Self-similar vorticity around the boundary and non-uniqueness of solutions to the two-dimensional Navier-Stokes equations in the half space

2025/07/03 by Aoki, Motofumi, Maekawa, Yasunori
#35A02 #35B30 #35Q30 #35Q35 #76D05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.02338

Abstract

In this paper we show the non-uniqueness of mild solutions to the two-dimensional forced Navier-Stokes equations in the half space under the noslip boundary condition, following the program established by Albritton, Brué, and Colombo in 2022. Our construction of non-unique solutions is based on the instability of self-similar vorticity at high Reynolds numbers which concentrates around the boundary at the initial time. In our construction, therefore, a kind of boundary layer has to be taken into account in the analysis, contrasting to the known results where the unstable self-similar vorticity is located away from the boundary with O(1) distance around the initial time.

Citations

• Uniqueness of mild solutions to the Navier-Stokes equations in weak-type L^d space
• Gluing non-unique Navier-Stokes solutions
• Instability and nonuniqueness for the 2d Euler equations in vorticity form, after M. Vishik
• Non-uniqueness of Leray solutions of the forced Navier-Stokes equations
• Optimal Prandtl expansion around concave boundary layer
• Instability and non-uniqueness in the Cauchy problem for the Euler\n equations of an ideal incompressible fluid. Part II
• Remark on the strong solvability of the Navier-Stokes equations in the weak Lⁿ space
• Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space?

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• Vanishing viscosity non-unique solutions to the forced 2D Euler Equations
• Uniqueness Criteria for the Oseen Vortex in the 3d Navier-Stokes\n Equations