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On the Existence and Regularity of Solutions for Degenerate Power-Law Fluids Josef Málek, Dalibor Pražák, Mark Steinhauer

no. 333

Diese Arbeit erscheint in: Differential and Integral Equations

Sie ist mit Unterstützung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Mai 2007

ON THE EXISTENCE AND REGULARITY OF SOLUTIONS FOR DEGENERATE POWER-LAW FLUIDS Abstract. We study time-dependent flows of incompressible degenerate power-law fluids characterized by the power-law index p − 2 with p > 2. In this case, the generalized viscosity vanishes as (the modulus of) the shear rate tends to zero. We prove global-in-time existence of a weak solution if p > max{ 3d−4 d , 2}. 3d+2 This improves the range p > d+2 for which the existence result was obtained by O.A. Ladyzhenskaya and J.L. Lions, via standard monotone operator theory. Since we apply higher differentiability techniques, certain regularity results are also established. The key step of the proof is an estimate of the velocity gradient in a suitable Nikol0 ski˘ı space. To make the presentation of the method transparent, we restrict ourselves to the spatially periodic problem. A possible extension of the approach to no-slip boundary conditions is however discussed, as well.

´lek1 J. Ma Charles University in Prague Faculty of Mathematics and Physics Mathematical Institute Sokolovsk´ a 83, 186 75 Prague 8 Czech Republic

´ k1 D. Praˇ za Charles University in Prague Faculty of Mathematics and Physics Department of Mathematical Analysis Sokolovsk´ a 83, 186 75 Prague 8 Czech Republic

M. Steinhauer Rheinische Friedrich-Wilhelm Universit¨ at Bonn Mathematisches Seminar d. Landwirtsch. Fakult¨ at Nussallee 15, D-53115 Bonn Germany

2000 Mathematics Subject Classification. 76A05, 35K65. Key words and phrases. Non-Newtonian fluid, existence, regularity, weak solution, degenerate parabolic problem, Nikol0 ski˘ı space. 1 This work is a part of the research project MSM 0021620839 financed by MSMT. The support of the Czech Science foundation, the project 201/03/0934, and of SFB611 at the University of Bonn is also acknowledged. 1

2

´ ˇ AK ´ AND M. STEINHAUER J. MALEK, D. PRAZ

1. Introduction In this paper, we are concerned with the question of global existence of a weak solution to the system ∂t v + div(vv ⊗ v ) − div T (Dvv ) + ∇π = f , div v = 0, (1) v |t=0 = v 0 , where (2)

T (Dvv ) = ν|Dvv |p−2 Dvv ,

ν > 0.

The system comes from non-Newtonian fluid mechanics: v : Ω × (0, T ) → Rd and π : Ω × (0, T ) → R are the unknown velocity and pressure, while the external forces f : Ω × (0, T ) → Rd and the initial velocity v 0 : Ω → Rd are given. In the present paper we only consider the spatially periodic setting, i.e., we eliminate the presence of the boundary by setting Ω = (0, L)d and assuming that all the functions are L-periodic in every direction and have zero mean value. Possible extensions to other boundary conditions are discussed in section 4. In (2), Dvv = 21 (∇vv + ∇vv T ) denotes the symmetric part of the velocity gradient. The system (1) describes unsteady flow of an incompressible fluid whose rheological properties are encoded in (2). This constitutive equations has the structure (3)

T (Dvv ) = a(|Dvv |)Dvv

with

a(s) = νsp−2 ,

and thus, for p 6= 2, the model (2) falls into the class of fluids with shear-dependent viscosity. Note that for p = 2, we obtain the NavierStokes equations. Fluids given through (2) are called power-law fluids, the exponent p−2 being then the power-law index. If p > 2 the fluid has the ability to shear thicken (such fluids are then called shear thickening or dilatant fluids). If p < 2, the fluid has the ability to shear thin (these fluids are called shear thinning or pseudoplastic fluids). Note that for p > 2, the generalized viscosity ν|Dvv |p−2 vanishes (degenerates) as |Dvv | → 0, while for p < 2 the viscosity tends to +∞ (i.e., becomes singular.). A more detailed exposition of incompressible fluids with shear dependent viscosity can be found in [12], for example. In this article, we deal with power-law fluids given through (2) with positive power-law index, i. e. p > 2. We clarify reasons for this choice in what follows. There are quite a number of papers devoted to questions of existence, uniqueness, regularity and further properties (as large time behaviour) of solutions to the system of equations (1) with (3), where the constitutive equation (2) is a special case. We refer to the survey article [13] where an overview and plenty of references are provided. Yet, even the question of (global) existence of solutions is not completely solved

ON THE EXISTENCE AND REGULARITY

3

(meaning for all p 6= 2), though a variety of techniques have been developed, depending on whether p < 2 or p > 2, see for example [4] for an overview of available techniques which we also briefly sketch below. The usual strategy of the existence theory is to find v n , solutions of a suitably chosen approximating problem (a Galerkin approximation, for example). With help of uniform estimates in proper spaces one then obtains v , a solution to (1), as a “weak” limit of a suitable chosen subsequence to v n . The central problem is the passage to the limit in the nonlinear terms. From standard energy estimates one concludes (regardless the dimension d) that

(4)

v n is bounded in ∂tv n is bounded in

1,p Lp (0, T ; Wdiv ) ∩ L∞ (0, T ; L2div ), 3,2 0 L2 (0, T ; [Wdiv ] ).

This implies weak convergence of ∇vv n and strong convergence of v n , which is enough to handle the quadratic convective term div(vv ⊗ v ) 2d provided that p > d+2 , but would not suffice to deal with the stress tensor T (Dvv ). The fact that T forms a monotone operator can help. The first method (Approach 1) considered in Ladyzhenskaya [6, 7] and Lions [8], where all the details can be found, is based on Minty’s trick implying that T (Dvv n ) → T (Dvv ). This, however, requires that one can take as a test function in (1) the difference v n − v , having only those regularity properties stated in (4). Here, the convective term gives a lower bound which, unfortunately, is strictly greater than 2 if d ≥ 3. p ≥ 3d+2 d+2 One can overcome this difficulty by testing with a suitable truncation of v n −vv . In [5] this method (Approach 2) is presented and the existence theory is extended up to p > 2(d+1) . The disadvantage of this result d+2 is its unclear extension to homogeneous Dirichlet (no-slip) boundary conditions. The method also provides no regularity statements. In this paper we follow another method (Approach 3) based on higher differentiability techniques, introduced to non-degenerate systems with spatially period setting in [10] and [1], and is presented with all the details in [9]. Roughly speaking, the aim is to obtain estimates for v n in Lp˜(0, T ; W 1+σ,p ) with suitable σ > 0 and p˜ ∈ (1, p). By standard embedding one obtains then strong convergence of ∇vv n , which enables to identify the limit of T (Dvv n ). In order to obtain this higher-order derivative estimate, we simply test the equation by a second difference of v n , which thanks to the absence of the boundary simplifies to testing by −∆vv n . This approach, however, has to overcome two difficulties.

4


(1) For power-law fluids, taking the scalar product of T (Dvv ) with −∆vv leads to the term Z (5) Ip (vv ) := |Dvv |p−2 |∇Dvv |2 . It is not completely obvious how this quantity ensures the compactness of gradients if p > 2. On the contrary, the “singular” case p < 2 can be treated easily since Ip (vv ) ≥ ckD(∇vv )k2p , see [9] for details. This is the reason why we concentrate on the “degenerate” case p > 2 here. Note also that this difficulty would not occur in the case of non-degenerate stresses of the type T (Dvv ) = ν(1 + |Dvv |p−2 )Dvv . In such a case one directly comes to Ip (vv ) ≥ ck∇Dvv k22 . See again [9] for details. It is worth mentioning that this procedure can be, at least for p > 2, extended to the homogeneous Dirichlet boundary value problem, see [11]. (2) The second problem comes from the convective term. After taking the scalar product of the convective term with −∆vv one obtains k∇vv k33 which is not integrable if p < 3. The standard tricks here give the same lower bound p ≥ 3d+2 that follows from the monotone operd+2 ator theory. To overcome the first difficulty one observes that for p > 2 (5) estimates the norm of Dvv in a certain Nikol0 ski˘ı space. This gives the desired compact embedding. For a more general development of this idea treating the p-Laplacian using the Nikol0 ski˘ı spaces, see [3]. The second difficulty is treated using the technique from [9]. Roughly speaking, the equation can be integrated after dividing by (1+k∇vv k22 )λ with a suitable λ > 1. This, of course, weakens our estimate coming from (5). Still, this weakened estimate keeps enough information to conclude the compactness of gradients. To clearly formulate the novelty of this article we summarize the above discussion for the most interesting case d = 3. The theory developed by Ladyzhenskaya gives the existence to (1)both for the spatially periodic and the homogeneous (3) for p ≥ 11 5 Dirichlet problem. The long-time and large-data existence for p ∈ [2, 11 ) is treated in [5] in case of the spatially periodic problem (in fact 5 the case p ∈ ( 58 , 11 ) is treated therein). The extension of the result 5 from [5] to the homogeneous Dirichlet problem is open. In [9], the spatially periodic problem for p ∈ [2, 11 ) is analyzed via 5 a higher-differentiabilty approach, however, for non-degenerate operators only. It is worth mentioning that this approach has been already extended, again only for non-degenerate operators, to the case of the homogeneous Dirichlet problem, see [11] for details. In the here presented article, dealing with a degenerate elliptic operator for the spatially periodic problem, we establish an existence theory ) using Approch 3 and obtain new fractional estimates; for p ∈ (2, 11 5


5

if p ≥ 11 , we even strengthen the regularity results proved in [9] and 5 show that for all > 0 2

v ∈ Lp (0, T ; W 1+ p −,p (Ω)). Because of the result stated in [11] we think Approach 3 (higher differentiability) is more suitable for possible extension to Dirichelt boundary conditions than Approch 2 (strict monotonicity combined with truncation operators). The paper is organized as follows: in Section 2, we introduce some function spaces and establish several auxiliary lemmas. In Section 3 we state and prove our main result, Theorem 3.1. Concluding remarks, in particular those related to other boundary conditions, are presented in the last Section 4. 2. Preliminaries Set

∞

d

V = φ ∈ C (R ); φ(x + Lej ) = φ(x) ∀j,

Z

φ=0 Ω

Then Lp , W s,p are the closures of V with respect to the corresponding norms k·kp , k·ks,p, restricted (thanks to periodicity) to Ω = (0, L)d . We allow also for spaces W s,p with a noninteger s and the functions can be scalar or vector valued. In the latter case the subscript div as in Lpdiv indicates that the functions are free of divergence. It is worth recalling that for any > 0 W s+,p ,→,→ W s,p ,→ Lq , provided that 1q = 1p − ds and sp < d. Further, for p ∈ [1, ∞) and s = m + σ, where m ≥ 0 is an integer and σ ∈ (0, 1) we introduce the Nikol0 ski˘ı space N s,p as the subspace of Lp - functions for which the norm kukpN s,p = kukpp + |u|pN s,p Z X |∂ α u(x + h) − ∂ α u(x)|p p dx = kukp + sup |h|σp 0 0 the embeddings (see [15]) (6)

N s,p ,→ W s−,p ,→ N s−,p.

Remark that thanks to the zero mean condition, one can take the highest order derivative seminorm as an equivalent norm in each of the above spaces. We complete this section with several auxiliary lemmas. The first lemma is a key step in exploiting the estimates of Ip (vv ) defined in (5) in terms of the norm in a Nikol0 ski˘ı space, see [3, Eq (3.7)] for its discrete analogue.


6

Lemma 2.1. Let u ∈ W 1,1 be a scalar or vector valued function and p > 2. Let Z Ip (u) = |u|p−2 |∇u|2 < ∞ . Ω

Then u ∈ N

2 ,p p

and

kukp

2 ,p

Np

≤ c Ip (u)

with c depending only on p and Ω. Proof. For a ≥ 1, we start with the inequality (7)

|u − v|a ≤ c1 ||u|a−1 u − |v|a−1 v|

for all u, v ∈ Rd

which holds with a suitable c1 = c1 (a). Inequality (7) follows from

|u − v|a+1 ≤ c1 |u|a−1 u − |v|a−1 v, u − v ≤ c1 ||u|a−1 u − |v|a−1 v| |u − v| ,

whereas the first inequality is proven in [2, chapter I, Lemma 4.4, p.13]. Taking δ > 0 fixed and x ∈ Ω, and considering h ∈ Rd such that 0 < |h| < δ we obtain using inequality (7) p

p

p

|u(x + h) − u(x)| 2 ≤ c1 ||u(x + h)| 2 −1 u(x + h) − |u(x)| 2 −1 u(x)| Z 1 ∂ n o p = c1 |u(x + sh)| 2 −1 u(x + sh) dt ∂s 0 Z 1 p ≤ c2 |h| |u(x + sh)| 2 −1 |∇u(x + sh)| ds . 0

By Hölder’s inequality we conclude that Z 1 p 2 2 |u(x + h) − u(x)| ≤ c2 |h| |u(x + sh)|p−2 |∇u(x + sh)|2 ds . 0

Integrating the result over x ∈ Ω, and applying then Fubini’s Theorem we come to the inequality Z |u(x + h) − u(x)|p ≤ c3 Ip (u) . 2 Ω |h| p ·p 2

Now the left-hand side is the seminorm of N p ,p , which is enough to finish the proof thanks to the spatially periodic setting. The well-known Aubin-Lions lemma about the compact embedding of Bochner spaces will be also needed. Lemma 2.2. Let Y ,→,→ X ,→ Z be Banach spaces, let X be reflexive. Let p > 1, q ∈ [1, ∞]. Then u ∈ Lp (0, T ; Y ), ∂t u ∈ Lq (0, T ; Z) ,→,→ Lp (0, T ; X) Proof. See [16], for example.

We also need a generalized version of the well-known Korn-inequality.


7

Lemma 2.3. Let (vector-valued) v ∈ W s,p , p ∈ (1, ∞). Then for any s ∈ [0, 1] one has k∇vv ks,p ≤ c kDvv ks,p . where Dvv = 12 (∇vv + ∇vv T ) and c depends only on s, p and Ω. Proof. For s = 0 (the standard version) see Neˇcas [14] or for example [9] and the references therein. The case s = 1 is in fact elementary as every second derivative of v can be expressed in terms of first derivatives of Dvv : ∂ ∂ ∂ ∂ 2 vj = Dij (vv ) + Dkj (vv ) − Dik (vv ) i, j, k = 1, . . . , d. ∂xi ∂xk ∂xk ∂xi ∂xj The general case is then obtained by interpolation. Finally, the following lemma concerning the passage to the limit under the integral sign will be needed. Lemma 2.4. Let M ⊂ Rm be measurable and bounded. Let the sequence {f n }n∈N be uniformly bounded in Lq (M ) for some q > 1. Finally let f n → f a.e. in M for some f ∈ Lq (M ). Then Z Z n f → f. M

M

Proof. It is a straightforward consequence of Vitali’s Theorem.

3. Main theorem In this section we formulate and prove our main result. Theorem 3.1. Let p ≥ 2 and d ≥ 2. Assume that v 0 ∈ L2div and dp 0 f ∈ Lp (0, T ; W 1, dp−d+2 ). Then the following hold: (i) If 3d − 4 4 p> =3− , d d then there exist 1,p v ∈ Lp (0, T ; Wdiv ) ∩ L∞ (0, T ; L2div )

and

π∈L

(d+2)p 2d

(0, T ; L

(d+2)p 2d

)

being together a weak solution to (1) with v (0) = v 0 . (ii) If moreover p is such that % :=

p2 (dp − 3d + 4) ≥1 p2 d − 3dp + 12

then v ∈ L% (0, T ; W 1+σ,p) with σ > 0 fulfilling the relation (16) below. (iii) Finally, if 3d + 2 p≥ d+2 ∞ 1,2 p then v ∈ L (η, T ; W ) ∩ L (η, T ; N 1+s,p) for any s ∈ (0, 2p ) and any 1,2 η ∈ (0, T ). One can take η = 0 if v 0 ∈ Wdiv .


8

Proof. (of Theorem 3.1) Assume v n solve the Galerkin system related to (1) as described in [9, chapter 5, page 207]. Note that the functions v n are based on the eigenfunctions of the Stokes operator, which justifies to use −∆vv n as test function. Let us first test by v n . Since the pressure and the convective term cancel out, one obtains Z 1d n 2 n p kvv k2 + νkDvv kp ≤ f · vn , 2 dt Ω which by Lemma 2.3 and some standard estimates gives Z T Z T n 2 n p (8) sup kvv (t)k2 + kDvv kp + k∇vv n kpp ≤ K . t∈(0,T )

0

0

This together with the equation implies Z T k∂tv n k2(W 3,2 )0 ≤ K . (9) 0

div

See [9, p. 207 ff.] for details. Here and in what follows K stands for a generic constant that can depend on kvv n (0)k22 and a suitable norm of f , but which is independent of n. As a consequence of the estimates (8) and (9), there exists v , belonging to the corresponding spaces, such that v n → v weakly in 1,p Lp (0, T ; Wdiv ) and by Lemma 2.2 also strongly in Lp (0, T ; Lpdiv ), for example. This is enough, as p > 2, to conclude that div(vv n ⊗ v n ) → div(vv ⊗ v ) at least in D 0 (Ω×(0, T )). A more difficult problem, however, is whether also T (Dvv n ) → T (Dvv )

(10)

in

D 0 (Ω × (0, T )) .

As already mentioned, this can proved using monotone operator theory (Minty’s trick) provided that p ≥ 3d+2 , see e.g. [7, 8] for details. d+2 Our first aim is to improve this lower bound. Consequently, we can restrict ourselves to the case p < 3d+2 = 2 + d−2 . Since T is of the form d+2 d+2 (2) we only consider p > 2 and thus we take d ≥ 3. Our intermediate goal is to obtain the estimate Z T (11) k∇vv n krσ,p ≤ K , with some r > 1, σ > 0. 0

Let us first observe that (11) together with (9) implies (10). Indeed, by 3,2 0 Lemma 2.2 with X = W 1,p , Y = W 1+σ,p , Z = (Wdiv ) , p = r and q = 2 n r one obtains that ∇vv → ∇vv (strongly) in L (0, T ; Lp ). In particular, one can assume Dvv n → Dvv a.e. in Ω × (0, T ). Let now φ ∈ D(Ω × (0, T )) be arbitrary. One has Z Z 0 n p0 T (Dvv ) · ∇φ| ≤ c |Dvv n |p |∇φ|p ≤ K |T Ω×(0,T )

Ω×(0,T )


9

independently of n. Hence by Lemma 2.4 with M = Ω × (0, T ), f n = T (Dvv n ) · ∇φ and q = p0 one sees that Z Z T (Dvv n ) · ∇φ → T (Dvv ) · ∇φ Ω×(0,T )

Ω×(0,T )

and (10) holds. To obtain (11) we incorporate −∆vv n as a test function in the relevant Galerkin system and perform the following operation with the particular terms separately. Note that we strongly rely on the fact that all functions are spatially periodic. (From now on, we drop the index n for simplicity.) We have Z T (Dvv )} : D(−∆vv ) Ω Z = |Dvv |p−2 Dvv : D(−∆vv ) ZΩ = ∇ |Dvv |p−2 Dvv : D(∇vv ) ZΩ Z p−2 2 = |Dvv | |D(∇vv )| + (p − 2) |Dvv |p−4 (Dvv : D(∇vv ))2 Ω

Ω

≥ Ip (Dvv ),

(p > 2),

where Ip (Dvv ) =

Z

|Dvv |p−2 |∇(Dvv )|2 . Ω

This term gives us two important estimates. On one hand, by Lemmas 2.1 and 2.3 and the embedding (6) one has Ip (Dvv ) ≥ ckDvv kp

2 ,p

Np

≥ ckDvv kps,p ≥ ck∇vv kps,p

where s ∈ (0, p2 ) is an arbitrary (but from now on a fixed) number. On the other hand, as p p ∂vv ∂ |Dvv |p/2 = |Dvv | 2 −2 Dij ( )Dij (vv ) , ∂xk 2 ∂xk

we have Ip (Dvv ) ≥ c

Z

Ω

∇|Dvv | p2 2 = ck∇|Dvv | 2p k2 . 2 p

Adding to both sides the term ckDvv kpp = ck|Dvv | 2 k22 and using then the 2d

embedding W 1,2 ,→ L d−2 (recall that d ≥ 3) and Lemma 2.3, we obtain ckDvv kpp + Ip (Dvv ) ≥ c k∇vv kpdp , d−2

which together with (5) imply Ip (Dvv ) ≥ c k∇vv kpdp − K . d−2


10

¿From the convective term one has Z Z Z ∂vi ∂ ∂vj ∂vi ∂vi ∇vv · ∇vv + − vj ∆vi = vj ∂xj ∂xj ∂xk ∂xj ∂xk Ω Ω Z Z Ω ∂vj ∂vi ∂vi 1 ∂ = vj |∇vv |2 + 2 Ω ∂xj ∂xk ∂xj ∂xk Z ZΩ ∂vj ∂vi ∂vi 1 =− div v |∇vv |2 + 2 Ω Ω ∂xk ∂xj ∂xk Z ∂vj ∂vi ∂vi = ≤ k∇vv k33 . Ω ∂xk ∂xj ∂xk Since ∆vv is divergence-free, the term involving the pressure vanishes. Finally, the right hand side gives Z Z f · (−∆vv ) = ∇ff · ∇vv ≤ k∇ff k dp k∇vv k dp Ω

dp−d+2

Ω

≤ C(ε)k∇ff k

p0 dp dp−d+2

+ εk∇vv k

d−2

p dp d−2

.

Altogether we have 0 1d k∇vv k22 + c1 k∇vv kps,p + (c1 − ε)k∇vv kpdp ≤ k∇vv k33 + C(ε)k∇ff kp dp . 2 dt d−2 dp−d+2

The last term is integrable over (0, T ) by our assumptions and to simplify the subsequent formulas we put f ≡ 0. Hence we have 1d k∇vv k22 + c1 k∇vv kps,p + c2 k∇vv kpdp ≤ k∇vv k33 . 2 dt d−2

(12)

v k33(1−α) and using the interpolation Considering k∇vv k33 = k∇vv k3α 3 k∇v inequalities2 kzk3 ≤ kzkβ2 kzk1−β dp

and

d−2

kzk3 ≤ kzkγp kzk1−γ dp , d−2

and Young’s inequality, we arrive at (see [9, pages 234-235]) (13)

c2 1d v kpp , k∇vv k22 + c1 k∇vv kps,p + k∇vv kpdp ≤ c3 k∇vv k2λ 2 k∇v 2 dt 2 d−2

where λ=

2(3 − p) . dp − 3d + 4

Note that here the condition p > 3d−4 comes out. Note also that d 3d+2 λ ≤ 1 ⇐⇒ p ≥ d+2 . Thus part (iii) of the Theorem follows by Gronwall’s lemma. 2Here,

β=

2((d+3)p−2d) 3((d+2)p−2d)

and γ =

(d+3)p−3d . 3p


11

As p < 3d+2 implies λ > 1, a trick to obtain uniform estimates from d+2 (13) (described again in [9]) is to divide (13) by k∇vv k2λ 2 . Rewriting the last inequality as λ d 1 + k∇vv k22 + c1 k∇vv kps,p ≤ c3 1 + k∇vv k22 k∇vv kpp , dt and dividing the result by (1 + k∇vv k22 )λ leads to d A(t) + c1 k∇vv kps,p (1 + k∇vv k22 )−λ ≤ c3 k∇vv kpp dt where A(t) = (1 − λ)−1 (1 + k∇vv (t)k22 )1−λ . Note that A(t) is bounded; in particular, one does not need uniform bounds for k∇vv n (0)k2 . Integrating the last inequality over (0, T ) and using (8) gives Z T −λ (14) k∇vv kps,p 1 + k∇vv k22 1 while s ∈ (0, 2p ) is an arbitrary, fixed number. By Hölder’s inequality one has for β ∈ (0, 1) Z T Z T −βλ βλ βp v k22 k∇vv ks,p = k∇vv kβp 1 + k∇vv k22 s,p 1 + k∇v 0

2(3−p) , dp−3d+4

0

≤

Z

T

0

k∇vv kps,p

1+

−λ k∇vv k22

β Z

T

1+ 0

k∇vv k22

βλ 1−β

1−β

.

The first integral is bounded by (14). The second is bounded by (8) for the largest possible value β given through the relation 2βλ =p 1−β

⇐⇒

β=

p(dp − 3d + 4) . 4(3 − p) + p(dp − 3d + 4)

Hence Z

(15)

T 0

k∇vv kβp s,p < K .

Note that pβ = % as stated in part (ii) of the Theorem, which thus follows. Also, for pβ > 1 (15) is just (11), and the proof is completed. It remains to treat the case pβ ≤ 1. Fix r ∈ (1, p) arbitrary. Due to the interpolation σ 1− σ s kukσ,p ≤ ckukp s kuks,p and Hölder’s inequality we have Z T Z T rσ r(1− σ ) r k∇vv kσ,p ≤ c k∇vv kp s k∇vv ks,ps 0

0

≤

Z

T 0

δr(1− σs ) k∇vv kp

1δ Z

T 0

δ0 r σ k∇vv kp s

δ10

.


12

Thus, by virtue of (8) and (15) we obtain (11) provided that δ, δ 0 > 1 are such that σ σ δr 1 − =p δ 0 r = βp . s s This means 1 σ r σ r 1 + 1− . (16) 1= + 0 = · δ δ s βp s p Since p > r > 1 ≥ βp, there exists a uniquely determined σ ∈ (0, s) such that this equality holds. The proof of Theorem 3.1 is finished.

4. Concluding remarks Let us first summarize what is the novelty of the present paper concerning the spatially-periodic problem. • The existence theory covers degenerate viscosities for a significantly larger range of the parameter p. In particular, in three spatial dimensions, all p > 2 are included. Earlier results based on monotone operator theory were established only for p > 11 . 5 • Part (ii) of Theorem 3.1 is new both for the degenerate and the non-degenerate case. The use of Nikol0 ski˘ı spaces is essential in both of these improvements. Next, we comment on possible extensions of our result to the homogeneous Dirichlet problem. In, [11] the authors treat the non-degenerate case and prove global-in-time existence of weak solutions for p ≥ 2, and regularity results similar to part (iii) of Theorem 3.1 for p ≥ 49 . In p−2 particular, for T (Dvv ) := (1 + |Dvv |2 ) 2 Dvv the interior estimate Z T Z p−2 2 −λ (17) (1 + k∇vv k2 ) (1 + |Dvv |2 ) 2 |D(∇vv )|2 dx dt < ∞ 0

Ω0

for any Ω0 ⊂⊂ Ω is established. Assume that using similar investigations as in [11], it is possible to p−2 show that for every > 0 and for T (Dvv ) := ( + |Dvv |2 ) 2 Dvv , the corresponding weak solutions v fulfill Z T Z p−2 2 −λ (18) (1 + k∇vv k2 ) ( + |Dvv |2 ) 2 |D(∇vv )|2 dx dt ≤ C < ∞ , 0

Ω0

with C independent of > 0. Then, neglecting in (18), we can proceed as in the spatially periodic case. This time, however, only locally for any Ω0 ⊂⊂ Ω.


13

References [1] H. Bellout, F. Bloom and J. Neˇcas, Young measure-valued solutions for nonNewtonian incompressible fluids, Comm. Partial Differential Equations 19, No.11-12, p. 1763-1803 (1994). [2] E. DiBenedetto, Degenerate Parabolic Equations, Springer, New York Berlin, 1993. [3] C. Ebmeyer, WB. Liu and M. Steinhauer, Global regularity in fractional order Sobolev spaces for the p-Laplace equation on polyhedral domains, Zeitschrift f. Analysis u. ihre Anwendungen 24, No. 2, p. 353-374 (2005) [4] J. Frehse and J. M´ alek, Problems due to the no-slip boundary in incompressible fluid dynamics, in S. Hildebrandt and H. Karcher: “Geometric analysis and nonlinear partial differential equations”, p. 559–571, Springer, Berlin, 2003. [5] J. Frehse, J. M´ alek, M. Steinhauer, On existence results for fluids with shear dependent viscosity- unsteady flows, in W. J¨ ager, J. Neˇcas, O. John, K. Najzar, and J. Star´ a (Editors): “Partial differential equations, Theory and Numerical solution”, Chapman and Hall/CRC, Research Notes in Mathematics 406, p. 121-129, 2000 [6] O. A. Ladyzhenskaya, On modifications of Navier-Stokes Stokes equations for large gradients of the velocity, Zapiski naukhnych seminarov LOMI 5, p. 126154 (in Russian, 1968) [7] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach Science Publishers, New York, 1969, xviii+224 pages. [8] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969, xx+554 pages. [9] J. M´ alek, J. Neˇcas, M. Rokyta and M. R˚ uˇziˇcka, “Weak and measure-valued solutions to evolutionary PDE’s”, Chapman and Hall/CRC, Applied Mathematics and Mathematical Computation 13, 1996. [10] J. M´ alek, J. Neˇcas and M. R˚ uˇziˇcka, On the non-Newtonian incompressible fluids, Math. Models Methods Appl. Sci. 3, No.1, p. 35-63 (1993). [11] J. M´ alek, J. Neˇcas and M. R˚ uˇziˇcka, On weak solutions to a class of nonNewtonian incompressible fluids in bounded three-dimensional domains: the case p ≥ 2, Adv. Differential Equations 6, No. 3, p. 257-302 (2001). [12] J. M´ alek and K. R. Rajagopal, Mathematical Issues Concerning the Navier— Stokes Equations and Some of Its Generalizations, Evolutionary Equations, Handbook of Differential Equations (eds. C. Dafermos and E. Feireisl) vol.2, p.1-91, Elsevier B. V., 2005 [13] J. M´ alek, K. R. Rajagopal and M. R˚ uˇziˇcka, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity, Math. Models Methods Appl. Sci. 5, No.6, p. 789-812 (1995). [14] J. Neˇcas, Sur les normes équivalentes dans Wpk (Ω) et sur la coercivité des formes formelllement positives, in: Séminaire Equations aux Dérivées Partielles, Les Presses de l’Université de Montréal, Montréal, 1967, p. 102-128. [15] S. M. Nikol’skiˇi, “Approximation of functions of several variables and imbedding theorems”, Springer, Berlin, 1975. [16] J. Simon, Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl., IV. Ser. 146, 65-96 (1987). E-mail address: [email protected], [email protected], mark [email protected]

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Verzeichnis der erschienenen Preprints ab No. 310

310. Eberle, Andreas; Marinelli, Carlo: Stability of Sequential Markov Chain Monte Carlo Methods 311. Eberle, Andreas; Marinelli, Carlo: Convergence of Sequential Markov Chain Monte Carlo Methods: I. Nonlinear Flow of Probability Measures 312. Albeverio, Sergio; Hryniv, Rostyslav; Mykytyuk, Yaroslav: Reconstruction of Radial Dirac and Schrödinger Operators from Two Spectra 313. Eppler, Karsten; Harbrecht, Helmut: Tracking Neumann Data for Stationary Free Boundary Problems 314. Albeverio, Sergio; Mandrekar, Vidyadhar; Rüdiger, Barbara: Existence of Mild Solutions for Stochastic Differential Equations and Semilinear Equations with Non-Gaussian Lévy Noise 315. Albeverio, Sergio; Baranovskyi, Oleksandr; Pratsiovytyi, Mykola; Torbin, Grygoriy: The Set of Incomplete Sums of the First Ostrogradsky Series and Anomalously Fractal Probability Distributions on it 316. Gottschalk, Hanno; Smii, Boubaker: How to Determine the Law of the Noise Driving a SPDE 317. Gottschalk, Hanno; Thaler, Horst: AdS/CFT Correspondence in the Euclidean Context 318. Gottschalk, Hanno; Hack, Thomas: On a Third S-Matrix in the Theory of Quantized Fields on Curved Spacetimes 319. Müller, Werner; Salomonsen, Gorm: Scattering Theory for the Laplacian on Manifolds with Bounded Curvature 320. Ignat, Radu; Otto, Felix: 2-d Compactness of the Néel Wall 321. Harbrecht, Helmut: A Newton Method for Bernoulli’s Free Boundary Problem in Three Dimensions 322. Albeverio, Sergio; Mitoma, Itaru: Asymptotic Expansion of Perturbative Chern-Simons Theory via Wiener Space 323. Marinelli, Carlo: Well-Posedness and Invariant Measures for HJM Models with Deterministic Volatility and Lévy Noise

324. Albeverio, Sergio; Ayupov, Sh. A.; Kudaybergenov, K. K.: Derivations on the Algebra of τ-Compact Operators Affiliated with a Type I von Neumann Algebra 325. Castaño Díez, Daniel; Gunzburger, Max; Kunoth, Angela: An Adaptive Wavelet Viscosity Method for Hyperbolic Conservation Laws 326. Albeverio, Sergio; Ayupov, Sh. A.; Omirov, B. A.; Khudoyberdiyev, A. Kh: n-Dimensional Filiform Leibniz Algebras of Length (n-1) and Their Derivations 327. Albeverio, Sergio; Rabanovich, Slavik: On a Class of Unitary Representations of the Braid Groups B3 and B4 328. Husseini, Ryad; Kassmann, Moritz: Markov Chain Approximations for Symmetric Jump Processes 329. Marinelli, Carlo: Local Well-Posedness of Musiela's SPDE with Lévy Noise 330. Frehse, Jens; Steinhauer, Mark; Weigant, Wladimir: On Stationary Solutions for 2 - D Viscous Compressible Isothermal Navier-Stokes Equations; erscheint in: Journal of Mathematical Fluid Mechanics 331. Müller, Werner: A Spectral Interpretation of the Zeros of the Constant Term of Certain Eisenstein Series 332. Albeverio, Sergio; Mazzucchi, Sonia: The Trace Formula for the Heat Semigroup with Polynomial Potential 333. Málek, Josef; Pražák, Dalibor; Steinhauer, Mark: On the Existence and Regularity of Solutions for Degenerate Power-Law Fluids; erscheint in: Differential and Integral Equations