Aug 9, 2013 - 1422-6928/14/020225-17. DOI 10.1007/s00021-013-0149-y. Journal of Mathematical. Fluid Mechanics. Regularity of Non-Newtonian Fluids.
J. Math. Fluid Mech. 16 (2014), 225–241 c 2013 Springer Basel 1422-6928/14/020225-17 DOI 10.1007/s00021-013-0149-y
Journal of Mathematical Fluid Mechanics
Regularity of Non-Newtonian Fluids Hyeong-Ohk Bae and Bum Ja Jin Communicated by H. Beir˜ ao da Veiga
Abstract. In this paper, we consider a non-Newtonian fluids with shear dependent viscosity in a bounded domain Ω ⊂ Rn , n = 2, 3. For the power-law model with the viscosity as in (1.4), we show the global in time existence of a weak solution 3n when n = 3 (see Theorem 1.1), and the local in time existence of a weak solution for 2 > q > n+2 , when n = 2, 3 for q ≥ 11 5 (see Theorem 1.2). Keywords. Navier–Stokes, non-Newtonian, weak, regular, strong, existence.
1. Introduction Let Ω ⊂ Rn , n = 2, 3 be a smooth bounded domain and let T > 0. This paper deals with unsteady flows of an incompressible fluid in Ω × [0, T ], which are described by the system of equations ut + (u · ∇)u − divT E + ∇p = 0,
divu = 0
in Ω × (0, T ).
(1.1)
The initial condition is given by u|t=0 = a in Ω
(1.2)
and the Dirichlet boundary condition is given by u|∂Ω×[0,T ] = 0.
(1.3)
Here u = (u1 , · · · , un ) is the velocity, p is the pressure, T E is the extra stress tensor, and a is the initial velocity of the fluid satisfying that a|∂Ω = 0 and diva = 0.For the non-Newtonian fluid with shear dependent viscosity T E is given as the form T E = ν E (|D|)D, where D denotes the symmetric nvelocity 2 gradient and |D| denotes the usual Euclidean matrix-norm: D = (Dij )i,j=1,...,n and |D|2 = i,j=1 Dij
for Dij = Dij (u) = 12 ( ∂xji + ∂u
∂ui ∂xj ).
In this paper we consider the degenerate power-law model ν E (|D|) = |D|q−2 .
(1.4)
There is abundant literature on the power-law model ν E (|D|) = (μ0 + |D|)q−2 + μ1 ,
μ0 , μ1 ≥ 0.
3n+2 n+2
had first appeared in [13–15], which is unique for q > n+2 The existence of weak solutions for q ≥ 2 (see 2n also [17]). The existence of measure-valued solutions was shown for q > n+2 in [18,23]. Later, the existence 2n of weak solution had been investigated for q > n+2 in [4,10,19–21,28]. Those results hold for both the space-periodic problem and the Dirichlet problem. This work was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology 2010-0016694.
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For the space-periodic problem, the global in time strong solution had been obtained for q ≥ 11 5 if 5 n = 3, and for q > 1 if n = 2 in [4,19](see also [26,27]); the local in time strong solution for q > 3 in [19], for 2 > q > 75 in [9], and for 2 > q > 1 in [5] when n = 3. There are fewer regularity results for the Dirichlet problem than for the space-periodic problem. However, as far as the case μ0 > 0 or μ1 > 0 had been considered, regularity for the Dirichlet problem has been well studied: the global in time strong solution was obtained for q ≥ 94 if n = 3 and q ≥ 2 if n = 2 in [21], and the regularity result was extended to the case q ≥ 11 5 in [3] when n = 3; the local in time strong solution was obtained for q > 1 in [1,6] (see also [2,24]). Up to now, the regularity of the weak solution of the equations (1.1)–(1.4) had been an open problem. Our aim of this paper is to investigate the global in time regularity properties of a weak solution for the 3n case 3 > q ≥ 11 5 when n = 3, and local in time regularity properties of a weak solution for 2 > q > n+2 when n = 2, 3. 1,q Theorem 1.1. Let Ω be a smooth bounded domain in R3 and let T > 0. Assume that a ∈ L2σ (Ω)∩W0,σ (Ω). 1,q ∞ 2 q . Then there exists a weak solution u ∈ L (0, T ; L (Ω)) ∩ L (0, T ; W (Ω)) Further, suppose 3 > q ≥ 11 0,σ 5 to (1.1)–(1.4)satisfying
|∇u| dx +
sup
0 0 and for any φ ∈ C0,σ ([0, T ) × Ω). Then u is called to be a weak solution to (1.1)–(1.4).
The following theorems and lemma are well known facts and will be used in proving Theorem 1.1– Theorem 1.2. Theorem 2.2 (Aubin-Lions compactness theorem). Let X, B and Y be Banach spaces so that X is compactly embedded into B and B is continuously embedded into Y . If {fn } is bounded in Lq (I, B) ∩ d L1 (I, X), 1 < q ≤ ∞, and if { dt fn } is bounded in L1 (I, Y ), then {fn } is relatively compact in Lp (I, B) for any 1 ≤ p < q. Theorem 2.3 (Vitali convergence theorem). Let {fn } be a sequence of measurable functions satisfying that fn (x) → f (x) a.e. in a measurable set M ⊂ Rn . Then fn converges strongly to f in L1 (X) if and only if the followings hold true. (i) for each > 0, there is δ > 0 so that sup |fn |dx < whenever |E| < δ, n
E
(ii) for any > 0, there is E ⊂ M of finite measurable set such that sup |fn |dx < . n
M \E
Lemma 2.4 (Algebraic Lemma). Let q > 1. There is a positive constant c so that for all ξ, η ∈ Rn \{0}, we have (δ + |D(ξ)|)q−2 D(ξ) − (δ + |D(η)|)q−2 D(η) : D(ξ) − D(η) c|D(ξ) − D(η)|q if q ≥ 2, ≥ c(δ + |D(ξ)| + D(η|)q−2 |D(ξ) − D(η)|2 if 1 < q < 2. ∞ n Define the mollification of f by Jδ f = Ω δ1n ζ( x−y δ )f (y)dy, where ζ ∈ C0 (R ). By the property of mollification, note that δ k ∇k Jδ f s ≤ cf s ,
1 ≤ s ≤ ∞.
The following inequalities also hold(for the completeness we give its proof in the Appendix). Proposition 2.5. ∇Jδ f q ≤ c∇f q , and ρ2 ∇Jδ f Lq (Ω) ≤ cρ2 ∇f Lq (Ω) + cf Lq (Ω) .
3. Proof of Theorem 1.1 Let Ω be a bounded domain in R3 with smooth boundary so that the distance function ρ = ρ(x) = dist(x, ∂Ω) satisfies |∇k ρ| ≤ cρ1−k ≤ c, k ≥ 0, for some positive constant c depending only on Ω. In the subsequent sections we consider the following approximating system ∂t uδ + (Jδ uδ · ∇)uδ − divTδE + ∇pδ = 0,
divuδ = 0
in Ω × (0, T )
(3.1)
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with the boundary condition uδ |∂Ω×[0,T ] = 0
(3.2)
and the initial condition uδ |t=0 = a
in Ω,
(3.3)
where TδE = (δ + |Dδ |)q−2 Dδ ,
Dδ = D(uδ ).
(3.4)
1,q Definition 3.1 (Strong solution). Let q ≥ 2. Suppose that uδ ∈ C(0, T ; L2σ (Ω)) ∩ Lq (0, T ; W0,σ (Ω)) ∩ 2
6
L q−1 (0, T ; W 2, q+1 (Ω)), ∂t uδ ∈ L2 ((0, T )×Ω) for any T > 0 satisfies the following variational formulation: ∂t uδ φ + (Jδ uδ · ∇uδ ) · φdx + (δ + |D(uδ )|)q−2 D(uδ ) : D(φ)dx = 0 (3.5) Ω
for any φ ∈
Ω
∞ (Ω) C0,σ
and for almost every t ∈ (0, T ). Then uδ is called to be a strong solution to (3.1)–(3.4).
According to the result of [21], there is a strong solution of the above approximating system (3.1) with the global energy inequality
Moreover, u ∈ L δ
2 q−1
T
|u | dx +
sup
0
