Liouville theorem for steady-state non-newtonian Navier-Stokes equations in two dimensions

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1 iouville theorem for steady-state non-newtonian Navier-Stokes euations in two dimensions Bum Ja Jin and Kyungkeun Kang Abstract: We consider the iouville type problem of stationary non-newtonian Navier-Stokes euations in the plane. We prove that weak solutions become trivial for both cases of some shear thickening and thinning flows. Introduction In this article, we study the iouville type problem of the steady state non-newtonian Navier-Stokes euations in the plane (u )u div T (E(u)) + π =, div u = in, (.) where u : is the velocity field and π : is the pressure. Under our consideration the shear stress tensor, T (E(u)), is given as the form T (E(u)) = µ E(u) + ( µ + E(u) ) E(u), (.) E(u) = E ij (u) = ( ) xi u j + xj u i, where µ and µ are nonnegative constants, i.e. µ, µ. We remark that the form of shear stress tensor in (.) can be extended to more generalized types of the models T (ξ) = σ( ξ )ξ, where σ( ) is given to satisfy σ( ξ ) µ + (µ + ξ ), σ( ξ )η : η µ η + (µ + ξ ) η, ( ) σ( ξ )ξ σ( η )η : (ξ η) µ ξ η + (µ + ξ + η ) ξ η. In addition, if >, then ( ) σ( ξ )ξ σ( η )η : (ξ η) (µ + µ ) ξ η + ξ η. If the shear stress is proportional to the strain tensor, i.e., = in (.), the system (.) becomes the (Newtonian) Navier-Stokes euations. For a certain types

2 of fluids, the relation between the shear stress and the strain rate is nonlinear and we refer to such fluids as non-newtonian flows. The viscous part of the stress tensor for non-newtonian fluids is often described in the form of power law of the symmetric deformation tensor as given in (.). The case that > describes shear thickening (dilatant) fluids whose viscosity increases with the rate of shear (see e.g. [3]). On the other hand, shear thinning (pseudo-plastic) fluids correspond to the case that < <, where viscosity decreases as the rate of shear increases (see e.g. [3]). Depending on the values of µ and µ, the fluid is called Newtonian, abinowitsch, Ellis, Ostwald-de Waele and Bingham, respectively (see e.g. [3]). The existence of weak solutions has been shown by ions [8] and adyzhenskaya [5], [6], [7] for 3n n n+ and this result was improved in [4] and [] for n+. The motivation of this paper is to study the iouville type theorem of weak solutions for the incompressible non Newtonian Navier-Stokes euations (.). Very recently, in [] Bildhauer, Fuchs and Zhang proved iouville type theorem for degenerate power law fluid in when a weak solution u is C ( ) (see also [5], [7], [6], [8] and [5] related to iouville type problem of non-newtonian fluid). As indicated in [], for some cases, due to improved regularity of weak solutions, weak solutions become C, for example 3/ < < in the interior case (see [4]). To the best of authors knowledge, it is, however, unknown whether or not, in general, weak solutions of (.)-(.) are C. Weak solutions will be defined in Section (see Definition.). Our main concern is to obtain iouville type theorem without additional assumption of higher regularity for weak solutions. To be more precise, our main result is read as follows: Theorem. Suppose that u is a weak solution of (.)-(.) and we assume one of following cases: (a) µ = µ = and u < with < 3. (b) µ >, µ and u < with < <. (c) µ = µ =, u < with 6, 5 < < and u Wloc ( ). Then, u is a constant vector in. Here we make some comments on Theorem.. Under the assumption that a weak solution u is C ( ), very recently, results in Theorem. was proved (see []). Compared to results of [], the improvement that we have made in this article is that we assume that u is just a weak solution for the case of (a), (b). For the case of (c), a weak solution u is assumed to be in W, loc ( ). In any case, we do not reuire that u is in the class of C ( ).

3 . For the case of (c), if we restrict with 3/ < <, the assumption that W, loc ( ) is not necessary. Indeed, in [4], it was shown that weak solutions become C,α in the interior. Hence, due to results of [] it is direct that u turns out to be trivial for such case. Therefore, in case of (c), our main concern is focused on the case that 6/5 < 3/. 3. We remark that a weak solution u in Theorem. is continuous, i.e. u C( ). Indeed, in case of (a) and (c), from Sobolev embedding theorem it is immediate that u C( ). On the other hand, for the case (b), the conditions u W, loc ( ), < < and µ > imply u W, loc ( ) (see emma 3. in Section 3). Again, due to Sobolev embedding theorem, we have u C( ). This paper is organized as follows: In Section we provide some preliminary results. In section 3 we present the proof of Theorem.. Preliminaries In this section we introduce the notation, define the notion of weak solutions, and present preparatory results that are useful to our analysis. We start with the notation. et Ω be a domain with smooth boundary in. For, we denote by W k, (Ω) the usual Sobolev spaces, namely W k, (Ω) = {f (Ω) : D α f (Ω), α k}. The set of th power ebesgue integrable functions on Ω is denoted by (Ω) and loc (Ω) indicates the set of locally th power ebesgue integrable functions defined on Ω. et s = k + α for < α <, then W s, (Ω) denotes the fractional Sobolev spaces with norm f W s, (Ω) = f W k, (Ω)+ f k,α,ω, where f k,α,ω = 3 ( j= β =k Ω Ω D β f(x) D β f(y) x y +α dy When Ω Ω is an open domain and its closure is compact in Ω, i.e., Ω Ω, it is well known that f k,α,ω is euivalent to the following: f k,α,ω = 3 ( l x Ω D β f(x + he j ) D β f(x) dh j= β =k h +α where l = dist(ω, Ω) (see e.g. []). B r (x) indicates a ball of radius r centered at x and we denote the mean value of f in a bounded domain D Ω by f D = D D f(y)dy. The letter C is used to represent a generic constant, which may change from line to line, and C(,, ) is considered a positive constant depending on,,. Next we define the notion of weak solutions.., 3

4 Definition. We say that u W, loc ( ) is a weak solution of the system (.) if the following identity is satisfied: For all ψ C ( ) with div ψ = ( ) T (E(u)) : E(ψ) + (u u) ψ =. (.) emark. It is well-known, due to the variational formulation, that there exists π loc ( ) such that the weak solution u in Definition. satisfies ( ) T (E(u)) : E(φ) + (u u) φ π div φ =, (.) where φ C ( ). Therefore, (.) can be replaced by (.). As mentioned earlier, it was proven in [4] that in case ( that 3 < <, µ = and σ(e(u))e(u) ) µ, any weak solution u in Definition. satisfies loc (Ω) and u C,α loc (Ω) with Ω. In such case, if µ >, solutions become smooth via the regularity theory of elliptic system (see e.g. []). For the Stokes system, the range of can be extended as < < (see [4]). Kaplicky, Malek, and Stara [4] established the existence of C,α ( Ω) and C,α (Ω) solutions of the two dimensional problem for > 3 and > 6 5, respectively, when µ and µ >. The subseuent lemmas are localized version of well-known ineualities such as Sobolev ineuality, adyzhenskaya ineuality and Korn s ineuality. emma.3 et u W, loc ( ). Then for each > u (B ) c + s u s (B ) + c s u s (B ) s s for B, then if s < ; < if s = ; for = if s >. Moreover, if u = on u (B ) c + s u s (B ). emma.4 et u W,4 loc ( ). Then for each > we have u 4 (B ) c u (B ) u (B ) + c u (B ). emma.5 et u W, loc ( ). Then for each > we have u (B ) c E(u) (B ) + c u (B ). Next, we recall the following lemma in [8], which is an extension of a result in Giuinta-Modica []. 4

5 emma.6 et f, f,, f l be nonnegative functions in loc ( ) and α,, α l given nonnegative numbers. There exists ɛ such that if for some ɛ ɛ the following ineuality l f c(ɛ) α j f j + ɛ f B (z) B (z) j= B (z) holds for any choice of B (z), then there exists a constant c such that B (z) f c l j= α j B (z) f j. We remind an existence result for divergence euation div w = f in a bounded domain Ω n (see [] and [9]). emma.7 et < <. Suppose that Ω n is a domain which contains B (x ) and is star-shaped with respect to each point of B (x ). Then for any f (Ω) with, Ω f =, there exists a vector field w W (Ω) such that w = f in Ω and w C f, Ω where C = C (n, )( + δ(ω)/) n+ and δ(ω) denotes the diameter of Ω. Proof. See emma 3. in [9]. Next we show decay estimate of ratio of the angular integral of f and some growing factor of radial variable r when f ( ) with < <. In case =, the estimate shown below was proven in []. Similar results have been very recently shown in [] when f C ( ). In our case, since we assume that f C( ), the details are presented. emma.8 et < <. Suppose that f C( ) and f ( ), i.e. f <. Then lim r Ω π r f(r, θ) dθ = if >, (.3) π lim r ln r f(r, θ) dθ = if =, (.4) π lim sup f(r, θ) dθ < c(r ) if <, (.5) r>r where c(r ) is a positive constant depending on r. 5

6 Proof. First, we assume that f C ( ). Noting first that we have d ( π ) π f(r, θ) dθ = f(r, θ) f(r, θ) f(r, θ)dθ, dr r d ( π f(r, θ) dθ dr ( π r f(r, θ) dθ We fix r > and let r > r. Integrating (.6) from r to r, we obtain ( π f(r, θ) dθ ( π Via Hölder s ineuality, we have. (.6) r ( f(r, θ) π dθ + r s f(s, θ) dθ π π r ( f(r, θ) ( f(r, θ) + ( f ( s r x r r Direct computations show that r s ds = r r s r r s ds = r ( r r ) cr if >, ds = ln (r/r ) c ln r if =, ( ) r r cr if <. ds) ds.. (.7) We first divide the both sides of (.7) by r if > and by ln r if =, respectively, and sending r to, we then have the following ineualities: and Since ( π lim r r f(r, θ) dθ ( π lim f(r, θ) dθ r ln r x r π ( π c f x r ( π c f x r if >, if =. f as r because of the hypothesis, we conclude that two ineualities, (.3) and (.4) of the emma are valid. In case that <, the estimate (.7) can be reduced to ( π f(r, θ) dθ ( π f(r, θ) dθ 6 +cr ( x r π f. (.8)

7 Taking as c(r ) the righthand side in (.8), we conclude the last ineuality of the emma. Now, we consider the case f C( ). et f ɛ be the mollification of f by a cut-off function ζ C (B ). Note that f ɛ ( ) c f ( ) for all ɛ >. Moreover, the condition f C( ) implies f ɛ C ( ) with lim ɛ f ɛ = f uniformly in. As in the case f C ( ), f ɛ satisfies (.7), that is, ( π f ɛ (r, θ) ( π f ɛ (r, θ) ( + f ɛ ( r ) s ds. r x r r Sending ɛ to the zero, we have ( π f(r, θ) ( π f(r, θ) ( + f ( r s r x r r ) ds This leads to the same result as in case f C ( ). This completes the proof. The direct conseuence of emma.8 is the following: Corollary.9 et f C( ) and f < with < <. Then cr 3 if >, f cr ln r if =, B r cr if <. Therefore, for any s > Proof. et r >. We note that r π f = f(s, θ) sdθds c B r From emma.8, we observe that ( π if >, and for any s > if lim r r s+ f =. (.9) B r f(s, θ) dθ r ( π f(s, θ) dθ cs if >, c ln s if =, c if <. sds. Integrating in radial variable, we obtain r ( π c r s+ ds = cr 3 if >, f(s, θ) dθ ds c r s ln sds cr ln r if =, c r sds = cr if <. We complete the proof.. 7

8 3 iouville theorem in two dimensions In this section, we present the proof of Theorem.. We often use the method of difference uotients and thus, for convenience, we denote h k f(x) = f(x+he k) f(x) h for < h < dist(x, Ω). First, we consider the shear thickening case. 3. Shear thickening flows : > We consider the degenerate case, i. e. µ = µ =. In next lemma we show that weak solutions of shear thickening flows are in fractional Sobolev space, which seems to be of independent interest. emma 3. et < < and µ = µ =. If u W, loc ( ) is a weak solution, +α, then E(u) Wloc ( ) for α < and therefore, u 3 loc ( ). Furthermore, as h tends to, h k u converges to x k u in 3 (B ) for all >, k =,. Proof. By density argument, the choice of the test function φ C ( ) can be extended to φ W, ( ). et η C ( ) be a cut-of function with η = on B and η = on B c. Taking test function φ = h k (η h ku) and noting that φ W, ( ), we have T ij (E(u))E ij ( h k (η h k u)) = u i u j xi h k (η h k u j) + πdiv h k (η h ku). (3.) We note that div(η h k u) = ( h k u )η and h k (fg) = f h,k h k g + g h kf, where f h,k (x) = f(x + he k ). By change of variables and integrations by parts, we obtain h k T ij(e(u)) h k E ij(u)η = h k T ij(e(u))( h k u j)( xi η ) ( h k u i)( xi u h,k j )(η h k u j) + (u η ) h k + h k π( h k u )η := I + J + K +. (3.) et h. We set X = h k T ij(e(u)) h k E ij(u)η. By the property of the constitutive relation, we recall that h k T ij(e(u)) h k E ij(u) c( E(u) + E(u h,k ) ) h k E(u) (3.3) and h k T ij(e(u)) c( E(u) + E(u h,k ) ) h ke(u). (3.4) 8

9 By Hölder s ineuality I c() X u W, (B 5 ) c() X. (3.5) Since u (B ) via Sobolev ineuality, we have K c() u 3 W, (B 5 ) c(). (3.6) Observing that ( h k u )η = div(η h ku) =, we get = h k π( h k u )η = (η h k π η h k π B )( h k u )η c() η h k π η h k π B (B \B ) (B 5 ) c() η h k π η h k π B (B ). Using emma.7, for each ψ (B ) with B ψ =, there is φ W, (B ) with divφ = ψ, and we have the following identity: < η h k π η h k π B, ψ >=< η h kπ, divφ > = < π, h k (ηφ) > +(π, h (( η) φ) > = (η h k T (E), E(φ)) + (T ij(e), h k (( x i η)φ j )) k ( h k (u iu j ), xi (ηφ j )) + (π, h k (( η) φ)). (3.7) We note that (π, h k (( η) φ)) = (π π B, h 5 k (( η) φ)), since h k (( η) φ) =. This implies that η h k π η h k π B (B ) c η h k (T (E(u)) (B ) ( ) +c() u W, (B 5 ) + u W, (B 5 ) + π π B 5. (B 5 ) Again due to emma.7, we have Therefore, π π B 5 (B 5 ) c()( u W, (B 5 ) + u W, (B 5 )). η h k π η h k π B (B ) c η h k (T (E(u)) (B ) ( ) +c() u W, (B 5 ) + u W, (B 5 ) c() X + c(), 9

10 and thus we obtain c() X + c(). (3.8) It remains to estimate J. By Hölder s, Soboelev and Korn s ineuality, we see that J c() u W, (B 5 ) (η h k u) ( ) c() u W, (B 5 ) E(η h k u) ( ) c() u W, (B 5 ) η h k E(u) ( ) + c() u 3 (B 5 ) ecalling that c() η h k E(u) ( ) + c(). (T (E(u h,k )) T (E(u)) : (E(u h,k ) E(u)) c E(u h,k ) E(u), we observe that η h k E(u) ( ) ch X and therefore, Combining all the above estimates (3.5)-(3.9), we have By Young s ineuality we have J c()h X + c(). (3.9) X c() X + c()h X + c(). X c()h + c(). From the above ineuality, if α <, we obtain B E(u h,k )(x) E(u)(x) h +α dh = This implies that E W Xh α dh c() +α, loc B h k E(u) h α dh h α + h α dh < c(). ( ). Via Sobolev embedding theorem, this again im- α plies that E loc ( α ) and by Korn s ineuality we conclude that u loc ( ). If α 3, then u 3 loc ( ). We note that such α always exists since > 3. This completes the proof. We are now ready to present the proof of (a) in Theorem.. Proof of Theorem. (a) As in emma 3., we can have eualities (3.) and (3.), where I, J, K and are defined. et h and we define X = η ( E(u) + E(u h,k ) ) h k E(u). We then have via Hölder s ineuality and (3.4) I c η h k (T (E(u)) ( ) h k u (B \B ) c X u (B 5 \B ).

11 By emma 3., we have u 3 loc ( ) and h k u converges strongly to x k u in 3 (B ), k =,. Hence, as h, we observe that J = η ( h k u i)( xi u h,k j )( h k u j) η ( xk u i )( xi u j )( xk u j ). Since i,j,k= ( x k u i )( xi u j )( xk u j ) =, we have J = ( h k u i)( xi u h,k j )(η h k u j) = o,h (), where v = o,h () means that for each ɛ > there is h = h () such that v < ɛ for all h h (). Using Sobolev ineuality and Hölder s ineuality, we get K c u (B ) h k u B \B c 6 ( u (B ) + u (B )) B 5 \B u eminding that u C( ) because u W, ( ) with >, we note by Poincaré s ineuality and by emma.9 that u (B ) c u (B ) + c u B + c u (B ) + c c.. Hence, we have K c 6 B 5 \B u. Using ( h k u )η = div(η h ku) =, we get = h k π( h k u )η = (η h k π η h k π B \B )( h k u )η c η h k π η h k π B \B (B \B ) u (B 5 \B ). Here we recall the identity (3.7) in the proof of emma 3.. Similarly as in emma 3., we estimate η h k π η h k π B \B (B \B ) c η h k (T (E(u)) (B \B ) + c h k (u iu j ) (B \B ) + c T (E) (B \B ) + c π π B \B (B \B ).

12 By emma.7, we see that π π B \B (B \B ) c T (E) (B \B ) + c u iu j (B \B ). Therefore, we obtain η h k π η h k π B \B (B \B ) c η h k (T (E(u)) (B \B ) +c η h k (u iu j ) (B \B ) + c T (E) (B \B ) + c u iu j (B \B ) c X u 6 (B 5 B )+c3 u (B 5 B ) + c 6 u (B 5 B )+c3, where we used the following estimates: η h k (T (E(u)) c X u (B \B ) (B 5 B ), η h k (u iu j ) (B \B ) c u (B \B ) h k u (B \B ) c 3 6 ( u (B )+ u (B )) u (B 5 B ) c 3 6 u (B 5 B ), and T (E) (B \B ) c u (B 5 B ), u i u j (B \B ) c4 6 ( u (B ) + u (B )) c 4 6. Combining all the above ineuality, we have X c X u 6 (B 5 \B ) +c u (B 5 \B )+ c u (B 5 \B ) +o (). Young s ineuality leads to X c 6 u (B 5 \B ) + c u (B 5 \B ) + o (). Sending h and then, we see that lim sup lim sup ( E(u) + E(u h,k ) ) h k E(u) =. h B Since we have h k ( E(u) E(u)) c( E(u) + E(u h,k ) ) h k E(u), lim sup ower semi-continuity implies that lim sup h k ( E(u) E(u)) =. h B ( E(u) E(u)) =. Therefore, we conclude that E(u) E(u) is a constant. Since u <, it is immediate that E(u) =, and thus u is a constant. This completes the proof.

13 3. Shear thinning flows : < < In this subsection, we consider the shear thinning flow. We first recall that in case that 3 < < and µ =, weak solutions are indeed in the class W, loc ( ) (see []). We remark that we can also have existence of second derivatives of any weak solution in the class W, loc ( ) in case < < and µ >. It may be known to experts but, for clarity, we present its proof. emma 3. If u be a weak solution of (.)-(.) for the case that < < and µ >, then u W, loc ( ). Proof. As we did in emma 3., we have eualities (3.) and (3.), where I, J, K and are defined. et h and we set X = h k T ij(e(u)) h k E ij(u)η. Due to Hölder s ineuality, we have I c()( u W, (B 5 ) + u W, (B 5 )) h k u (B 5 ) +c()( u W, (B 5 ) + u W, (B 5 )) c() h k u (B 5 ) + c(), K c() u W, (B 5 ) η h k u (B 5 ) c() η h k u (B 5 ). Next we estimate. Using that ( h k u )η = div(η h ku) =, = (π π B ) h k ( h k u )η c() π π B (B ) η h k u ( ) + c() π π B (B ) u W, (B 5 ) c() π π B (B ) η h k u ( ) + c() π π B (B ). ecalling emma.7, we obtain π π B (B 5 ) c() u W, (B ) +c() u W, (B )+c() u W, (B ) c(). Therefore, c() η h k u ( ) + c(). Finally, it remains to estimate J. By Hölder s ineuality and Soboelev ineuality J c() u W, (B 5 ) η h k u ( ) + c() u 3 W, (B 5 ) Due to Korn s ineuality, we see that c() η h k u ( ) + c(). η h k u ( ) c η h k E(u) ( ) + c() u (B 5 ) c X + c(). 3

14 Combining all the above estimates, we have X c() X + c(), which implies that via Young s ineuality X c(). This completes the proof. We first start with the non-degenerate case, namely µ > and µ. Proof of Theorem. (b) Fix z. et η C ( ) be a standard cut-off function such that η = if x z and η = if x z. We note by emma 3. that u W, loc ( ), and taking test function φ = (η u) and integrating it by parts, we have η T : E = T ij ( ( xk η)( xk E ij ) + xi ([( η) ( u)] j ) + ( xi η) u j ) + u i (( xi u j )[( η) ( u)] j + ) ( x i η) u j +( xk η)( xi u j )( xk u j ). (3.) From the above identity, we obtain ( µ E(u) + (µ + E(u) ) E ) B (z) c ( µ u + u ) ( B (z)\b (z) u + u ) + c u u := I + J. B (z)\b (z) For convenience, we set X() = ) B (z) (µ E(u) + (µ + E(u) ) E. Via Hölder s ineuality, we estimate I as follows: I c µ u + c ( ( B (z)\b (z) µ u u B (z)\b (z) B(z) + c u + c ( u B (z)\b (z) B (z)\b (z) We note that ( B (z) and by Korn s ineuality, u c B (z) Therefore, we have B 4 (z) u I cµ ( B u (z)\b (z) ) c ( u B (z) ( u B(z). E(u) + c u c E(u) + c B 4 (z) B 4 (z). + cµ ( u X(4) B (z)\b (z) 4.

15 + c ( B u ) + c ( u ) X(4). (3.) (z)\b (z) B (z)\b (z) et A := B (z) \B (z) B (z) \B (z) u. We then have J c u A u + c A u = J + J. B (z)\b (z) B (z) \B (z) We remind that u C( ), since u W, loc ( ). Noting that A C ln in Corollary.9, we obtain ln J c u. (3.) B (z)\b (z) On the other hand, via Hölder s and Poincare s ineuality, J is estimated as follows: J c ( u A ( u 4 B (z)\b (z) ( B (z)\b (z) B (z)\b (z) u ( u 4. B (z) Using emma.3 and emma.4, we observe that ( u 4 c u (B 4 (z)) u (B 4 (z)) + c u (B 4 (z)) B (z)\b (z) c u (B 4 (z)) E (B 4 (z)) + c u (B 4 (z)) Summing up above estimates, we obtain c c X(4) + µ. ( ( J c u ) X(4) +. (3.3) B (z)\b (z) µ Combining the ineualities (3.)-(3.3), we obtain X() cµ ( B u (z)\b (z) + c ( B (z)\b (z) ln +c B (z)\b (z) By Young s ineuality, we have + cµ ( u X(4) B (z)\b (z) u ) + c ( u B (z)\b (z) u + ) X(4) c ( u X(4). µ B (z)\b (z) X() ɛx(4) + c(ɛ)µ ( B u + c(ɛ)µ (z)\b (z) 5 B (z)\b (z) u

16 + c(ɛ) ( B (z)\b (z) ln +c(ɛ) u ) + c 4 4( u ) ( ) B (z)\b (z) B (z)\b (z) u + c(ɛ) µ Applying emma.6 to (3.4), we obtain for all > B (z)\b (z) X() cµ ( B u + cµ u (z) B + c ( B (z) ln +c ) u + c 4 4( ) ( ) u B (z) B (z) u. (3.4) u + c u. (3.5) µ B (z) Passing to to the both sides of (3.5), we conclude that X := µ E(u) + (µ + E ) E = lim X() is bounded. Applying this result to (3.4), we conclude that X ɛx, which implies that X =. Therefore, it is immediate that E(u) is a constant and then from the hypothesis that u ( ), we have E(u) =. We then conclude that u is a constant. This completes the proof. astly, we present the proof of the degenerate case (c). Proof of Theorem. (c) We take test function φ by φ = (η u). As in the proof of Theorem. (b), we have (3.), which implies that E(u) E c u ( u + u ) B (z) + c B (z)\b (z) We note that via Hölder s ineuality ( E (B 4 (z)) B 4 (z) B (z)\b (z) u u := I + J. (3.6) ( ) E E u B4(z) ( c E E B 4 (z) This implies, due to Korn s ineuality, that. u (B (z)) c E (B 4 (z)) + c u (B (z)) c ( E E B 4 (z) 6 + c. (3.7)

17 Using Hölder s ineuality and above estimate, we have I c u (B (z)\b (z)) u (B (z)) + c u (B (z)\b (z)). c ( u (B (z)\b (z)) E E B 4 (z) + c u (B (z)\b (z)). (3.8) Next, we estimate J. Using Sobolev ineuality and (3.7), we observe first that u (B (z)) c u (B (z)) + c u (B (z)) ( c E E + c B 4 (z). We remind that u C( ) and set A = B B u. By Corollary.9, we have A c. Using Sobolev ineuality, we note that u (B (z)) c u (B (z)) + c u (B (z)) c u (B (z)) + c u A (B (z)) + c A + c u (B (z)) + c + c + c +. Combining above estimates and Hölder s ineuality, we obtain c J c u (B (z)) ( ) + + u 5 6 u 5 6 (B (z)\b (z)) 3( ) (B (z)\b (z)) u ( B 4 (z) (B (z)) E E + ) 3( ) c( ( ) 3( ) ) u (B (z)\b (z)) E E B 4 (z) +c( Combine the ineualities (3.8)-(3.9), we have B (z) 5 6 ) u (B (z)\b (z)). (3.9) E E c ( u (B (z)\b (z)) E E B 4(z) +c( ( ) 3( ) ) u (B (z)\b (z)) E E B 4 (z) 7

18 + c 4 u (B (z)\b (z)) + c( ) u (B (z)\b (z)) (3.) Then, applying Young s ineuality to (3.), we obtain E E ɛ E E B (z) B 4 (z) + c(ɛ) u ( ) (B (z)\b (z)) + c u (B (z)\b (z)) +c ( 4( ) 5 6 +c( 4 Applying emma.6, we have E E +c( 4( ) 5 6 B (z) ) u (B (z)\b (z)) ) u 4 (B (z)) + c( 5 6 ) u (B (z)\b (z)) (3.) c u ( ) (B (z)) + c u (B (z)) ) u (B (z)). (3.) Passing to for the both sides of (3.), we obtain that E E =. As before, we can see that u is a constant. This completes the proof. Acknowledgments Bum Ja Jin s work is supported by NF -57 and Kyungkeun Kang s work was supported by NF eferences [] M. Bildhauer, M. Fuchs and G. Zhang, iouville-type theorems for steady flows of degenerate power law fluids in the plane, preprint, arxiv:.48 [] M. E. Bogovskiı, Solution of the first boundary value problem for an euation of continuity of an incompressible medium, Dokl. Akad. Nauk SSS 48 (979) no. 5, [3] G. Bohme, Non-Newtonian fluid mechanics, North-Holland Series in Applied Mathematics and Mechanics, Amsterdam, 987. [4] J. Frehse, J. Málek and M. Steinhauer, An existence result for fluids with shear dependent viscosity-steady flows, Nonlinear Anal., 3 (997), [5] M. Fuchs, iouville Theorems for Stationary Flows of Shear Thickening Fluids in the Plane, J. Math. Fluid Mech. 4 (), no. 3,

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20 [9] J. Málek, J. Nečas, M. okyta and M. uzička, Weak and measure-valued solutions to Evolutionary PDEs, Chapman & Hall, ondon, 996. [] J. Naumann and J. Wolf, Interior differentiability of weak solutions to the euations of stationary motion of a class of a non-newtonian fluid, J. Math. Fluid. Mech., 7 (5), [] M. užička, A note on steady flow of fluids with shear dependent viscosity, Nonlinear Anal., 3 (997), [] H. Triebel, Theory of function spaces II, Birkhauser Verlag, Basel, 99. [3] S. Whitaker, Introduction to Fluid Mechanics, Krieger, 986. [4] J. Wolf, Interior C,α -regularity of weak solutions to the euations of stationary motions of certain non-newtonian fluids in two dimension, Bollettino U.M.I. (8)-B (7), [5] G. Zhang, iouville theorems for stationary flows of shear thickening fluids in D, preprint, arxiv: Bum Ja Jin Department of Mathematics Mokpo National University Mokpo, epublic of Korea bumjajin@mokpo.ac.kr Kyungkeun Kang Department of Mathematics Yonsei University Seoul, epublic of Korea kkang@yonsei.ac.kr