On the w2,2-Regularity of incompressible fluids with shear and pressure dependent viscosity in the case of flat boundary

JOURNAL OF SCIENCE OF HNUE 2011, Vol. 56, N◦. 1, pp. 3-10 ON THE W 2,2-REGULARITY OF INCOMPRESSIBLE FLUIDS WITH SHEAR AND PRESSURE DEPENDENT VISCOSITY IN THE CASE OF FLAT BOUNDARY Nguyen Duc Huy Hanoi National University of Education E-mail: ndhuyuk@gmail.com Abstract. We study the higher differentiability of stationary solutions up to the flat boundary for a class of systems of fluid mechanics modelling flows of incompressible fluids with shear and pressure dependent viscosity in 2D or 3D. We consider systems which are a kind of generalized Navier-Stokes system where stress tensor T has linear growth in symmetric velocity gradient and dependence of viscosity on the pressure is small with respect to an ellipticity constant. Keywords: incompressible fluid, elliptic system, regularity up to the boundary 1. Introduction Let Ω ⊂ Rd, (d = 2, 3), be a bounded domain with boundary ∂Ω. We study a following problem: For given f = (f1, · · · , fd) : Ω −→ Rd and stress tensor T (Dv, p) : Rd×d × R −→ Rd×d we look for v = (v1, · · · , vd) : Ω −→ Rd and p : Ω −→ R solving d∑ k=1 vk ∂v ∂xk − div T (p,Dv) +∇p = f in Ω, div v = 0 in Ω, (1.1) v = 0 on ∂Ω, where Dv denotes the symmetric part of the velocity gradient ∇v: Dv = 1 2 (∇v +∇Tv) with Dijv = 1 2 ( ∂vi ∂xj + ∂vj ∂xi ). We assume throughout this section that T (p,Dv) = ν(p, |D|2) Dv, (1.2) 3 Nguyen Duc Huy where a generalized viscosity ν is supposed to be continuously differentiable function of both variables. Moreover, there exist positive constants λ0, λ1 and ν0 such that for arbitrary symmetric d× d- matrices ξ, D and any p ∈ R the following estimates hold λ0|ξ|2 ≤ ∂T ∂D (p,D)ξ : ξ ≤ λ1|ξ|2,∣∣∣∣∂ν∂p (p, |D|)D ∣∣∣∣ ≤ ν0. (1.3) The existence of solutions to problem (1.1) is proved in [2] under assumptions on growth conditions of T . For the regularity problems, the smoothness of u and p is a more delicate problem. As we deal with a system of nonlinear elliptic PDEs we can not expect full regularity in space dimensions d ≥ 3. The local regularity of solutions for problem (1.1) studied in [3]. In this paper we study the higher differentiability of these solutions up to the boundary. For simplicity, we consider problem (1.1) only m = 2 and Ω = B+1 (0) := B1(0) ∩ {x ∈ Rd; xd > 0} and ν0 small enough. We obtain the result about derivatives up to order 2 of solutions of problem (1.1). 2. Preliminaries We begin with some definitions and notations. Let Ω be a domain in Rd (d ≥ 2), x = (x1, · · · , xd) ∈ Rd, f ∈W 2,2(Ω), v = (v1, · · · , vd) ∈W 1,2(Ω)d, ∇f := ( ∂f ∂xj )dj=1, ∇2f := ( ∂2f ∂xj∂xl )dj,l=1, Dv := 1 2 (∇v +∇Tv) x = (x′, xd), x ′ = (x1, · · · , xd−1), u = (u′, ud), u′ = (u1, · · · , ud−1), ∇f = (∇′f, ∂f ∂xd ), ∇′ = ( ∂f ∂x1 , · · · , ∂f ∂xd−1 ), ∂v ∂xj := ( ∂v1 ∂xj · · · , ∂vd ∂xj ). Br(x0) := { x ∈ Rd; |x− x0| < r } , B+r (x0) := Br(x0) ∩ {x ∈ Rd; xd > 0}, B∗r (x0) := Br(x0) ∩ {x ∈ Rd; xd ≥ 0}, x0 ∈ Rd, r > 0. Let ξ be d× d matrix and A be d2 × d2 matrix. Denote |x| := ( d∑ i=1 |xi|2) 12 , |ξ| := ( m∑ i,j=1 |ξij|2) 12 ;Aξ : ξ := d∑ k,l,i,j=1 Aklijξklξij Now we recall the results on solvability of equations div v = g, ∇p = f. Lemma 2.1. ( See [6, Ch. 2, Lemma 2.1.1, 2.2.2.]) Let Ω be a bounded Lipschitz domain in Rd, let Ω0 be a nonempty subdomain of Ω and let 1 < q < ∞, q′ = qq−1 . 4 On the W 2,2-regularity of incompressible fluids with shear and... Then it holds: a)There is a constant C = C(q,Ω) > 0 such that for each g ∈ Lq(Ω) with ∫ Ω g dx = 0 there exists at least one v ∈W 1,q0 (Ω)d satisfying div v = g in Ω, ‖∇v‖q ≤ C‖g‖q. (2.1) b)There is a constant C = C(q,Ω,Ω0) > 0 such that for each f ∈ W−1,q(Ω)d sat- isfying condition [f, v] = 0 for all v ∈ W 1,q′0,div(Ω) there exists a unique p ∈ Lq(Ω) satisfying ∇p = f in Ω, ∫ Ω0 p dx = 0 and ‖p‖q ≤ C‖f‖−1,q. (2.2) Remark. It is easy to see that C does not depend on translations and rotations of the couple Ω,Ω0. By scaling argument we deduce that for Ω = Ω0 = BR(x0) or Ω = Ω0 = B + R(x0)(x0 ∈ Γ∞) the constant C does not depend on R. In this case we denote Cdiv as the infimum of the constants C from the inequality (2.1). In connection with estimates of full gradient and symmetric gradient we will use Korn's inequality Lemma 2.2. (Korn's inequality, See [5, Ch. 6, Theorem 3.1.]) Let Ω be a bounded Lipschitz domain in Rd. Then for every u ∈ W 1,20 (Ω) it holds ‖∇u‖2;Ω ≤ √ 2‖Du‖2;Ω. (2.3) Due to the relation ∂2ui ∂uj∂uk = ∂Diku ∂xj + ∂Diju ∂xk − ∂Djku ∂xi we have the lemma: Lemma 2.3. Let u ∈W 2,q(Ω), then it holds ‖∇2u‖q ≤ 3‖D(∇u)‖q. (2.4) We conclude this part by providing a simple result from the linear algebra about a positive definite matrix. Lemma 2.4. Let M be d× d matrix : Ω→ Rd ×Rd, M ∈ L∞(Ω). If M is positive definite i.e. there exists a constant λ > 0 such that d∑ i,j=1 Mijxixj ≥ λ|x|2 in Ω for all x ∈ Rd, (2.5) then M is regular and the inequality |det M| ≥ C (2.6) holds with a constant C = C(λ, d) > 0. 5 Nguyen Duc Huy 3. Higher differentiability of solutions The higher differentiability of solutions to the system (1.1) in the interior is studied in [3] for T having subquadratic growth m. In [3] under the suitable conditions the authors show that v ∈ W 2,2loc (Ω)d, p ∈ W 1,2loc (Ω). By a similar way we get the following result: Theorem 3.1. (Interior estimates) Let the assumption (1.2), (1.3) be satisfied, ν0 < λ0 (λ0+Cdiv,Ωλ1)Cdiv,Ω and f ∈ L2(Ω)d. Let (v, p) ∈W 1,20 (Ω)d×L2(Ω) be a weak solu- tion to problem (1.1). Then v ∈W 2,2loc (Ω), p ∈ W 1,2loc (Ω). Moreover, for any subdomain Ω′ ⊂⊂ Ω the inequality ‖∇2v‖2;Ω′ + ‖∇p‖2;Ω′ ≤ C (3.1) holds with a positive constant C which depends on ‖v‖2, ‖p‖2, ‖f‖2 and dist (Ω′,Ω). In this section, we prove higher differentiability of solution up to the boundary. Suppose that Ω = B+1 (0). Then we have the following theorem Theorem 3.2. Let the assumption (1.2), (1.3) be satisfied, ν0 < min ( λ0 (λ0+Cdivλ1)Cdiv , λ0 λ0+3(d−1)(λ1−λ0) ) with (Cdiv defined in Remark after Lemma 2.1), f ∈ L2(B+1 (0))d. Let (v, p) ∈ W 1,20 (B+1 (0))d × L2(B+1 (0)) be a weak solution to problem (1.1) and supp v, supp p ⊂ B∗1(0). Then v ∈ W 2,2(B+1 (0)), p ∈ W 1,2(B+1 (0)). Moreover, we have an estimate ‖∇2v‖2;B+ 1 (0) + ‖∇p‖2;B+ 1 (0) ≤ C (3.2) with a positive constant C which depends on ‖v‖2, ‖p‖2, and ‖f‖2. Proof. By the same procedure as in the proof of higher differentiability of solution to (1.1) in the interior of domain Ω (see Theorem 5.1 in [3]) we get D ∂v ∂xs ∈ L2(B+1 (0))d 2 , ∂p ∂xs ∈ L2(B+1 (0)) for all s = 1, · · · , d− 1, and we have estimate ‖D∇′v‖2 + ‖∇′p‖2 ≤ C (3.3) with a constant C = C(‖v‖2, ‖p‖2, ‖f‖2) > 0. From the assumption supp v ⊂ B∗1(0), v = 0 on Γ, it follows that ∂v∂xs ∈W 1,2 0 (B + 1 (0)) d for all s = 1, · · · , d − 1. We have from Korn's inequality that ∇ ∂v ∂xs ∈ L2(B+1 (0))d2 for all s = 1, · · · , d− 1 . Since div v = 0, ∇ ∂v ∂xs ∈ L2(B+1 (0))d2 for all s = 1, · · · , d− 1, we get ∂2vd ∂2xd ∈ L2(B+1 (0)). (3.4) Therefore, it is sufficient to prove that ∂Didv ∂xd ∈ L2(B+1 (0)), i = 1, · · · , d − 1 and ∂p ∂xd ∈ L2(B+1 (0)). 6 On the W 2,2-regularity of incompressible fluids with shear and... Next, Theorem 3.1 guarantees the existence of second derivatives of v and first derivatives of p which are locally square integrable on B+1 (0). Thus a.e. on B + 1 (0) it holds d∑ k=1 vk ∂vi ∂xk − d−1∑ j=1 ∂Tij(p,Dv) ∂D ∂Dv ∂xj − d−1∑ j=1 ∂Tij(p,Dv) ∂p ∂p ∂xj − ∂Tid(p,Dv) ∂xd +∇p = f , i = 1, · · · , d (3.5) or −∂Tid(p,Dv) ∂D ∂Dv ∂xd − ∂Tid(p,Dv) ∂p ∂p ∂xd + d∑ k=1 vk ∂vi ∂xk − d−1∑ j=1 ∂Tij(p,Dv) ∂D ∂Dv ∂xj − d−1∑ j=1 ∂Tij(p,Dv) ∂p ∂p ∂xj +∇p = f , i = 1, · · · , d. (3.6) Hence d−1∑ k=1 2 ∂Tid(p,Dv) ∂Dkd ∂Dkdv ∂xd +( ∂Tid(p,Dv) ∂p −δid) ∂p ∂xd = d∑ k=1 vk ∂vi ∂xk +Fi , i = 1, · · · , d, (3.7) where Fi are given by Fi := − d−1∑ k,l=1 2 ∂Tid(p,Dv) ∂Dkl ∂Dklv ∂xd − ∂Tid(p,Dv) ∂Ddd ∂Dddv ∂xd − d−1∑ j=1 ∂Tij(p,Dv) ∂D ∂Dv ∂xj − d−1∑ j=1 ∂Tij(p,Dv) ∂p ∂p ∂xj + (1− δid) ∂p ∂xi − fi , i = 1, · · · , d. (3.8) It is easy to see that (3.3) and (3.4) implies Fi ∈ L2(B+1 (0)); i = 1, · · · , d. (3.9) Now, we consider the system (3.7) at first as a linear system in the unknowns ∂Dkdv ∂xd , k = 1, · · · , d− 1, ∂p ∂xd . Denote (Rik) d i,k=1 the matrix of system (3.7) i.e. Rik = 2 ∂Tid(p,Dv) ∂Dkd if k < d, i < d; Rid = ∂Tid(p,Dv) ∂p if k = d, i < d; Rdd = ∂Tdd(p,Dv) ∂p − 1. 7 Nguyen Duc Huy Multiply dth row of R by (1 − ∂Tdd(p,Dv) ∂p )−1 ∂Tid(p,Dv) ∂p and then subtract it from ith row (i = 1, · · · , d − 1). We obtain a new matrix whose determinant det (R) = (∂Tdd(p,Dv) ∂p − 1)2d−1det S where S is a (d− 1)× (d− 1) matrix given by Sik = ∂Tid(p,Dv) ∂Dkd + (1− ∂Tdd(p,Dv) ∂p )−1 ∂Tid(p,Dv) ∂p ∂Tdd(p,Dv) ∂Dkd . Moreover, we have d−1∑ k=1 Sik ∂Dkdv ∂xd = d∑ k=1 vk ∂vi ∂xk + Fi + [ d∑ k=1 vk ∂vd ∂xk + Fd] [1− ∂Tdd(p,Dv) ∂p ]−1 ∂Tid(p,Dv) ∂p =: Hi, i = 1, · · · , d− 1. (3.10) Next, we show that S is positive definite matrix. In fact, we have from (1.3) d−1∑ i,k=1 Sikζiζk = d−1∑ i,k=1 ∂Tid(p,Dv) ∂Dkd ζiζk + d−1∑ i,k=1 [(1− ∂Tdd(p,Dv) ∂p )−1 ∂Tid(p,Dv) ∂p ∂Tdd(p,Dv) ∂Dkd ]ζiζk ≥ λ0 4 |ζ |2 − d−1∑ i,k=1 ν0 1− ν0 | ∂Tdd(p,Dv) ∂Dkd ||ζi||ζk| for all ζ ∈ Rd−1. On the other hand, for every k ∈ {1, · · · , d − 1} by choosing ξdd = ξkd = ξdk = 1, ξij = 0 if (i, j) 6= (d, d), (d, k) and (k, d) the inequality (1.3) shows that 3λ1 ≥ 4∂Tkd(p,Dv) ∂Dkd + 4 ∂Tdd(p,Dv) ∂Dkd + ∂Tdd(p,Dv) ∂Ddd ≥ 3λ0. We also have λ1 ≥ ∂Tdd(p,Dv) ∂Ddd ≥ λ0; 2λ1 ≥ 4∂Tkd(p,Dv) ∂Dkd ≥ 2λ0. Hence |∂Tdd(p,Dv) ∂Dkd | ≤ 3 4 (λ1 − λ0). Therefore, we obtain an inequality d−1∑ i,k=1 Sikζiζk ≥ [ λ0 4 − 3 4 (d− 1) ν0 1− ν0 (λ1 − λ0) ] |ζ |2 = λ|ζ |2 for all ζ ∈ Rd−1. (3.11) 8 On the W 2,2-regularity of incompressible fluids with shear and... Thanks to ν0 < λ0 λ0+3(d−1)(λ1−λ0) we get λ > 0. From this we conclude that matrix S is positive definite. Thus there exists a positive constant C such that det (N) ≥ C (thanks to Lemma 2.4). Therefore det(R) ≥ 2d−1C (3.12) and system (3.7) can be solved for the unknowns ∂Dkdv ∂xd , k = 1, · · · , d− 1, ∂p ∂xd , for almost all x ∈ B+1 (0). Sobolev embedding theorem shows that L6(B+1 (0)) d ⊂W 1,20 (B+1 (0))d. Using Young's inequality we get∫ B+ 1 (0) (v.∇v) 32 ≤ 1 4 ∫ B+ 1 (0) |v|6 dx+ 3 4 ∫ B+ 1 (0) |∇v|2 dx ≤ C. It implies v.∇v ∈ L 32 (B+1 (0))d. From this, (3.12) and the calculation of ∂Dkdv∂xd , k = 1, · · · , d− 1, ∂p ∂xd from (3.6) where Fi ∈ L2(B+1 (0)), i = 1, · · · , d; v.∇v ∈ L 32 (B+1 (0)), we deduce that ∂Dkdv ∂xd ∈ L 32 (B+1 (0)), k = 1, · · · , d− 1, ∂p ∂xd ∈ L 32 (B+1 (0)). It implies v ∈W 2, 3 2 0 (B + 1 (0)) d . (i) d = 2 Sobolev embedding theorem implies that ∇2v ∈ L 32 (B+1 (0)), then ∇v ∈ L6(B+1 (0)). Hence v ∈ L∞(B+1 (0)) and v.∇v ∈ L2(B+1 (0))d. (ii) d = 3 Sobolev embedding theorem implies that∇v ∈ L3(B+1 (0)) , v ∈ L6(B+1 (0)). Using Young's inequality we get∫ B+ 1 (0) (v.∇v)2 ≤ 1 3 ∫ B+ 1 (0) |v|6 dx+ 2 3 ∫ B+ 1 (0) |∇v|3 dx ≤ C. It implies v.∇v ∈ L2(B+1 (0))d and Hi ∈ L2(B+1 (0)) for all i = 1, · · · , d− 1. By a similar way, we obtain ∂Dkdv ∂xd ∈ L2(B+1 (0)), k = 1, · · · , d− 1 , ∂p ∂xd ∈ L2(B+1 (0)). (3.13) Next, we estimate for ∂Dkdv ∂xd , k = 1, · · · , d− 1, ∂p ∂xd . By setting ζk = ∂Didv ∂xd , k = 1, · · · , d− 1; , we get from (3.10) d−1∑ k=1 Sikζk = Hi, i = 1, · · · , d− 1. (3.14) Thus d−1∑ i,k=1 Sikζkζi = d−1∑ i=1 Hiζi. (3.15) 9 Nguyen Duc Huy Consequently, λ|ζ |2 ≤ |H||ζ | or λ d−1∑ k=1 |∂Dkdv ∂xd | ≤ |H| a.e. in B+1 (0). Hence d−1∑ k=1 ‖∂Dkdv ∂xd ‖2,B+ 1 (0) ≤ C(‖v‖2, ‖p‖2, ‖f‖2). Now the dth equation of system (3.7) implies ‖ ∂p ∂xd ‖2,B+ 1 (0) ≤ C(‖v‖2, ‖p‖2, ‖f‖2), and we obtain the inequality (3.2). Theorem is proved. REFERENCES [1] H. Beir¢o da Veiga, 2005. On the regularity of flows with Ladyzhenskaya shear dependent viscosity and slip or non-slip boundary conditions. Com. Pure Appl. Math. 58, No. 4, pp. 552-577. [2] M. Franta, J. M¡lek, K. R. Rajagopal, 2005. On steady flows of fluids with pressure- and shear-dependent viscosities. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461, No. 2055, pp. 651-670. [3] J. M¡lek, G. Mingione, J. Star¡, Fluids with pressure dependent viscosity: partial regularity of steady flows. to appear. [4] J. M¡lek, J. Necas, K. R. Rajagopal, 2002. 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