流形）上的方程为不同数学分支和不同科学之间的相互作用提供了丰富的领域。弯曲空间方程的研究对几何学家、分析学家、数论学家以及研究量子混沌和广义相对论的理论物理学家都很有兴趣。四阶方程自然出现在物理和数学的许多不同分支中，特别是那些与弹性有关的，对于理解各种现象至关重要，如桥梁的运动（以及可能的振荡和破裂）或血管结构以及膜与生物流体之间的相互作用。

This project aims at developing the study of three fundamental equations from physics. First, stability issues for the 2-fluids Euler-Maxwell equation, which is one of the fundamental equations in plasma physics, are considered. The goal is to prove that, under certain conditions, small perturbations of an equilibrium will not develop shocks, and that, actually, the plasma will get back to equilibrium, even in the absence of dissipation. Such a result would be of great physical and mathematical importance as it is known to be false for the compressible Euler equation in the absence of a self-consistent electromagnetic field. In a second part, a study is made of the Schrodinger equation on a curved background. In this case, many classical tools from the Euclidean theory break down and one expects the appearance of many new phenomena due to the influence of the geometry of the background. In particular, a study will be made of the effect of the growth of the volume on the global existence and global behavior of the solutions to the energy-critical equation. Finally, in a third part, ideas developed before are used to upgrade some known results about the homogeneous fourth-order equation to the more physical inhomogeneous equation. Understanding how a fluid can be stabilized by a self-consistent electromagnetic field in the absence of any friction represents a cross-disciplinary collaboration between pure mathematics, applied mathematics, and physics, with applications in fluid engineering. More specifically, proving that a plasma at rest is stable under small perturbations would be a major physical discovery and would most certainly greatly enhance our ways to control plasma. This is especially fitting since plasma stability is one of the main factor limiting performances in tokamaks. Finally, beyond industrial applications, plasma represents the state of more than 99% of the matter in the universe and any work providing better understanding of this state would be of great importance. Dispersive equations and equations on curved spaces (manifolds) provide a rich area of interaction between various branches of mathematics as well as between different sciences. The study of equations on curved spaces is of interest to geometers, analysts, and number theorists in mathematics, as well as to theoretical physicists working in quantum chaos and general relativity. Fourth-order equations naturally arise in many different branches of physics and mathematics, especially those linked with elasticity and are critical to understand phenomena as diverse as the movement (and possible oscillations and breakdown) of bridges or the structure of blood vessels and the interaction between their membranes and the biofluids.