Regularity vs. Singularity for Elliptic and Parabolic Systems

这个项目研究在纯数学和应用数学的几个领域中出现的偏微分方程式，包括气象学、弹性力学、微分和复杂几何。所提出的问题涉及解的正则性(光滑性)。这些问题共有的关键挑战是椭圆性(正则性的代理)退化，并且解可以是向量值的。克服这些挑战不仅需要重要的新数学思想，还可以帮助我们更好地预测天气模式，并帮助设计汽车。结果将通过发表在同行评议的期刊上，并通过撰写说明性注释来传达，PI打算公开提供这些注释，并将对研究人员和学生同样有用。第一个项目解决真实的Monge-Ampere方程的正则性问题，其动机是应用于气象学和微分几何。PI将研究模拟大尺度大气流动的半地转系统的奇点形成。半平面中不规则定常解的某些例子可能是边界点上井喷的有用模型。第二个项目涉及复Monge-Ampere方程的局部正则性，该方程是在复杂几何中产生的。困难包括解的非凸性，以及在添加某些二次多项式时的不变性。国际和平研究所已经启动了对模型方程的研究，该模型方程捕捉到了这些困难。第三个项目涉及非凹一致椭圆型方程。构造低维奇异解是一个具有挑战性的问题。一个障碍是，在3D中，具有粗糙系数的线性方程没有非平凡的一次齐次解；PI将转而考虑具有“螺旋形”一次齐次对称性的解。第四个项目涉及带凸被积的变分积分的极小化问题。一个猜想是标量极小值与被积函数的勒让德变换具有相同的正则性。PI建议在被积函数的退化集满足某些几何条件时予以确认，并在不满足这些条件时构造反例。最后一个项目旨在回答抛物系统的经典正则性和稳定性问题，特别是在低维情况下。PI最近构造了平面上线性抛物型方程组的奇异解的例子，回答了一个长期存在的问题。这可能会为解决相关问题开辟道路，例如确定在线性情况下防止奇点的系数条件，以及确定多孔介质类型的非线性结构的正则性与奇异性。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。

This project investigates partial differential equations that arise in several areas of pure and applied mathematics including meteorology, elasticity, and differential and complex geometry. The proposed problems concern the regularity (smoothness properties) of solutions. Key challenges that these problems share are that ellipticity (an agent of regularity) degenerates, and that solutions can be vector-valued. Overcoming these challenges will not only require significant new mathematical ideas, but could also help us better predict weather patterns and assist in the design of cars. The results will be communicated by publication in peer-reviewed journals, and through the writing of expository notes that the PI intends to make publicly available and will be useful to researchers and students alike.The first project tackles regularity questions for the real Monge-Ampere equation, motivated by applications to meteorology and differential geometry. The PI will investigate singularity formation for the semi-geostrophic system, which models large scale atmospheric flows. Certain examples of irregular stationary solutions in a half-plane may be useful models for blowup at boundary points. The second project concerns local regularity for the complex Monge-Ampere equation, which arises in complex geometry. Difficulties include the non-convexity of solutions, and invariance under adding certain quadratic polynomials. The PI has initiated the study of a model equation that captures these difficulties. The third project concerns non-concave uniformly elliptic equations. A challenging problem is to construct singular solutions in low dimensions. An obstruction is that in 3d, linear equations with rough coefficients have no nontrivial one-homogeneous solutions; the PI will instead consider solutions with "spiraling" one-homogenous symmetry. The fourth project concerns minimizers of variational integrals with convex integrand. A conjecture is that scalar-valued minimizers have the same regularity as the Legendre transform of the integrand. The PI proposes to confirm this when the degeneracy set of the integrand satisfies certain geometric conditions, and to construct a counterexample when these conditions aren't met. The last project aims to answer classical regularity and stability questions for parabolic systems, especially in low dimensions. The PI recently constructed examples of singular solutions to linear parabolic systems in the plane, answering a long-standing question. This may open the way to tackling related problems, e.g. to identify conditions on coefficients that prevent singularities in the linear case, and to determine regularity vs. singularity for nonlinear structures of porous medium type.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.