Nonlinear Stability of Multidimensional Structures in Fluid Dynamics

流体动力学中多维结构的非线性稳定性拟议研究摘要mark Williams本项目旨在对可压缩流体数学研究中出现的某些重要多维结构的存在性及其非线性稳定性进行严格的研究。这些结构包括激波、爆轰锋面和旋涡片。1987年，Artola和Majda提出了一个严谨的研究过程，通过这个过程，扭结模式导致了超音速涡旋片的卷起。在这种情况下，提出的策略利用了涡旋片问题的新简并Kreiss对称子和奇异副微分算子的微积分，这在研究高振荡的多维冲击中已经被证明是有用的。我们还将研究多维弯曲冲击的长期稳定性，以及曲率对稳定性的影响。另一个目标是研究Chapman-Jouget和ZND燃烧模型的解（如强爆）的存在性、唯一性和稳定性，并阐明这些简化模型的解何时以及如何很好地接近完整的Navier-Stokes燃烧系统的真正精确解。关于这些主题有大量的应用文献，其中的论点往往只是形式上的，并不严谨。直到最近，这些主题还无法进行仔细的数学分析，特别是在更复杂的多维情况下。用严格的分析来补充正式工作是很重要的，这不仅是为了确保正式工作的正确性，也是因为严格的分析提供了新的见解，新的分析工具，有时还揭示了意想不到的现象。在最近的工作中，提案人、他的合作者和其他新工具已经可用，可以对在研究冲击、爆炸和漩涡片的稳定性时出现的高度奇异摄动问题进行严格的研究。

Nonlinear Stability of Multidimensional Structures in Fluid DynamicsAbstract of Proposed ResearchMark Williams This project is to conduct a rigorous investigation of the existence and nonlinear stability of certain important multidimensional structures arising in the mathematical study of compressible fluids. These structures include shock waves, detonation fronts, and vortex sheets. One example of a problem that is now within reach is the rigorous investigation of the process, proposed in 1987 by Artola and Majda, through which kink modes lead to roll-up in supersonic vortex sheets. The proposed strategy in this case draws on new degenerate Kreiss symmetrizers for the vortex sheet problem and a calculus of singular paradifferential operators that has already proved useful in studying highly oscillatory multidimensional shocks. We would also study long-time stability of multidimensional curved shocks, and investigate the effects of curvature on stability. Another goal is to study the existence, uniqueness, and stability of solutions (like strong detonations) to the Chapman-Jouget and ZND models of combustion, and to clarify when and how well solutions to these simplified models approximate true exact solutions of the full Navier-Stokes combustion system. There is a vast applied literature on these topics in which the arguments are often merely formal and not rigorous. Until relatively recently these topics have resisted careful mathematical analysis, particularly in the more complex multidimensional case. It is important to complement the formal work with rigorous analysis not only to insure the correctness of the formal work, but also because the rigorous analysis provides new insight, new analytical tools, and sometimes uncovers unexpected phenomena. In recent work by the proposer, his collaborators, and others new tools have become available that permit a rigorous study of the highly singular perturbation problems that arise in investigating the stability of shocks, detonations, and vortex sheets.