Asymptotic instability for the forced Navier--Stokes equations in critical Besov spaces

2025/09/25 by Fujii, Mikihiro, Tsurumi, Hiroyuki · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.21272

Abstract

The asymptotic stability is one of the classical problems in the field of mathematical analysis of fluid mechanics. In ℝⁿ with n ≥ 3, it is easily proved by the standard argument that if the given small external force decays at temporal infinity, then the small forced Navier--Stokes flow also strongly converges to zero as time tends to infinity in the framework of the critical Besov spaces B_p,q^n/p-1(ℝⁿ) with 1 ≤ p < n and 1 ≤ q < ∞. In the present paper, we show that this asymptotic stability fails for p ≥ n with n ≥ 3 in the sense that there exist arbitrary small external forces whose critical Besov norm decays in large time, whereas the corresponding Navier--Stokes flows oscillate and do not strongly converge as t → ∞ in the framework of the critical Besov spaces B_p,q^n/p-1(ℝⁿ). Moreover, we find that the situation is different in the two-dimensional case n=2 and show the forced Navier--Stokes flow is asymptotically unstable in B_p,1^2/p-1(ℝ²) for all 1 ≤ p ≤ ∞. Our instability does not appear in the linear level but is caused by the nonlinear interaction from external forces.

Citations

• Ill-posedness of the two-dimensional stationary Navier--Stokes equations on the whole plane
• Ill-posedness for the stationary Navier-Stokes equations in critical Besov spaces
• Remark on the strong solvability of the Navier-Stokes equations in the weak Lⁿ space
• Ill-posedness of the Navier-Stokes equations in a critical space in 3D
• StrongL p -solutions of the Navier-Stokes equation inR m , with applications to weak solutions

Cited by

• Non-Algebraic Decay for Solutions to the Navier-Stokes Equations

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