中国科学院数学与系统科学研究院
数学研究所
数学科学全国重点实验室
几何分析研讨班
Speaker: 寿凌云 副教授（南京师范大学）
Inviter: 杨明轩
Language: 中文
Title: Optimal Decay, Gevrey Analyticity, and Low-Frequency Regularity for Viscous Compressible Fluids with Capillarity
Time & Venue：2026年8月14日（星期五） 10:00-11:00 ZoomID：869-584-100
Abstract: We study the Cauchy problem for the multi-dimensional Navier--Stokes equations of viscous compressible fluids with capillarity. A key point in the capillary model is its hidden diffusion mechanism: due to the coupling between viscosity and capillarity, the density fluctuation acquires a heat-like diffusive behavior. This enables us to establish an equivalent connection between optimal decay, Gevrey analyticity, and low-frequency regularity. For given global solutions, we prove that the boundedness of the low-frequency part of the initial perturbation in a Besov space with negative regularity exponent is not only sufficient but also necessary for obtaining Gevrey analyticity and upper decay bounds for arbitrary higher-order derivatives. We further establish a sharp two-sided decay characterization: the matching upper and lower decay bounds hold if and only if the low-frequency initial data belong to a suitable nondegenerate subset of this Besov space. The proof combines a refined low-frequency spectral analysis, inspired in part by the work of Hoff and Zumbrun, with nonlinear estimates that track the growth of the analyticity radius and quantify the faster decay of the discrepancy between the nonlinear solution and the associated diffusion profile.
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