Separation Rates for Dissipative Nonlinear Partial Differential Equations

耗散非线性偏微分方程的分离率

• 批准号: 2307097
• 负责人: Zachary Bradshaw
• 金额: $ 19.11万
• 依托单位: University of Arkansas
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307097&HistoricalAwards=false
• 关键词: Separation; Rates; Dissipative; Nonlinear; Partial

Fluid models are used to make predictions about critical real-world systems arising in diverse fields including but not limited to meteorology, climate science, mechanical engineering, and geophysics. Simulations based on fluid models can, for example, be used to make predictions about the strength of a tornado or the stresses on an aircraft wing passing through turbulent air. The possibility that a mathematical model does not capture the full range of possible real-world scenarios is concerning if the predictions do not account for extreme events. Mathematically, this may occur if the model is unstable. This can be evident through a butterfly effect, in which a seemingly negligible change in a parameter leads to a wildly different outcome. Even more concerning is the specter of non-uniqueness, in which the same set of parameters may generate different dynamics. Consequently, a simulation could accurately describe one real-world scenario but not account for another possibly catastrophic one. In this project, the investigator will develop a broad understanding of the possible severity of these unstable dynamics. Students will be involved in this project. The project will include outreach efforts to promote mathematics in secondary schools and community colleges. The investigator will address the issue of how rapidly solutions to partial differential equations can and possibly must separate, primarily in the context of the Navier-Stokes equations, a system which models viscous incompressible fluid flow. Recent results suggest non-uniqueness for this system in physical classes of solutions. To assess the severity of non-uniqueness, estimates will be developed for the difference of two solutions having the same initial data. If the difference grows slowly, then one solution can be approximated from the other. If it grows rapidly, then the solutions quickly become uncorrelated, which is concerning when making predictions. New approaches for these estimates will be developed, e.g., through higher-order local time-regularity results. The investigator will additionally explore connections to an experimentally supported view of predictability, in which small scale instabilities do not instantly ruin macroscopic forecasts. Similar questions about non-uniqueness and separation arise in other partial differential equations models, which can come from the world of fluids, as with the surface quasi-geostrophic equation, but are not limited to it, as with the semi-linear heat or complex Ginzburg-Landau equations. The results for the Navier-Stokes equations will be adapted to these models, providing valuable information about the evolution of non-uniqueness in these equations and shedding light on how robust they are across partial differential equations with significantly varying structures.This project is jointly funded by the DMS Applied Mathematics Program and the Established Program to Stimulate Competitive Research (EPSCoR).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.

流体模型用于对各种领域中出现的关键真实世界系统进行预测，包括但不限于气象学、气候科学、机械工程和地球物理。例如，基于流体模型的模拟可以用来预测龙卷风的强度或飞机机翼通过湍流空气时的应力。如果预测没有考虑到极端事件，数学模型可能无法捕捉到现实世界中所有可能的情景，这是令人担忧的。从数学上讲，如果模型不稳定，就可能发生这种情况。这可以通过蝴蝶效应很明显，在蝴蝶效应中，参数的一个看似可以忽略不计的变化会导致截然不同的结果。更令人担忧的是不唯一性的幽灵，在这种情况下，相同的参数集可能会产生不同的动态。因此，模拟可以准确地描述一个真实世界的场景，但不能解释另一个可能是灾难性的场景。在这个项目中，研究人员将对这些不稳定动力学可能的严重性有一个广泛的了解。学生将参与到这个项目中来。该项目将包括在中学和社区大学推广数学的外联工作。研究人员将讨论偏微分方程组的解可以多快且可能必须分离的问题，主要是在N-S方程的背景下，N-S方程是一个模拟粘性不可压缩流体流动的系统。最近的结果表明，这个系统在解的物理类别中不是唯一的。为了评估不唯一性的严重程度，将对具有相同初始数据的两个解决方案的差异进行估计。如果差异缓慢增长，则可以从一个解近似到另一个解。如果它快速增长，那么解决方案很快就会变得不相关，这是做出预测时需要考虑的问题。将开发这些估计的新方法，例如，通过更高阶的当地时间规律性结果。研究人员还将探索与实验支持的可预测性观点的联系，在这种观点中，小范围的不稳定不会立即摧毁宏观预测。类似的关于非唯一性和分离性的问题也出现在其他偏微分方程组模型中，它们可以来自流体世界，如地表准地转方程，但不限于此，如半线性热或复杂的Ginzburg-Landau方程。Navier-Stokes方程的结果将适用于这些模型，提供关于这些方程中非唯一性演变的有价值的信息，并揭示它们在结构显著变化的偏微分方程中的健壮性。该项目由DMS应用数学计划和既定的刺激竞争研究计划(EPSCoR)联合资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。