Separation Rates for Dissipative Nonlinear Partial Differential Equations

流体模型用于对不同领域出现的关键现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学。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在湍流空气中所受的压力。如果数学模型的预测没有考虑到极端事件，那么数学模型不能捕捉到可能的真实世界情景的全部范围的可能性令人担忧。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体现出来，在这种效应中，一个看似微不足道的参数变化会导致截然不同的结果。更令人担忧的是非唯一性的幽灵，在这种情况下，同一组参数可能产生不同的动态。因此，模拟可以准确地描述一个现实世界的场景，但不能解释另一个可能的灾难性场景。在这个项目中，研究者将对这些不稳定动力学的可能严重程度有一个广泛的了解。学生将参与这个项目。该项目将包括在中学和社区大学推广数学。研究者将解决偏微分方程的解如何快速分离的问题，主要是在Navier-Stokes方程的背景下，这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，该系统在物理类的解决方案的非唯一性。为了评估非唯一性的严重程度，将对具有相同初始数据的两个解决方案的差异进行估计。如果差值增长缓慢，那么一个解可以近似于另一个解。如果它增长迅速，那么解决方案很快就会变得不相关，这在进行预测时是令人担忧的。将发展这些估计的新方法，例如通过高阶局部时间规律性结果。研究者还将探索与实验支持的可预测性观点的联系，其中小规模的不稳定性不会立即破坏宏观预测。类似的关于非唯一性和分离的问题也出现在其他偏微分方程模型中，这些模型可能来自流体世界，如表面准地转方程，但并不局限于它，如半线性热方程或复杂的金兹堡-朗道方程。Navier-Stokes方程的结果将适用于这些模型，为这些方程的非唯一性演化提供有价值的信息，并揭示它们在具有显著变化结构的偏微分方程中的鲁棒性。该项目由DMS应用数学项目和促进竞争性研究的既定项目（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.