一些重要不等式在四维空间中的推广

Generalizations of Some Important Inequalities in Four-Dimensional Space

摘要: 本文专注于四维Sobolev空间，综合运用调和分析技术，Sobolev嵌入定理及不等式等估计技巧，推导并证明了一些重要不等式，如推广的Gagliardo-Nirenberg不等式，Sobolev嵌入不等式及复合函数各阶导数的L^∞范数估计。研究结果进一步丰富了高维Sobolev空间相关数学理论，为解决四维可压缩Navier-Stokes方程，四维可压缩MHD方程等偏微分方程的适定性和研究解的性质提供了相应的数学工具。

Abstract: This paper focuses on the four-dimensional Sobolev space. By comprehensively using techniques of harmonic analysis, the Sobolev embedding theorem, and estimation skills such as inequalities, some important inequalities are deduced and proved, such as the generalized Gagliardo-Nirenberg inequality, the Sobolev embedding inequality, and the L^∞-norm estimation of derivatives of all orders of composite functions. The research results further enrich the relevant mathematical theory of high-dimensional Sobolev spaces and provide corresponding mathematical tools for solving the well-posedness of partial differential equations, such as the four-dimensional compressible Navier-Stokes equations and the four-dimensional compressible MHD equations, as well as for studying the properties of solutions.