On the refined analyticity radius of 3-D generalized Navier-Stokes equations

We analyze the instantaneous growth of analyticity radius for three dimensional generalized Navier-Stokes equations. For the subcritical Hγ( R³) case with γ>12, we prove that there exists a positive time t₀ so that for any t∈]0, t₀], the radius of analyticity of the solution u satisfies the pointwise-in-time lower bound rad(u)(t)≥ √(2γ-1)t(|ln t|+ln|ln t|+Kₜ), where Kₜ → ∞ as t→ 0⁺. This in particular gives a nontrivial improvement of the previous result by Herbst and Skibsted in {HS} for the case γ∈ ]1/2,3/2[ and also settles the decade-long open question in {HS}, namely, whether or not t→ 0⁺ rad(u)(t)√t|ln t|≥ √2γ-1 for all γ≥ 32. For the critical case H¹²( R³), we prove that there exists t₁>0 so that for any t∈ ]0, t₁], rad(u)(t)≥ λ(t)√t with λ(t) satisfying limt→ 0⁺λ(t)=∞.