CACCIOPPOLI TYPE INEQUALITY FOR NON-NEWTONIAN STOKES SYSTEM AND A LOCAL ENERGY INEQUALITY OF NON-NEWTONIAN NAVIER-STOKES EQUATIONS WITHOUT PRESSURE

. We prove a Caccioppoli type inequality for the solution of a parabolic system related to the nonlinear Stokes problem. Using the method of Caccioppoli type inequality, we also establish the existence of weak solutions satisfying a local energy inequality without pressure for the non-Newtonian Navier-Stokes equations.


Introduction.
Let Ω be a domain in R 3 and T > 0. We consider a nonstationary and nonlinear variant of the Stokes system in Q T = Ω × (0, T ): where u is the velocity field of the fluid, π is the pressure of the fluid, f is the external force, and S(D(u)) = (S ij (D(u))) i,j=1,··· ,n is the extra stress tensor given as S ij (D(u)) = ν(|D(u)|)D ij (u), D ij (u) = 1 2 Here viscosity ν(|D(u)|) is a function of |D(u)|, given as ν(s) = µ 0 + (µ 1 + s) q−2 .
B x0,r = {x ∈ R n : |x − x 0 | < r}, S x0,r = {x ∈ R n : |x − x 0 | = r}, Q z0,r = B x0,r × (t 0 − r 2 , t 0 ). For simplicity, we assume that f = 0 and z 0 = (0, 0) via translation and we denote B r = B x0,r , S r = S x0,r and Q r = Q z0,r , unless there is any confusion to be expected. If we consider a similar situation for the heat equation, ∂ t v − ∆v = 0, it is well-known that the following Caccippoli inequality holds: However, due to non-local effect of the pressure, Caccippoli inequality of the system (1) doesn't seem to be obvious. Nevertheless, introducing the notion of local pressure and using a suitable decomposition of the pressure in [27], the Caccippoli type inequality of the Navier-Stokes equations was obtained in the interior (see also [28]). The first author in [17] proved independently the Caccippoli inequality for the Stokes system, i.e. for the case ν(s) = µ 0 , by testing a suitable divergence free vector field. We remak that the methods of proofs in [17] and [27] are different, although the same Caccippoli inequality can be derived. Our main motivation is to establish a Caccippoli type inequality of the nonlinear Stokes system (1) as an extension of the result in [17] and also show existence of weak solutions of the non-Newtonian Navier-Stokes equations satisfying local energy inequality without pressure.
For example, it was shown in [13,14] that there is unique solution u of the system (1) with ν(s) = (1 + s 2 ) q− 2 2 such that for any bounded subdomain Ω Ω ⊂ R n and for any δ > 0 In section 2, we present the proof of the existence of unique strong solution in Definition 1.1 including the case of degenerate power-law type as well as q > 1 (see Proposition 1). Such results may be known to experts but we cannot find it in the literature.
We are now ready to state our main results. The first result is Caccioppoli type inequality and the main point is that pressure does not appear on the righthand side.
Suppose that u is a strong solution of non-Newtonian Stokes system in Definition 1.1. Then, for any r > 0 with 2r < ρ The second result is the estimates of the higher regularity regarding spatial derivative of the second order. We treat separately the shear thinning case, i.e. 1 < q < 2, and shear thickening case, i.e. q > 2, since structures of two cases are dissimilar, which cause quite different methods of proofs. Next theorem deals with shear thinning case (1 < q < 2).
Suppose that u is a strong solution of non-Newtonian Stokes system in Definition 1.1. Then,the following estimates hold for any r > 0 with 2r < ρ: For shear-thickening case (2 < q < ∞), we obtain the following: Suppose that u is a strong solution of non-Newtonian Stokes system in Definition 1.1. Then,the following estimates hold for any r > 0 with 2r < ρ: As an application of Caccioppoli type inequality in Theorem 1.2, we can construct a weak solution of non-Newtonian Navier-Stokes equations satisfying a local energy inequality without pressure. To be more precise, we consider the non-Newtonian Navier-Stokes equations with no slip boundary condition, i.e. u = 0 on ∂Ω × [0, T ] and divergence free initial data u(·, 0) = u 0 in L 2 (Ω). Next, we recall the notion of weak solution of the non-Newtonian Navier-Stokes equations (13).
For the spatially periodic domains, the strong solution had been obtained globally in time for q ≥ 11 5 if n = 3, and for q > 1 if n = 2 in [6,22]; the local-in-time strong solution for q > 5 3 in [22], for 2 > q > 7 5 in [11], and for 2 > q > 1 in [7] when n = 3. When the case µ 1 > 0 had been considered, regularity for the Dirichlet problem has also been well studied: the global-in-time strong solution was obtained for q ≥ 9 4 if n = 3 and q ≥ 2 if n = 2 in [24], and the regularity result was extended to the case q ≥ 11 5 in [5,9] when n = 3; the local-in-time strong solution was obtained for q > 1 in [1,8]. The uniqueness of weak solution is known for q > 5 2 in [18] and for q > 11 5 in [9]. Regularity of weak solution for the degenerate case is considered in [2,25].
As mentioned earlier, we can show the existence of weak solutions satisfying a certain type of local energy inequality, where pressure term does not appear. To be more precise, our last main result reads as follows: Theorem 1.6. Let q ≥ 9/5 and N (x) be the fundamental solution of −∆ in R 3 . Then, there exists a weak solution u of the non-Newtonian Navier-Stokes equations (13) satisfying the following local energy inequality: For almost every τ ∈ (0, T ) and Remark 3. In Theorem 1.6, if µ 0 > 0, then the assumption q ≥ 9/5 is not required. The condition u ∈ L ∞ ((0, T ) : L 2 (Ω)) ∩ L q (0, T ; W 1,q 0 (Ω)) in Definition 1.5 implies u ∈ L 5q 3 (Ω × (0, T )). From the right hand side of (15), u ∈ L 3 loc (Q T ) is necessary, and this holds for q ≥ 9 5 .
Next corollary is direct consequence of the local energy inequality Corollary 1. The weak solution u constructed in Theorem 1.6 satisfies where v is given in (16).

Remark 4.
We recall that the non-Newtonian Navier-Stokes equations (13) satisfy the following scaling invariance: We also note that v in (16) hold . Therefore, the local energy inequality (17) holds for u λ , p λ and v λ with replacement of Q r bỹ Q r = B r × (−r 2 3−q , 0). From the above scaling invariance, since r 5q−11 3−q Q r |∇u| q dz is scaling invariant, we conjecture that in case that 2 < q < 11 5 , the size of a possible singular set is of 11−5q 3−q -dimensional Hausdorff measure zero, and we leave it as an open question.
Remark 5. For the magnetohydrodynamic equations, the local energy estimate similar to (15) was recently constructed in [10] by using the Caccioppoli type inequality of the Stokes system proved in [17]. One of our motivation is to extend such estimate to the non-Newtonian fluid flow. The inequality (15) contains the non-local effect of pressure, which appears as a different form v defined in (16). The advantage of the inequality (15) is that the non-local effect does not present itself on the righthand side but it is included on the left hand side, which yields the Caccioppoli type inequality (17). The price to pay is, however, lack of control of the local L ∞ t L 2 x −norm in the inequality, although it can be estimated in terms of the right side together with itself in a bigger parabolic cylinder (see Corollary 2). On the other hand, as mentioned earlier, another type of local energy inequality was established in [27] and [28], which contains a part of pressure on the right hand of the inequality (see [28,Definition 3.1]), but such pressure can be controlled in terms of velocity fields similarly as in right hand side of (15) and it leads to the Caccioppoli type inequality (see [28,Lemma 4.1]). This paper is organized as follows. In section 2 we recall some useful lemmas and existence of strong solution is shown. The section 3 and section 4 are devoted to providing the proofs of Theorem 1.2-Theorem 1.6.

2.
Preliminaries. In this section we introduce the notations and present preparatory results that are useful to our analysis. We start with the notations. Let Ω be a domain with smooth boundary in R 3 . For 1 ≤ q ≤ ∞, we denote by W k,q (Ω) the usual Sobolev spaces, namely The set of q−th power Lebesgue integrable functions on Ω is denoted by L q (Ω) and L q loc (Ω) indicates the set of locally q−th power Lebesgue integrable functions defined on Ω. 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 present a local version of Sobolev inequality and Korn's inequality.
Lemma 2.2 (Local Korn's inequality). If u ∈ W 1,p loc (R 3 ) with 1 < p < ∞ and x 0 ∈ R 3 , then there exists r 0 > 0 such that for any r, s with 0 < r < s < r 0 , We omit the details of the above lemmas since proofs are rather straightforward via the Sobolev embedding theorem or Korn's inequality for v = uφ, where φ ∈ C ∞ 0 (B s ) is a standard cut-off function satisfying φ = 1 in B r and |∇φ| ≤ C s−r . Next, we recall the following lemma proved in [15], which is parabolic analog to a result of elliptic cases (see e.g. [16]).

Lemma 2.3.
Let f, f 1 , · · · , f l be nonnegative functions in L 1 loc (Q ρ ) and α 1 , · · · , α l given nonnegative numbers. There exists 0 , depending on α 1 , · · · , α l , such that for any with ≤ 0 and for any 2r < ρ if for some constantc, then there exists a constant c > 0, independent of such that following inequality holds: Next, we prove an auxiliary lemma, which we will use often in proofs of Theorems. Let 2r < ρ and φ r = φ r (x) indicate a standard cut-off function satisfying We denote by N (x) = 1 4π|x| the fundamental solution of −∆ in R 3 and for a given Next lemma shows some estimates of Sobolev norm for v.
and v is given as in (21).
Remark 6. We remark that B ρ in the left hand side of (22) and (23) could be replaced by R 3 , in case that q > 3 and q > 3/2, respectively. Indeed, if q > 3/2, due to Calderon-Zygmund estimates and Sobolev embedding, we have Using the above estimate, when q > 3, it is immediate via Sobolev embedding that where we used that 3q/(3 + q) > 3/2.
By standard argument, that is, by multiplying (25) with u and D t u , respectively, and by using the integration by parts, we then have d dt Due to Biot-Savart law and Korn's inequality, the above estimates imply that Via Bogovski's formula, this again implies that π ∈ L min{2, q q−1 } ((0, T ) × (Ω )) uniformly in .
Moreover, using Aubin-Lions theorem u k converges strongly toũ in L r loc , r < 5q 3 , and using Minty theorem D(u k ) converges strongly to D(ũ) in L q loc (see [14] and [26] for details). This again implies thatũ solves the equation (1) in the sense of distributions. Since uniqueness of the strong solution is straightforward due to the monotonicity property of S(D) : D, we skip its details.
• (The case 1 < q < 2) We note that Via integration by parts, the first three integrands of the right hand side in (29) becomes Hence, applying Hölder's inequalities and Young's inequalities, the right hand side of (29) is dominated by Taking δ small enough, (29) reduces to the inequality which yields by integrating in time that From Biot-Sawart law and Korn's inequality, (33) implies uniformly in that As in the case q > 2, we can find a subsequence { k } ∞ k=1 so that as k → ∞, k → 0 and {u k } ∞ k=1 which converges to someũ and satisfies the regularity assumption in Definition 1.1.
Moreover, sing Aubin-Lions theorem u k converges strongly toũ in L r loc , r < 5q 3 , and using Minty theorem D(u k ) converges strongly to D(ũ) in L q loc (see [14] and [26] for details). Therefore, it is straightforward thatũ solves the equation (1) in the sense of distributions. Uniqueness of the strong solution satisfying Definition 1.1 is rather standard.
3. Non-Newtonian Stokes system. In this section, we present the proofs of Theorem 1.2, Theorem 1.3 and Theorem 1.4. (see [17] for Newtonian case).
3.1. Proof of Theorem 1.2. This subsection is devoted to giving the proof of Theorem 1.2.
Proof of Theorem 1.2. According to Lemma 2.4, ∂ t u ∈ L q q−1 (0, T ; W −1, q q−1 (Ω)) and u ∈ L ∞ (0, T ; L 2 (Ω))∩L q (0, T ; W 1,q (Ω)) imply that We denote B r = B r,x0 and Q r = Q r (z 0 ). Let φ r be given in (20) and ψ r (t) a smooth cut off function in time variable such that We first show that τ −4r 2 a.e. τ ∈ (0, −r 2 ). Indeed, we note that Integrating the above with ψ r in time variable, we obtain (36). Next we consider the second term in (35). We note first that (37) Therefore, we obtain Summing up above estimates, we obtain that (38) It is direct via Hölder's and Young's inequalities that By Hölder's inequality, we estimate I 2 as follows: On the other hand, as in Lemma 2.4, we can observe that Hence, the terms I 3 and I 4 are controlled as follows: Via Lemma 2.4, we also have Summarizing all estimates above, we obtain sup −r 2 <τ <0 R 3 Due to Lemma 2.3, we obtain sup −r 2 <τ <0 R 3 Using Korn's inequality, we again obtain We complete the proof.
As mentioned earlier in Introduction, the Caccioppoli type inequality (10) doesn't control the local L ∞ t L 2 x −norm of u by right hand side of (10). Next corollary, however, shows that L ∞ t L 2 x −norm of u somehow can be estimated in terms of righthand side of (10) as well as itself in a bigger cylinder.
Corollary 2. Let 1 < q < ∞ and u be a strong solution of non-Newtonian Stokes system in Definition 1.1. Then, for any r > 0 with 2r < ρ sup −r 2 <τ <0 Br Proof. Let v be given in (21). We observe first that u − ∇ × v is harmonic in B r . By the mean value property of the harmonic functions, we have By triangle inequality and the above inequality, we have From (44) in the proof of Theorem 1.2, we also have the inequality Combining (47) and (48), we obtain (45). This completes the proof.

3.2.
Proofs of Theorem 1.3 and Theorem 1.4. This subsection is devoted to proving Theorem 1.3 and Theorem 1.4. Before providing the proof, we begin to recognize that u is the limit of the solution of the approximate system, which is considered in the proof of Proposition 1.
Taking inner product by ∇ × (φ 2 ∇ × u k ) to the equation (1) and integrating over Ω, due to the orthogonality of ∇ and ∇×, we have the identity that We note that Hence, due to (55) with a temporal test function ψ, we obtain • (Case 1 < q < 2) We observe that Integrating by parts, the first term of the right hand side of (56) becomes Hence (56) reduces to the inequality Passing to the limit, (59) reduces to the inequality Here we use (50), (51), (52), (54) and lower semi-continuity on the left hand side. First, we note that On the other hand, Hölder's inequality yields Choosing δ > 0 small enough and summing up all the estimates, we obtain This completes the proof of the case 1 < q < 2.
As in the case 1 < q < 2, we use (50), (51), (52), (53) and lower semi-continuity on the left hand side. We observe that Via Hölder's inequality, we get Combining all the estimates, we have the inequality Choosing δ > 0 in (65) small enough and summing up above estimates, we obtain This completes the proof of the case q > 2, and thus we deduce Theorem 1.3 and Theorem 1.4.
Following similarly the proof in Theorem 1.2 for the approximating system (67) and taking limit procedures, we establish the existence of weak solution satisfying (15). Indeed, let φ(x) ∈ C ∞ 0 (Ω), ψ(t) ∈ C ∞ 0 ([0, T )) be cut-off functions as in the proof of Theorem 1.2 and we set an auxiliary vector field v m as v m (x, t) := R 3 N (x − y) (φ(y)∇ y × u m (y, t)) dy.
Testing ψ∇ × (φv m ) to (67), for any 0 < τ < T we have the identity that 0 = As in the case of Stokes system, the first term and the third term in (72) can be computed as follows: