2024/06/16 by Li, Dong, Zhang, Ping
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2406.10865
We analyze the instantaneous growth of analyticity radius for three dimensional generalized Navier-Stokes equations. For the subcritical Hγ(\mathbb R3) case with γ>\frac12, 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 \citeHS for the case γ∈ ]1/2,3/2[ and also settles the decade-long open question in \citeHS, namely, whether or not \liminft→ 0+\frac rad(u)(t)√(t|ln t|)≥ √(2γ-1) for all γ≥ \frac32. For the critical case H\frac 12(\mathbb R3), we prove that there exists t1>0 so that for any t∈ ]0, t1], \mathrm rad(u)(t)≥ λ(t)√(t) with λ(t) satisfying limt→ 0+λ(t)=∞.