流体力学方程组的自由边值问题

The free boundary problem of high dimensional fluid dynamics system is widely used in aerodynamics, shallow water wave and astrophysics. It is classic mathematical and physical models for Navier-Stokes system. The free boundary problem have always been the core subjects of heory on nonlinear partial differential equation. This project is planning to studies: (1) Global well-posedness and decay properties of solutions of the free boundary to 3D compressible Navier-Stokes equations;(2)Local and global well-posedness of the vacuum free boundary to 3D Inhomogeneous Navier-Stokes equations. The progress achieved in this proposal will help to further improve the theory in the field of nonlinear partial differential equations.

流体力学方程组的自由边界问题广泛应用于空气动力学、浅水波、天体物理等领域。Navier-Stokes方程组是流体力学方程组的经典模型，其自由边值问题是非线性偏微分方程中的核心问题。本项目将利用拉格朗日坐标变换法和渐进匹配法研究: (1) 三维可压缩Navier-Stokes方程组自由边值问题的整体适定性，自由边界的运动规律及解的衰减性质; (2) 三维非齐次Navier-Stokes方程组自由边值问题的局部适定性及整体适定性。本项目的研究将有助于进一步丰富非线性偏微分方程的理论和方法。

本研究项目的结果包括高维Navier-Stokes方程组自由边值问题及相关模型的适定性数学理论。其结果为：(1) 非齐次不可压缩 Navier-Stokes 方程在上半空间的自由边界问题; (2) 无磁扩散不可压缩磁流体方程组的初边值问题; (3) 气-液两相流模在二维有界区域和三维外区域整体适定性。取得了一系列研究成果，在SCI源期刊上发表和接受发表论文3篇，发表和接受发表论文的杂志包括 《Journal of Mathematical Fluid Mechanics》，《Nonlinear Analysis: Real World Applications》，《Pure and Applied Mathematics》。