2019/11/29 by Oussama Ben Said, Jiahong Wu, Said, Oussama Ben +1
Engineering · Mathematics · #35A01 #35A02 #35Q35 #76D03 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1911.13236
openalex publication_date 2019/11/29 · openalex created_date 2019/12/05 · openalex updated_date 2026/07/28
This article examines the existence and uniqueness of weak solutions to the d-dimensional micropolar equations (d=2 or d=3) with general fractional dissipation (-Δ)αu and (-Δ)βw. The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when α≥ \frac12 and β≥ \frac12, any initial data (u0, w0) in the critical Besov space u0∈ B1+(d)/(2)-2α2,1(\mathbb Rd) and w0∈ B1+(d)/(2)-2β2,1(\mathbb Rd) yields a unique weak solution. For α≥ 1 and β=0, any initial data u0∈ B1+(d)/(2)-2α2,1(\mathbb Rd) and w0∈ B(d)/(2)2,1(\mathbb Rd) also leads to a unique weak solution as well. The regularity indices in these Besov spaces appear to be optimal and can not be lowered in order to achieve the uniqueness. Especially, the 2D micropolar equations with the standard Laplacian dissipation, namely α=β=1 have a unique weak solution for (u0, w0)∈ B02,1. The proof involves the construction of successive approximation sequences and extensive \it a priori estimates in Besov space settings.