Long time behaviour for nonlinear evolution equations

非线性演化方程的长时间行为

• 批准号: RGPIN-2018-06487
• 负责人: Pusateri, FabioGiuseppe
• 金额: $ 1.68万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758693
• 关键词: Long; time; behaviour; nonlinear; evolution

Partial Differential Equations (PDEs) are used to describe a wide variety of phenomena in the physical world, from the motion of fluids and plasma, to the description of elementary particles, to the motion of stars. In this proposal we are interested in nonlinear PDEs describing the complex nature of the nonlinear interactions of waves, particles, fields... In particular, we will investigate the dynamics of various physical systems, study their evolution over large periods of time, and their asymptotic states. The proposed research activities can be group into two main categories, corresponding to two fundamental issues in the analysis of PDEs: the long-time dynamics around special solutions, and the behavior of quasilinear evolution equations in periodic settings.Several nonlinear equations possess special solutions such as stationary solutions (that do not change with time) and traveling waves and solitons (solutions that keep their shape as they move). These coherent structures, emerging from the balance between dispersive effects and the focusing mechanisms of nonlinear interactions, can represent various physical phenomena (e.g., traveling waves on the surface of the sea, particle-like objects in field theories, or black-holes) and play a major role in the global dynamics of the system. Studying their stability is then a crucial task from both a mathematical and physical perspective. We propose projects related to the global behavior around (topological) solitons for some models in mathematical physics and fluid dynamics, such as the phi-4 model, long waves in shallow water, and equations from multidimensional field theories.Many of the fundamental equations that model inviscid fluids, plasmas, gravitational waves, are quasilinear, which means that the highest order derivatives of the unknowns appear nonlinearly in the equations. Building on the pioneering works of Shatah, Klainerman, and Christodoulou-Klainerman in the `80s, several powerful techniques have emerged in the past 5-10 years in the study of quasilinear equations. These led to important achievements, such as the first global existence results for several water waves systems and plasma models.One of the major open questions in the field is what happens in non-purely-dispersive settings, such as a bounded domain (typical for a water wave), or a periodic box (a suitable scenario for plasma experiments). The second part of our project concentrates on this type of questions. Our aim is to investigate the long-time existence of periodic solutions of quasilinear systems, with particular emphasis on the water waves problem, which describes the motion of an irrotational and incompressible fluid with a free moving boundary, such as waves on the surface of the ocean. Related problems for the nonlinear Schrodinger equations, and connections to the general theory of Hamiltonian system will also be sought.

偏微分方程（PDE）用于描述物理世界中的各种现象，从流体和等离子体的运动到基本粒子的描述，再到恒星的运动。在这个建议中，我们感兴趣的是非线性偏微分方程描述波，粒子，场的非线性相互作用的复杂性质。特别是，我们将研究各种物理系统的动力学，研究它们在很长一段时间内的演化，以及它们的渐近状态。拟开展的研究活动可分为两大类，分别对应于偏微分方程分析中的两个基本问题：特解周围的长时间动力学，以及周期性条件下准线性演化方程的行为。一些非线性方程具有特解，如定态解（不随时间变化）和行波与孤子（运动时保持形状的解）。这些相干结构，从色散效应和非线性相互作用的聚焦机制之间的平衡中出现，可以代表各种物理现象（例如，海洋表面的行波，场论中的粒子状物体或黑洞），并在系统的全球动力学中发挥重要作用。从数学和物理的角度来看，研究它们的稳定性是一项至关重要的任务。我们提出的项目涉及全球行为，数学物理和流体动力学中某些模型的（拓扑）孤子，例如phi-4模型，浅水中的长波，以及多维场论中的方程。许多模拟无粘流体，等离子体，引力波的基本方程都是准线性的，这意味着未知数的最高阶导数在方程中出现非线性。基于Shatah、Klainerman和Christodoulou-Klainerman在80年代的开创性工作，在过去的5 - 10年里，在拟线性方程的研究中出现了几种强大的技术。这些都导致了重要的成就，如第一个整体存在的结果，几个水波系统和等离子体模型。在该领域的主要开放问题之一是发生在非纯色散设置，如有界域（典型的水波），或周期盒（一个合适的场景等离子体实验）。 我们项目的第二部分集中在这类问题上。我们的目的是研究拟线性系统周期解的长期存在性，特别强调水波问题，它描述了一个无旋和不可压缩流体的运动与自由移动的边界，如海洋表面的波浪。也将探讨非线性薛定谔方程的相关问题，以及与哈密顿系统一般理论的联系。