Scaling limit in dispersive equations

这个项目旨在从物理学的角度发展对三个基本方程的研究。首先，研究了等离子体物理中的一个基本方程--双流体Euler-Maxwell方程的稳定性问题。我们的目标是证明，在某些条件下，平衡的微小扰动不会产生激波，实际上，即使没有耗散，等离子体也会回到平衡。这样的结果将具有重要的物理和数学意义，因为众所周知，在没有自洽电磁场的情况下，可压缩欧拉方程是错误的。第二部分研究了弯曲背景下的薛定谔方程。在这种情况下，欧几里德理论中的许多经典工具都崩溃了，人们预计会因为背景几何的影响而出现许多新的现象。特别是，我们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性态的影响。最后，在第三部分中，我们利用前人的思想，将齐次四阶方程的一些已知结果推广到更具物理意义的非齐次方程。理解如何在没有任何摩擦的情况下通过自洽的电磁场稳定流体，代表着纯数学、应用数学和物理学之间的跨学科合作，以及在流体工程中的应用。更具体地说，证明静止的等离子体在微小扰动下是稳定的将是一项重大的物理发现，肯定会极大地改进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，除了工业应用，等离子体代表了宇宙中99%以上的物质的状态，任何提供对这种状态的更好理解的工作都将是非常重要的。色散方程和曲面空间(流形)上的方程提供了数学各分支之间以及不同科学之间丰富的相互作用领域。几何学家、分析家和数学家以及从事量子混沌和广义相对论研究的理论物理学家对弯曲空间上的方程的研究很感兴趣。四阶方程自然而然地出现在许多不同的物理和数学分支中，特别是那些与弹性有关的分支，对于理解各种现象，如桥梁的运动(以及可能的振荡和破裂)或血管结构及其膜和生物流体之间的相互作用，是至关重要的。

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.