Critical Threshold Phenomena in Nonlinear Balance Laws

我们计划研究与不同的非线性平衡律，其中的持久性的解决方案的全球功能取决于非线性对流和各种强迫机制之间的微妙平衡，所谓的临界阈值现象。因此，例如，非线性双曲守恒律的解将在有限时间内产生一般奇点，而在其他时间依赖问题中，例如，二维不可压Euler方程组始终保持全局光滑性。本项目建议分析那些边界情况，其中的持久性的特点，如平滑和有限时间的崩溃与时间衰减等，并不属于任何一个特定的类别。相反，全局特征依赖于与我们的问题的内在配置相关联的临界阈值。非线性对流占优偏微分方程中各种强迫机制的存在改变了物理状况，并导致了问题的复杂性。我们提出的研究福尔斯分为四个主要的子类别，都与非线性平衡律中的临界阈值现象的中心问题有关：（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，并通过高分辨率数值模拟来加强这些研究;在适当的边界正则性空间中，依赖于初始构形，证明了Euler方程弱解的存在性（从而整体存在）;（iii）限制欧拉和Navier-Stokes动力学的全局与局部存在性问题;（iv）双曲松弛问题中临界次特征阈值条件的跨越问题，其中非线性对流波的不同阶数之间存在平衡。波在海滩上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，并且一般而言，它们取决于初始配置是否越过内在临界阈值，该阈值区分有限时间击穿和平滑波图案的长期持续性。这个项目的目标是研究各种临界阈值现象的问题，由时间依赖的问题。虽然许多这样的问题发展有限时间奇异性和其他问题保留全球光滑的解决方案，我们专注于边界的情况下，内在功能的解决方案，如平滑与通用有限时间故障，有界性，时间衰减等，铰链之间的微妙平衡的非线性对流和（可能是非线性的）强迫条款。特别是，这种全球性的功能的持久性不属于任何特定的类别，而是，这些功能取决于跨越临界阈值与我们的问题的一般配置，非常像有条件的崩溃波在海滩。这些正是在各种应用中具有巨大研究兴趣的问题。非线性平衡定律中的临界阈值现象还没有得到很好的理解，研究这种现象的现有技术需要进一步研究。在这方面，有许多问题需要澄清，相互联系进行分析-即使在简化的设置，并在现实情况下的临界阈值现象的一般理解，寻求在分析和数值研究方面。教授E。Tadmor和H.刘将继续他们正在进行的合作和个人研究的临界阈值现象的背景下，欧拉-泊松方程，不可压缩的欧拉方程，长时间存在和有限时间故障的限制欧拉和Navier-Stokes动力学，和双曲松弛系统。

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.