在本项目申请书，考虑p-变分与流体力学方程的适定性理论，主要内容为：(1) 研究相应于一般形式p-泛函的非齐次非线性椭圆方程的二阶正则性，(2) 研究带有misalignment效应的Euler-alignment方程的光滑解整体适定性，(3) 研究带有分数次耗散的非齐次Navier-Stokes方程与二维Boussinesq方程的密度补丁问题。这些数学问题既具有数学与物理背景，又具有强烈的研究动因。通过本项目的实施，我们期望能够对p-变分问题的二阶正则性理论与流体力学方程的整体适定性理论加深理解。

In this research proposal, we consider the wellposedness theory of p-variational problems and fluid mechanics equations. The main content includes: (1) the research on the second-order regularity of the nonhomogeneous nonlinear elliptic equations corresponding to p-functionals with general forms, (2) the research on the global wellposedness of smooth solutions for the Euler-alignment equation with misalignment effect, (3) the research on the density patch problems of the nonhomogeneous Navier-Stokes equation and 2D Boussinesq equation with fractional dissipation. Such mathematical problems have mathematical and physical background, and also have intense research motives. With the implement of this research proposal, we expect that we can deepen the understanding of the second-order regularity theory of p-variational problems and the global wellposedness theory of fluid mechanics equations.