Critical Threshold Phenomena in Nonlinear Balance Laws

非线性平衡定律中的临界阈值现象

• 批准号: 0107917
• 负责人: Eitan Tadmor
• 金额: $ 13.45万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-01 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0107917&HistoricalAwards=false
• 关键词: Critical; Threshold; Phenomena; Nonlinear; Balance

We plan to study the so-called critical threshold phenomena associated with different nonlinear balance laws, in which the persistence of global features of the solutions hinges on a delicate balance between nonlinear convection and a variety of forcing mechanisms. Thus, for example, solutions for nonlinear hyperbolic conservation laws will develop generic singularities in finite time, whereas the existence of balancing forces in other time-dependent problems, e.g., the 2-D incompressible Euler equations, retain global smoothness for all time. This project proposes to analyze those borderline cases, where the persistence of features such as smoothness and finite-time breakdown vs. time decay etc., does not fall into any one particular category. Instead, global features depend on crossing critical threshold associated with the intrinsic configurations of our problems. The presence of various forcing mechanisms in nonlinear convection-dominated PDEs changes the physical situation, and is responsible for the complexity of the underlying problem. Our proposed research falls into four major sub-categories, all tied to the central issue of critical threshold phenomena in nonlinear balance laws: (i) The question of global smoothness/finite-time breakdown for Euler-Poisson equations. We also plan to study the critical threshold phenomena for solutions of Euler-Poisson equations, its relation to semi-classical limits of nonlinear Schrodinger equations and to augment these studies by high-resolution numerical simulations; (ii) Lack of L2-concentrations (-- and hence global existence) of weak solution for Euler equations, depending on the initial configuration in appropriate borderline regularity spaces; (iii) Questions of global vs. local existence for restricted Euler and Navier-Stokes dynamics; and finally, (iv) The issue of crossing the critical sub-characteristic threshold condition in hyperbolic relaxation problems, where there is a balance between different orders of nonlinear convective waves.The breakdown of waves on the beach is a familiar phenomenon. This breakdown phenomenon depends on whether the waves accumulate sufficient strength, height etc. and in general, they depend on whether the initial configuration crosses intrinsic critical thresholds which distinguish between finite time breakdown and long term persistence of the smooth wave patterns. The goal of this project is to study a variety of critical threshold phenomena in problems governed by time-dependent problems. While many such problems develop finite-time singularities and other problems retain global smooth solutions, we focus on borderline cases, where intrinsic features of the solutions like smoothness vs. generic finite time breakdown, boundedness, time decay, etc, hinge on the delicate balance between the nonlinear convection and the (possibly nonlinear) forcing terms. In particular, the persistence of such global features does not fall to any particular category, but instead, these features depend on crossing critical threshold associated with the general configurations of our problems, very much like the conditional breakdown of waves on the beach. These are precisely the kind of problems that are of great research interest in various applications. The critical threshold phenomenon in nonlinear balance laws is not well understood, and the available techniques to study such phenomena need to be further investigated. In this context, there are many issues to be clarified, inter connections to be analyzed -- even in simplified settings, and general understanding of the critical threshold phenomena in realistic situations is sought in terms of both analytical and numerical studies. Professors E. Tadmor and H. Liu will continue their ongoing cooperative and individual research on the critical threshold phenomena in the context of Euler-Poisson equations, incompressible Euler equations, long time existence and finite time breakdown of restricted Euler and Navier-Stokes dynamics, and hyperbolic relaxation systems.

我们计划研究与不同非线性平衡定律相关的所谓临界阈值现象，其中解的全局特征的持久性取决于非线性对流和各种强迫机制之间的微妙平衡。因此，例如，非线性双曲守恒定律的解将在有限时间内产生通用奇点，而其他与时间相关的问题（例如二维不可压缩欧拉方程）中平衡力的存在始终保持全局平滑性。该项目建议分析那些边界情况，其中平滑度和有限时间故障与时间衰减等特征的持久性不属于任何一种特定类别。相反，全局特征取决于跨越与我们问题的内在配置相关的临界阈值。非线性对流主导的偏微分方程中各种强迫机制的存在改变了物理情况，并导致了潜在问题的复杂性。我们提出的研究分为四个主要子类别，所有子类别都与非线性平衡定律中临界阈值现象的核心问题相关：（i）欧拉-泊松方程的全局平滑/有限时间分解问题。我们还计划研究欧拉-泊松方程解的临界阈值现象、其与非线性薛定谔方程的半经典极限的关系，并通过高分辨率数值模拟来增强这些研究； (ii) 缺乏欧拉方程弱解的 L2 浓度（因此全局存在），这取决于适当的边界正则空间中的初始配置； (iii) 受限欧拉和纳维-斯托克斯动力学的全局与局部存在问题；最后，（iv）跨越双曲松弛问题中的临界子特征阈值条件的问题，其中不同阶的非线性对流波之间存在平衡。海滩上波浪的破裂是一种常见的现象。这种击穿现象取决于波是否积累了足够的强度、高度等，并且一般来说，它们取决于初始配置是否跨越了区分有限时间击穿和平滑波模式的长期持续性的内在临界阈值。该项目的目标是研究由时间相关问题控制的问题中的各种临界阈值现象。虽然许多此类问题会产生有限时间奇点，而其他问题保留全局平滑解，但我们重点关注边界情况，其中解的内在特征（例如平滑性与通用有限时间分解、有界性、时间衰减等）取决于非线性对流和（可能是非线性）强迫项之间的微妙平衡。特别是，这种全局特征的持久性不属于任何特定类别，而是这些特征取决于与我们问题的一般配置相关的临界阈值，非常类似于海滩上波浪的条件击穿。这些正是各种应用中引起极大研究兴趣的问题。非线性平衡定律中的临界阈值现象尚不清楚，需要进一步研究研究此类现象的可用技术。在这种背景下，有许多问题需要澄清，内部联系需要分析——即使是在简化的背景下，并且通过分析和数值研究寻求对现实情况中临界阈值现象的一般理解。 E. Tadmor 和 H. Liu 教授将继续他们正在进行的合作和个人研究，研究欧拉-泊松方程、不可压缩欧拉方程、受限欧拉和纳维-斯托克斯动力学的长期存在和有限时间分解以及双曲松弛系统中的临界阈值现象。