Partial Regularity for Stationary Navier-Stokes Systems by the Method of $$\mathcal{A}$$ A -Harmonic Approximation
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
PARTIAL REGULARITY FOR STATIONARY NAVIER-STOKES SYSTEMS BY THE METHOD OF A-HARMONIC APPROXIMATION∗
ÌÒ)
Yichen DAI (
School of Mathematical Sciences, Xiamen University, Fujian 361005, China E-mail : [email protected]
Zhong TAN (
§)
School of Mathematical Science and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Fujian 361005, China E-mail : [email protected] Abstract In this article, we prove a regularity result for weak solutions away from singular set of stationary Navier-Stokes systems with subquadratic growth under controllable growth condition. The proof is based on the A-harmonic approximation technique. In this article, we extend the result of Shuhong Chen and Zhong Tan [7] and Giaquinta and Modica [18] to the stationary Navier-Stokes system with subquadratic growth. Key words
Stationary Navier-Stokes systems; controllable growth condition; partial regularity; A-harmonic approximation
2010 MR Subject Classification
1
35J60; 35Q30; 76N10
Introduction and Statement of the Result
Throughout this article, on a domain Ω, where Ω is a bounded with Lipschitz boundary in R with dimension n ≥ 2, we consider weak solutions u : Ω → RN of stationary Navier-Stokes systems of the type ∇ · u = g, Z Z (1.1) α i Bi (x, u(x), ∇u(x))ϕi dx, Ai (x, u(x), ∇u(x))Dα ϕ dx = n
Ω
Ω
for all ϕ ∈ W01,p (Ω, RN ) with ∇ · ϕ = 0 in Ω. We shall assume that Aα i (x, u, ∇u) and Bi (x, u, ∇u) are measurable functions for all u ∈ 1,p N W (Ω, R ) satisfying the following growth conditions. 1.1
Assumptions on the structure functions Aα i
We assume that Aα i (x, u, ξ) are differentiable functions with respect to the ξ-variable with bounded and continuous derivatives. Moreover, Aα i (x, u, ξ) is uniformly strongly elliptic. That ∗ Received
February 11, 2019.
836
ACTA MATHEMATICA SCIENTIA
is, Aα i (x, u, ξ) satisfies the following ellipticity and growth conditions, for some ∂Aα (x, u, ξ) p−2 i ζ · ζ ≥ ν(1 + |ξ|2 ) 2 |ζ|2 , β ∂ξj
Vol.40 Ser.B 2n n+2
∂Aα (x, u, ξ) p−2 i ≤ L(1 + |ξ|2 ) 2 , β ∂ξj
< p < 2, (A1)
(A2)
for all x ∈ Ω, u ∈ RN , and ξ, ζ ∈ RN n , where ν, L > 0 are given constants. There exist β ∈ (0, 1) and K : [0, ∞) −→ [1, ∞) monotone nondecreasing such that α |Aα x, u e, ξ)| ≤ K(|u|)(|x − x e|p + |u − u e|p )β/p (1 + |ξ|)p/2 , i (x, u, ξ) − Ai (e
(A3)
for all x, x e ∈ Ω, u, u e ∈ RN , and ξ ∈ RN n . Furthermore, we assume that there is a function ω : [0, ∞)×[0, ∞) −→ [0, ∞) with ω(t, 0) = 0 for all t such that t → ω(t, s) is monotone nondecreasing for fixed s, where s 7→ ω(t, s) is concave and monotone nondecreasing for fixed t, and such that ∂Aα ∂Aα i i e (x, u, ξ) β (x, u, ξ) − β ∂ξj ∂ξj (A4) 2 2 (p−2)/2 e e ≤ c(1 + |ξ| + |ξ| ) ω(|ξ|, |ξ − ξ|), for all x ∈ Ω, u ∈ RN , and ξ, ξe ∈ RN n .
1.2
Assumptions on the inhomogeneity Bi
For the inhomogeneity Bi , we assume that Bi satisfies the controllable growth condition 1
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