Generic Singularities and Fine Regularity Structure for Nonlinear Partial Differential Equations

非线性偏微分方程（PDE）的数值解在许多应用中起着至关重要的作用。然而，如果解不具有足够的正则性，数值算法的性能和计算精度是一个具有挑战性的问题。在这个项目中，PI将研究一些基本问题，关于各种PDE模拟非线性波的解的规律性，也存在非局部源项。研究的一个主要重点将是奇异性的出现，如冲击波，通用的解决方案。预期的结果将提供一个准确的渐近描述如何形成新的奇点，以及它们如何相互作用，有效的几乎所有的初始数据。这将导致一类新的数值格式，具有高阶精度和广泛的应用。此外，本研究计画将提供一个训练基地给大学部及研究所的学生。第一部分是关于Hamilton-Jacobi方程解的精细正则性结构。在这里，我们的目标是（i）深化对Hamilton-Jacobi方程解集的度量熵的分析，研究SBV正则性，并开发覆盖更广泛的方程类的新技术;（ii）研究广义单调函数的精细性质，研究奇点的传播，并建立汉密尔顿Jacobi方程的新的正则性估计和可控性结果。该项目的第二部分将集中在非线性平衡定律的一般奇点，也在非局部条款的存在。对于一类广泛的此类方程，众所周知，具有光滑初始数据的解在有限时间内会失去正则性。然而，它们可以在弱意义上延伸到一阶导数爆破之后。理解熵弱解的一个主要困难是非局部源项对奇异性形成有巨大影响。由于这个原因，熵弱解的许多有趣的性质，如唯一性，激波形成，激波相互作用和奇点的结构，仍然远未得到很好的理解。在该项目的这一部分中，PI旨在（i）对熵弱解的激波数量进行定量分析，并研究各种非线性波模型和非线性双曲守恒律系统的一般规律性;以及（ii）详细描述非局部平衡律熵弱解的激波形成和波破碎，并追踪它们对BV正则性的影响，稳定性结果。将追求的方法，利用分段正则性的解决方案，旨在减少一个方程定义在一个空间上的低正则性的一个方程上的一个更规则的空间耦合的常微分方程的有限维流形。在已知感兴趣的解决方案是一般分段光滑的情况下，这可能会在更广泛的数值分析领域产生影响，提出新的高阶计算算法。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Numerical solutions of nonlinear partial differential equations (PDE) play a crucial role in a variety of applications. However, if the solutions do not have sufficient regularity, the performance of numerical algorithms and their computational accuracy is a challenging issue. In this project, the PI will study some fundamental questions regarding the regularity of solutions for various classes of PDE modeling nonlinear waves, also in the presence of nonlocal source terms. A major focus of the research will be on the emergence of singularities, such as shock waves, for generic solutions. The expected results will provide an accurate asymptotic description of how new singularities are formed, and how they interact with each other, valid for almost all initial data. This will lead to a new class of numerical schemes, with high-order accuracy and a wide range of applications. In addition, the project will provide a training ground for both undergraduate and graduate students.This research project contains two main parts. The first part is concerned with the fine regularity structure of solutions to Hamilton-Jacobi equations. Here, the goals are to (i) deepen the analysis of the metric entropy of sets of solutions to Hamilton-Jacobi equations, study SBV regularity, and develop new techniques that cover a wider class of equations; and (ii) investigate the fine properties of generalized monotone functions, study the propagation of singularities, and establish new regularity estimates and controllability results for Hamilton Jacobi equations. The second part of the project will focus on generic singularities for nonlinear balance laws, also in the presence of nonlocal terms. For a wide class of such equations, it is well known that solutions with smooth initial data can lose regularity in finite time. However, they can be extended in a weak sense beyond the time when the first derivatives blowup. A major difficulty in understanding entropy weak solutions is that nonlocal source terms have a huge influence on singularity formation. For this reason, many interesting properties of entropy weak solutions, such as uniqueness, shock formation, shock interactions, and the structure of singularities, are still far from being well understood. In this part of the project the PI aims to (i) develop a quantitative analysis of the number of shocks for entropy weak solutions and study their generic regularity for various models of nonlinear waves and for nonlinear hyperbolic systems of conservation laws; and (ii) provide a detailed description of shock formation and wave breaking of entropy weak solutions for nonlocal balance laws as well as trace their impact on BV regularity and stability results. The approach that will be pursued, taking advantage of the piecewise regularity of solutions, seeks to reduce an equation defined on a space with low regularity to an equation on a more regular space coupled with an ODE on a finite dimensional manifold. In cases where the solutions of interest are known to be generically piecewise smooth, this can have an impact in the broader field of numerical analysis, suggesting new high-order computational algorithms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.