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对基本颗粒的描述，再到恒星的运动。在此提案中，我们对描述波，颗粒，田地非线性相互作用的复杂性的非线性PDE感兴趣……特别是，我们将研究各种物理系统的动力学，研究它们在大时的演变及其渐近状态。拟议的研究活动可以分为两个主要类别，对应于PDE分析中的两个基本问题：特殊解决方案周围的长期动态，以及定期环境中的准连续性进化方程的行为。几个非线性方程都具有特殊的解决方案，例如平稳的解决方案（不会随着时间而变化）和行驶浪潮和独奏者（将其与他们的形状变化），以保持其状态（以保持他们的形状）。这些一致的结构是从分散效应和非线性相互作用的聚焦机制之间的平衡中出现的，可以代表各种物理现象（例如，海面上的行进波，田间理论中的粒子样物体或黑洞中的粒子样物体），在系统的整体动力学中起主要作用。从数学和物理角度来看，研究其稳定性是至关重要的任务。 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方程式。在80年代的Shatah，Klainerman和Christodoulou-Klainerman的开创性作品建立基础上，在过去的5 - 10年中，有几种强大的技术在对准方程式的研究中出现了。这些导致了重要的成就，例如几种水波系统和等离子体模型的第一个全球存在结果。该领域的主要开放问题之一是在非透明散发性环境中发生的情况，例如有界域（典型的水浪）或周期盒（适用于plasma实验的情况）。 我们项目的第二部分集中在此类问题上。我们的目的是研究准线性系统的周期性解决方案的长期存在，并特别强调了水波问题，该问题描述了具有自由移动边界的不旋转和不可压缩的流体的运动，例如海洋表面上的波。也将寻求非线性Schrodinger方程的相关问题，以及与哈密顿系统一般理论的联系。