非对角型拟线性退化抛物方程组弱解的正则性

由向量场构成的退化抛物方程组与物理学、生物学等学科有关，其正则性是研究抛物方程组解的存在性和唯一性的基础，因此对方程组弱解正则性的研究具有重要的意义。然而，目前为止，较少有学者利用先验估计法研究非对角型退化抛物方程组弱解的正则性。.本项目致力于研究由Hörmander向量场构成的非对角型拟线性退化抛物方程组弱解的全局正则性，并探讨系数和低阶项对弱解正则性的影响。.本项目的创新之处：(1) 将欧氏空间中非对角型椭圆方程组扩展到由Hörmander向量场构成的非对角型退化抛物方程组的情形，并将其可积性进行提升；(2) 将弱解的局部正则性改进到全局情形。.本项目将促进退化抛物方程组正则性的研究，为超抛物方程和带漂移项方程问题的进一步研究提供理论依据。

Degenerate parabolic systems constructed by vector fields are related to physics, biology and so on, and regularity of solutions to parabolic systems is the key for the existence and uniqueness, therefore it is important to study the regularity of weak solutions to parabolic systems. However, up to now, only a few results about the regularity of weak solutions to nondiagonal degenerate parabolic systems have been built by priori estimates..This project is devoted to the research of global regularity for weak solutions to nondiagonal quasilinear degenerate parabolic systems formed by Hörmander's vector fields, and investigate the influence to regularity for weak solution by the coefficients and the lower order items..The innovations of this project are: (1) Extending nondiagonal elliptic systems in Euclidean space to nondiagonal parabolic systems in Hörmander's vector fields, and promoting integrability; (2) Improving the local regularity of weak solutions to the global case..This project will promote the study of the regularity to degenerate parabolic systems, and provide a theoretical basis for the further study of ultraparabolic equations and the equations with drift.