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的非线性平衡律有关的所谓临界阈值现象，其中解的整体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，非线性双曲型守恒律的解将在有限时间内发展出一般奇性，而在其他依赖于时间的问题中平衡力的存在，例如二维不可压缩Euler方程，始终保持全局光滑性。这个项目建议分析那些边缘情况，其中特征的持久性，如光滑性和有限时间分解与时间衰减等，不属于任何一个特定的类别。相反，全局特征依赖于跨越与我们的问题的内在配置相关的临界阈值。在以非线性对流为主的偏微分方程组中，各种强迫机制的存在改变了物理状况，并导致了潜在问题的复杂性。我们的研究分为四个主要子类，它们都与非线性平衡律中的临界阈值现象有关：(I)Euler-Poisson方程的全局光滑性/有限时间崩溃问题。我们还计划研究Euler-Poisson方程解的临界阈值现象及其与非线性薛定谔方程半经典极限的关系，并通过高分辨率的数值模拟来加强这些研究：(Ii)Euler方程弱解的L2浓度的缺乏(因此整体存在)，这取决于适当的边界正则性空间中的初始构形；(Iii)受限Euler和Navier-Stokes动力学的整体与局部存在性问题；最后，(Iv)双曲松弛问题中跨越临界次特征阈值条件的问题，其中不同阶次的非线性对流波之间存在平衡。这种击穿现象取决于波是否积累了足够的强度、高度等，一般而言，它们取决于初始构型是否超过了区分平滑波形的有限时间击穿和长期持续的内在临界阈值。这个项目的目标是研究由时间依赖问题支配的问题中的各种临界阈值现象。虽然许多这样的问题发展了有限时间奇性，而其他问题保留了全局光滑解，但我们关注的是边界情况，在这些情况下，解的内在特征，如光滑性与一般的有限时间破裂、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，这种全球特征的持久性不属于任何特定类别，相反，这些特征取决于跨越与我们问题的一般配置相关的关键门槛，非常类似于海滩上的海浪有条件地破裂。这些正是在各种应用中引起极大研究兴趣的问题。非线性平衡律中的临界阈值现象还没有得到很好的理解，研究这类现象的有效方法还有待于进一步研究。在这方面，有许多问题需要澄清，相互联系需要分析--即使在简化的情况下也是如此，从分析和数字研究的角度寻求对现实情况中的临界阈值现象的普遍理解。TAdmor和H.Liu教授将继续在Euler-Poisson方程、不可压缩Euler方程、受限Euler和Navier-Stokes动力学的长期存在和有限时间崩溃以及双曲松弛系统的背景下对临界阈值现象进行合作和个人研究。