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October 1, 2025

Sharp non‐uniqueness for the 2D hyper‐dissipative Navier–Stokes equations

Key Points

This research shows that weak solutions of the hyper-dissipative Navier-Stokes equations are non-unique.
Key findings indicate sharp non-uniqueness at critical points based on viscosity exponent assessments.
The methodology involved extending prior work on non-uniqueness in two-dimensional Navier-Stokes equations to the hyper-dissipative case.
These results highlight the limitations of existing conditions, indicating possible new avenues for fluid dynamics investigations.

Abstract

Abstract In this paper, we study the non‐uniqueness of weak solutions for the two‐dimensional hyper‐dissipative Navier–Stokes equations (NSE) in the super‐critical spaces when the viscosity exponent , and obtain the conclusion that the non‐uniqueness of the weak solutions at the two endpoints is sharp in view of the generalized Ladyženskaya–Prodi–Serrin condition with the triplet and . By using the intermittency of the temporal concentrated function in an almost optimal way, we extend the recent elegant works on the non‐uniqueness of 2D NSE in Cheskidov and Luo Invent. Math. 229 (2022), no. 3, 987–1054 and Ann. PDE, 9 (2023), no. 2, Paper No. 13 to the hyper‐dissipative case . In particularly, the viscosity exponent is the upper limit for the one endpoint case when .

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Sharp non‐uniqueness for the 2D hyper‐dissipative Navier–Stokes equations

Key Points

Abstract

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