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
偏微分方程研讨班
Speaker: Jiahong Wu (University of Notre Dame)
Inviter: 张平
Language: English
Title: Stabilizing Mechanisms for the 2D Navier–Stokes Equations with Horizontal Dissipation
Time & Venue: 2026年8月16日（星期日） 15:00-16:00南楼N913
Abstract: The 2D Navier–Stokes equations with dissipation in only the horizontal direction sit between two very different regimes. With full dissipation, solutions on the whole space R^2 are stable near the trivial solution and decay algebraically in time. Without any dissipation, solutions to the 2D Euler equations can grow rapidly in Sobolev norms. In the intermediate case of purely horizontal dissipation on the whole space R^2, neither the stability near the trivial solution nor the large-time behavior is currently understood, and the small-data global well-posedness remains an outstanding open problem. This talk describes three mechanisms that help stabilize the horizontally dissipated 2D Navier--Stokes system. The first is a weakening of the dissipation itself: we prove that the equations with fractional horizontal dissipation $\Lambda_1^{2s}$ ($0 \le s < 1$) are stable, together with explicit decay rates (joint with Z. Wang and N. Zhu). The second is buoyancy: coupling the horizontally dissipated velocity to a temperature equation in the Boussinesq system, we establish global stability and enhanced decay, where the interplay of anisotropic dissipation and anisotropic dispersion is the driving mechanism (joint with F. Yan and N. Zhu). The third is a background magnetic field: coupling the system to the magnetic field in a 2D MHD system with only horizontal dissipation and magnetic diffusion, we prove global stability near a background magnetic field. Reformulating the system in Elsässer-type variables reveals an intrinsic dispersive interaction between the two coupled fields, and the analysis uncovers a hidden nonlinear dissipative structure generated by the background field and the coupling (joint with F. Yan, N. Zhu, and Y. Zhu).
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