AI 中文总结
该研究在三维环面上针对大数据证明了含退化粘性项和Korteweg项的可压缩Navier-Stokes方程的全局弱解存在性，结合先验估计与近似格式，为相关模型的解构造提供了理论依据。
AI 中文摘要
针对三维环面上的大数据情形，本文确立了一大类Navier-Stokes-Korteweg方程的全局弱解存在性，该类方程包含扩散界面模型和量子Navier-Stokes系统作为特殊情形。该模型由可压缩Navier-Stokes方程构成，其中包含依赖于密度的退化粘性项以及一般的非线性三阶Korteweg项。存在性证明结合了由能量不等式和Bresch-Desjardins（BD）熵不等式给出的先验估计，以及精心设计的近似格式。分析的关键要素是与Korteweg项相关的新耗散不等式，该不等式通过系统的分部积分方法得到。为构造解，本文引入了人工阻力项和量子Korteweg正则化项，随后通过建立重正化公式在极限过程中消去这些项。
英文摘要
The existence of global weak solutions to a broad class of Navier-Stokes-Korteweg equations is established for large data in the three-dimensional torus, including the diffuse-interface and quantum Navier-Stokes systems as special cases. The model consists of the compressible Navier-Stokes equations with degenerate density-dependent viscosity and a general nonlinear third-order Korteweg term. The existence proof combines a priori estimates provided by the energy and Bresch-Desjardins (BD) entropy inequalities with a carefully designed approximation scheme. A crucial ingredient of the analysis is a new dissipation inequality associated with the Korteweg term, obtained via the systematic integration-by-parts method. To construct the solutions, artificial drag terms and a quantum Korteweg regularization are introduced, which are subsequently removed in the limit by establishing a renormalized formulation.
