Lp -theory for a generalized nonlinear viscoelastic fluid model of differential type in various domains

Nonlinear Analysis 75 (2012) 5015–5026

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Lp -theory for a generalized nonlinear viscoelastic fluid model of differential type in various domains Matthias Geissert a,∗ , Dario Götz b,1 , Manuel Nesensohn b,1 a

Fachbereich Mathematik, Angewandte Analysis, Technische Universität Darmstadt, Schlossgartenstr. 7, D-64289 Darmstadt, Germany

b

Fachbereich Mathematik, International Research Training Group 1529, Technische Universität Darmstadt, Schlossgartenstr. 7, D-64289 Darmstadt, Germany

article

info

Article history: Received 8 September 2011 Accepted 9 April 2012 Communicated by S. Ahmad MSC: 76A10 76D03 35Q35

abstract This paper studies a coupled system of hyperbolic and parabolic equations governing the motion of viscoelastic, incompressible fluids on various domains. The model under consideration covers a wide range of nonlinear fluids including generalized Newtonian fluids, generalized Oldroyd-B fluids or Peterlin approximations. Existence and uniqueness of strong Lp -solutions for large times are proved for small data, for arbitrarily large data local well-posedness is shown. © 2012 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear viscoelastic fluid Strong Lp solutions Oldroyd-B fluid Maximal regularity

1. Introduction A commonly used model to describe the motion of a fluid in mathematical and engineering applications is the Navier–Stokes system which is based on a linear dependence of extra stress on the deformation tensor. In many cases, however, more complex fluids, such as polymeric liquids, biological fluids, suspensions or liquid crystals, exhibit behavior which cannot be characterized by this relation alone. Shear-thinning (or respectively shear-thickening), stress-relaxation, nonlinear creeping and more observed effects call for more general models. In this work, we consider a nonlinear model describing the motion of an incompressible viscoelastic fluid given by the following quasilinear system of coupled hyperbolic and parabolic partial differential equations:

ρ(∂ u + u.∇ u) − div S (Du) + ∇π = div µ(τ ) + f t v    div u = 0   ∂t τ + u.∇τ + bτ = g (∇ u, τ ) u|∂ Ω = 0     u(0) = u0 τ (0) = τ0 ∗

in (0, T0 ) × Ω , in (0, T0 ) × Ω , in (0, T0 ) × Ω , on (0, T0 ) × ∂ Ω , in Ω , in Ω .

Corresponding author. Tel.: +49 6151 16 3580; fax: +49 6151 16 4580. E-mail addresses: [email protected] (M. Geissert), [email protected] (D. Götz), [email protected] (M. Nesensohn). 1 Tel.: +49 6151 16 50184; fax: +49 6151 16 4580. 0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.04.016

(1.1)

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Here, the unknowns are the velocity field u, the pressure π and the elastic part of the stress τ . Furthermore, ρ is the density, Sv (Du) the viscous part of the stress tensor, f exterior body force and µ and g some given functions. The deformation tensor is denoted by Du = 12 (∇ u + ∇ uT ). For the viscous part of the stress, we impose the generalized Newtonian law Sv (Du) = 2α1 (tr((Du)2 ), tr((Du)3 ))Du + 2α2 (tr((Du)2 ), tr((Du)3 ))(Du)2

(1.2)

where α1 is the viscosity function and α2 relates to the cross-viscosity. Note that for divergence free functions tr((Du) ) = |Du|2 and tr((Du)3 ) = det Du. Roughly speaking, we prove existence and uniqueness of a strong solution (u, π , τ ) up to an arbitrary time T0 > 0 for a wide class of domains provided the data is sufficiently small, g (0, 0) = 0 (or smallness of |g (0, 0)| if the domain Ω is bounded) and α1 (0, 0) > 0. Note that – besides regularity – no further restrictions on the structure of µ, α1 , α2 and g are imposed. Moreover, for constant α1 > 0 and α2 = 0 local strong well-posedness is proved for arbitrarily large data in the same setting. In this case, the condition g (0, 0) = 0 can be omitted if Ω is bounded. Before we state our main result precisely, let us briefly discuss related results found in the literature. In [1], strong Lp -theory for generalized Newtonian fluids without elasticity, i.e. incompressible viscous fluids satisfying the generalized Newtonian law (1.2) and τ ≡ 0, is considered for small initial data u0 . Recently, Bothe and Prüß [2] have extended these results to the case of large initial data. The special case of Oldroyd-B fluids (cf. [3]), i.e. for constant α1 > 0, α2 = 0 and setting 2

µ(τ ) = τ and g (∇ u, τ ) = β Du − τ Wu + Wuτ + a(Duτ + τ Du)

(1.3)

for β > 0, −1 ≤ a ≤ 1 and Wu = (∇ u − ∇ u ), has been investigated by Guillaupé and Saut in the L2 -setting [4] in bounded domains. They proved the existence of local strong solutions for large data as well as global solutions for small data with a Schauder fixed point argument. Their method relies on a priori estimates and compactness arguments. Later, Fernández-Cara et al. [5] proved the existence of unique strong solutions in an Lp -setting similar to our approach for the same model problem as Guillaupé and Saut also in a bounded domain. They rely on a Schauder fixed point argument as well. A more general system than Oldroyd-B, where in (1.3) the constant term β is replaced by a shear-rate dependent function β(|Du|2 ), has been investigated in the steady L2 -setting on bounded and exterior domains by Arada and Sequeira [6,7]. This model is called generalized Oldroyd-B. Another generalization of the Oldroyd-B model is the so-called White–Metzner system (cf. [3]), where one takes constant α1 > 0, α2 = 0, b = 0, the identity µ(τ ) = τ and 1 2

T

g (∇ u, τ ) = β(|Du|2 )Du + γ (|Du|2 )τ − τ Wu + Wuτ + a(Duτ + τ Du) for some functions β and γ . Strong well-posedness of this model in 2D has been shown in the L2 -setting by Hakim [8] and later also in 3D by Molinet and Talhouk [9] in the non-stationary case in bounded domains. Note that, in particular, our main result shows well-posedness in Lp in all cases mentioned above. Bringing together a nonlinear viscosity function and elastic effects, Agranovich and Sobolevskii [10] and Dmitrienko et al. [11] studied a viscoelastic fluid model in the L2 -setting on a bounded domain. However, they replaced the frame-invariant objective derivative

Da τ Dt

= ∂t τ + u.∇τ + τ Wu − Wuτ − a(Duτ + τ Du)

by a partial derivative ∂t . This way, one can directly integrate the transport equation and insert the resulting elastic stress into the fluid equation. Finally, we would like to mention a work by Vorotnikov and Zvyagin [12] who considered a problem very similar to the one we consider in this article. They proved the existence of strong solutions in the L2 -setting where Ω = Rn , n = 2, 3. However, due to the L2 -approach using a priori estimates for a nonlinear system, they impose strong regularity assumptions on the initial data, i.e. u0 ∈ H23 (Rn ) and τ0 ∈ H23 (Rn ). It is thus likely that a generalization of their method to domains – if possible – would lead to additional compatibility conditions on the initial stress. In contrast to their approach, our proof relies on Lp -theory and thus allows to deal with a wide class of domains without further compatibility conditions and lower regularity assumptions on the initial data. Throughout this paper for p, q ∈ [1, ∞], s ∈ R+ and k ∈ N0 the spaces Hqs (Ω ), Wqs (Ω ), Bspq (Ω ) and  Hqk (Ω ) denote the usual Bessel potential spaces, Sobolev spaces, Besov spaces and homogeneous Sobolev spaces, respectively. If not said otherwise, Ω ⊂ Rn is a domain with a uniform C 2 -boundary (boundary regularity is to be understood in the sense of [13, Definition 4.10]). This guarantees the existence of a total extension operator for Ω , see [13, Theorem 5.24]. Hence, interpolation theory for Bessel potential spaces and Besov spaces, Sobolev embedding theorems and the mixed derivative theorem are valid in Ω . Moreover, for q ∈ (1, ∞) and a domain Ω ⊂ Rn we define the space of solenoidal vector fields by Lq,σ (Ω ) = {u ∈ Cc∞ (Ω )n : div u = 0}

Lq (Ω )

.

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In order to simplify notation, we shortly write ∥ · ∥q instead of ∥ · ∥Lq (Ω ) and ∥ · ∥T ,p,q instead of ∥ · ∥Lp (0,T ;Lq (Ω )) for T > 0. If the Helmholtz projection from Lq (Ω ) onto Lq,σ (Ω ) exists, we denote it by Pq . In this case, it is well-known that the Stokes operator defined by Aq = −Pq ∆,

D(Aq ) := Hq2 (Ω ) ∩ Hq1,0 (Ω ) ∩ Lq,σ (Ω )

(1.4)

generates an analytic semigroup on Lq,σ (Ω ) provided ∂ Ω is smooth enough, see [14] or [15]. Let us now state the main result of this work. Theorem 1.1. Fix n ∈ N, n ≥ 2, p, q, q′ ∈ (1, ∞) satisfying

= 1, T0 > 0, b ≥ 0. Let Ω ⊂ Rn be a uniform C -domain such that for r ∈ {q, q } the Helmholtz projection exists for Lr (Ω ) and a shift of the Stokes operator, i.e. λ + Ar for some λ ≥ 0 admits bounded imaginary powers with power angle smaller than π /2. Assume that 1 p

+

n 2q

<

1 2

and

1 q

+

1 q′

′

2

α1 , α2 ∈ C 1 (R2 , R) with α1 (0, 0) > 0, and g ∈ C 1 (Rn×n × Rn×n , Rn×n ) with g (0, 0) = 0, 2− 2p

as well as µ ∈ C 1 (Rn×n , Rn×n ). Furthermore, assume that f ∈ Lp (0, T0 ; Lq (Ω )), τ0 ∈ Hq1 (Ω ), and u0 ∈ Bqp (Ω ) ∩ Hq1,0 (Ω ) ∩ Lq,σ (Ω ). (a) Then there is a constant κ > 0 such that there exists a unique solution (u, π , τ ) of (1.1) within the regularity classes u ∈ Hp1 (0, T0 ; Lq (Ω )) ∩ Lp (0, T0 ; D(Aq )),

π ∈ Lp (0, T0 ;  Hq1 (Ω ))

and 1 τ ∈ W∞ (0, T0 ; Lq (Ω )) ∩ L∞ (0, T0 ; Hq1 (Ω )),

provided the smallness condition

∥f ∥Lp (0,T0 ;Lq (Ω )) + ∥u0 ∥

2− 2 p Bqp (Ω )

+ ∥τ0 ∥Hq1 (Ω ) + |∇ g (0, 0)| ≤ κ

is fulfilled. (b) Let α1 > 0 be constant and α2 = 0. Then there exists 0 < T ≤ T0 and a unique strong solution (u, π , τ ) of (1.1) in the time interval (0, T ) within the regularity classes u ∈ Hp1 (0, T ; Lq (Ω )) ∩ Lp (0, T ; D(Aq )),

π ∈ Lp (0, T ;  Hq1 (Ω ))

and 1 τ ∈ W∞ (0, T ; Lq (Ω )) ∩ L∞ (0, T ; Hq1 (Ω )).

Before we prove our main result, a few remarks are in order. Remark 1.2. (a) The condition α1 (0, 0) > 0 gives the parabolicity of the linearization of div Sv (Du). Furthermore, n the condition 1p + 2q < 12 ensures the maximal regularity space for the fluid to be continuously embedded into

1 (Ω )). Note also, that our assumption on p and q implies q > n and p > 2. L∞ (0, T ; W∞ (b) In Theorem 1.1 (a), the functions α1 , α2 , µ and g need not be defined on the whole space but may be indeed defined on some balls around zero. However, in that case, the smallness conditions on the data also depend on their radii. (c) Theorem 1.1 (a) includes the so-called generalized Newtonian fluid model by taking g = 0 and τ0 = 0 (and hence τ = 0). (d) In the case of bounded domains, the condition g (0, 0) = 0 may be relaxed to a smallness condition in Theorem 1.1 (a) and can be omitted in Theorem 1.1 (b). (e) The main theorem covers the whole space, half-spaces, bent half-spaces, layers, bent layers, bounded and exterior domains, since there the Stokes operator admits bounded imaginary powers (see [16,17,14]). (f) The result stated in [18, Theorem 3.4] is also true in our situation. Indeed, the only difference is that the assumptions of [18, Theorem 3.4] are not fulfilled for all r ∈ (1, ∞) but for q and q′ only. However, since the proof of [18, Theorem 3.4] n < 12 we relies on the assumptions for q and q′ only, this is not an issue. In particular, for p, q ∈ (1, ∞) satisfying 1p + 2q deduce 2− 2

(Lq,σ (Ω ), D(Aq ))1− 1 ,p = Bqp p (Ω ) ∩ Hq1,0 (Ω ) ∩ Lq,σ (Ω ), p

where (·, ·) denotes the real interpolation functor, as well as

[Lq,σ , D(Aq )] 1 = Hq1,0 (Ω ) ∩ Lq,σ (Ω ), 2

where [·, ·] denotes the complex interpolation functor.

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The present paper is organized as follows: in Section 2 we introduce a quasilinear operator formulation of (1.1) and the relevant function spaces as well as the necessary results on the Stokes problem and the transport equation. The proof of the main result is carried out in Section 3. 2. Preliminaries Without loss of generality, we set ρ = 1 in the following. Then, calculating div Sv (Du) explicitly, problem (1.1) can be written in the form

∂ u + u.∇ u + A(u)u + ∇π = div µ(τ ) + f t    div u = 0   ∂t τ + u.∇τ + bτ = g (∇ u, τ ) u|∂ Ω = 0    u(0) = u0 τ (0) = τ0

in (0, T0 ) × Ω , in (0, T0 ) × Ω , in (0, T0 ) × Ω , on (0, T0 ) × ∂ Ω , in Ω , in Ω ,

(2.1)

where u0 and τ0 are the initial values. Here, A(u)v is defined by

[A(u)v]i := −

n 1 

2 j,k,l=1

(∂(j,l) Sv,ik )(Du)(∂k ∂l vj + ∂k ∂j vl ),

i = 1, . . . , n,

where for X ∈ Rn×n it is

∂(j,l) Sv,ik (X ) = 2α1 (tr(X 2 ), tr(X 3 ))δji δlk + 2∂1 α1 (tr(X 2 ), tr(X 3 ))2Xlj Xik + 2∂2 α1 (tr(X 2 ), tr(X 3 ))3(X 2 )lj Xik + 2α2 (tr(X 2 ), tr(X 3 ))(δij Xlk + δlk Xij ) + 2∂1 α2 (tr(X 2 ), tr(X 3 ))2Xlj (X 2 )ik + ∂2 α2 (tr(X 2 ), tr(X 3 ))3(X 2 )lj (X 2 )ik .

(2.2)

In particular, we have

A(u)u = −div Sv (Du) and A(0)u = −α1 (0, 0)1u,

u ∈ Hq2 (Ω ), div u = 0.

(2.3)

Since our approach is based on a linearization of (2.1) and a fixed point argument, we start with estimates for solutions of the corresponding linearized problem. To be more precise, we first consider the Stokes system with homogeneous Dirichlet boundary conditions, i.e.

 ∂ u + A(0)u + ∇π = f ,   t div u = 0  u|∂ Ω = 0 u(0) = u0

in (0, T ) × Ω , in (0, T ) × Ω , on (0, T ) × ∂ Ω , in Ω .

(2.4)

Then the following proposition gives an a priori estimates of the solution (u, π ) of (2.4) in the following function spaces Xu (T ) := Hp1 (0, T ; Lq (Ω )) ∩ Lp (0, T ; D(Aq )),

Xπ (T ) := Lp (0, T ;  Hq1 (Ω )),

Yu (T ) := Hp1/2 (0, T ; Lq (Ω )) ∩ Lp (0, T ; Hq1 (Ω )). Note that Xu (T ) and Xπ (T ) are the usual spaces for maximal Lp -regularity estimates of the solution of (2.4) for f ∈ Lp (0, T ; Lq,σ (Ω )) and u0 in a suitable interpolation space. Estimates in the weaker norm Yu (T ) are used to carry out the fixed point argument, see Proposition 2.4. The next proposition is an application of the mixed derivative theorem and Sobolev embeddings. The proof follows the argumentation of [19, Lemma 4.2]. Proposition 2.1. Let p, q ∈ (1, ∞) satisfy

1 p

n + 2q < 12 , T0 > 0 and Ω ⊂ Rn be a uniform C 2 -domain. Then for 0 < T ≤ T0 it is

1 Xu (T ) ↩→ L∞ (0, T ; W∞ (Ω )) ∩ L∞ (0, T ; Lq (Ω ))

and

Yu (T ) ↩→ L∞ (0, T ; L∞ (Ω )).

Moreover, the embedding constants do not depend on T if one considers functions with zero time-trace, i.e. there is a constant C > 0 such that

∥u∥L∞ (0,T ;W∞ 1 (Ω )) + ∥u∥T ,∞,q ≤ C ∥u∥Xu (T ) ,

T ∈ (0, T0 ], u ∈ Xu (T ),

and

∥u∥T ,∞,∞ ≤ C ∥u∥Yu (T ) ,

T ∈ (0, T0 ], u ∈ Yu (T ),

u(0) = 0.

u(0) = 0,

M. Geissert et al. / Nonlinear Analysis 75 (2012) 5015–5026

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Proof. Let 0 < T ≤ T0 . It follows from the mixed derivative theorem that Hp1 (0, T ; Lq (Ω )) ∩ Lp (0, T ; Hq2 (Ω )) ↩→ Hpα (0, T ; Hq2(1−α) (Ω )),

α ∈ (0, 1).

By the Sobolev embedding theorem, we deduce 1

Hpα (0, T ; Hq2(1−α) (Ω )) ↩→ BUC ([0, T ]; BUC 1 (Ω )),

p

<α<

1 2

−

n 2q

.

Next, we show that the embedding constant does not depend on T , if one considers functions with zero time-trace. Let u ∈ Xu (T ) with u(0) = 0. We define u¯ (t ) =



0 u(t + T0 − T )

t ∈ (0, T0 − T ], t ∈ (T0 − T , T0 ),

t ∈ (0, T ).

Then, by construction, it follows that u¯ ∈ Xu (T0 ) and

∥u∥L∞ (0,T ;W∞ u∥L∞ (0,T0 ;W 1 (Ω )) ≤ C ∥¯u∥Xu (T0 ) ≤ C ∥u∥Xu (T ) , 1 (Ω )) = ∥¯ ∞ where C is independent of T , 0 < T ≤ T0 . The embeddings Xu (T ) ↩→ L∞ (0, T ; Lq (Ω ))

and Yu (T ) ↩→ L∞ (0, T ; L∞ (Ω )),

as well as the estimates

∥u∥L∞ (0,T ;Lq (Ω )) ≤ C ∥u∥Xu (T ) , u ∈ Xu (T ), ∥u∥L∞ (0,T ;L∞ (Ω )) ≤ C ∥u∥Yu (T ) , u ∈ Yu (T ), can be proved by the same arguments.

u(0) = 0, u(0) = 0,



Although the next proposition is a variant of [20, Corollary 4.2], for the convenience of the reader, we give a sketch of its proof.

= 1, and 0 < T ≤ T0 . Let Ω be a uniform C 2 -domain, such that for r ∈ {q, q } the Helmholtz projection Pr : Lr (Ω ) → Lr ,σ (Ω ) exists and for some λ ≥ 0 the shifted Stokes operator λ + Ar admits bounded imaginary powers with power angle smaller than π /2. Then for f ∈ Lp (0, T ; Lq (Ω )) and u0 ∈ (Lq,σ (Ω ), D(Aq ))1− 1 ,p , p there exists a unique solution (u, π ) ∈ Xu × Xπ of (2.4). Moreover, there exists C1 > 0, independent of f and u0 , such that   ∥u∥Xu (T ) + ∥π ∥Xπ (T ) ≤ C1 ∥f ∥T ,p,q + ∥u0 ∥(Lq,σ (Ω ),D(Aq )) 1 . Proposition 2.2. Fix 1 < p, q, q′ < ∞, with

1 q

+

1 q′

′

1− p ,p

If u0 = 0, the constant C1 can be chosen uniformly in T , 0 < T ≤ T0 . Finally, if u0 = 0 and f = div F , F ∈ Lp (0, T ; Hq1 (Ω )) then

∥u∥Yu (T ) ≤ C2 ∥F ∥T ,p,q , where C2 > 0 is independent of F and T , 0 < T ≤ T0 . Proof. The first estimate follows from [21], since λ + Aq (w.l.o.g. we may assume α1 (0, 0) = 1) admits bounded imaginary powers with power angle smaller than π /2. Let λ0 > λ, such that λ0 ∈ ρ(Aq ). For the proof of the second estimate let u 1

denote the solution of (2.4) with f = div F and u0 = 0 and set w := (λ0 + Aq )− 2 u. Then, since w ∈ Hp1 (0, T ; Lq (Ω )) ∩ Lp (0, T ; D(Aq )), an easy calculation shows that 1

w ′ (t ) + Aq w(t ) = (λ0 + Aq )− 2 Pq div F (t ) in (0, T ), w(0) = 0. Next, we prove 1

(λ0 + Aq )− 2 Pq div ∈ L(Lp (0, T ; Lq (Ω ))), by a duality argument. Let 1 < p′ , q′ < ∞ with

1 p

+ p1′ = 1 and

1 q′

+ 1q = 1 and g ∈ Lp′ (0, T ; Lq′ (Ω )). Due to [22, Proposition

2.6], the operator (λ0 + Aq ) = λ0 + Aq′ (see Remark 1.2(f)) admits bounded imaginary powers. It follows that ′

1

1

((λ0 + Aq )− 2 Pq div F , g ) = (F : ∇(λ0 + Aq′ )− 2 g ) ≤ C ∥F ∥T ,p,q ∥g ∥T ,p′ ,q′ , 1

where we used the Dirichlet boundary data of (λ0 + Aq′ )− 2 g and the continuity of the operator 1

(λ0 + Aq′ )− 2 ∈ L(Lp′ (0, T ; Lq′ (Ω )), Lp′ (0, T ; Hq1′ ,0 (Ω ) ∩ Lq′ ,σ (Ω ))),

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M. Geissert et al. / Nonlinear Analysis 75 (2012) 5015–5026 1 2

which follows from D(Aq′ ) = Hq1′ ,0 (Ω ) ∩ Lq′ ,σ (Ω ) (cf. Remark 1.2(f)). Since λ0 + Aq has bounded imaginary powers and [Lq,σ (Ω ), D(Aq )] 1 = Hq1,0 (Ω ) ∩ Lq,σ (Ω ), standard arguments show 2

1 2

(λ0 + Aq ) ∈ L(Hp1 (0, T ; Lq,σ (Ω )) ∩ Lp (0, T ; D(Aq )), Yu ). We thus obtain 1

1

∥u∥Yu (T ) = ∥(λ0 + Aq ) 2 w∥Yu (T ) ≤ C ∥w∥Xu (T ) ≤ C ∥(λ0 + Aq )− 2 Pq div F ∥T ,p,q ≤ C ∥F ∥T ,p,q . With arguments similar to the proof of Proposition 2.1, one can show that the embedding constant is independent of T , 0 < T ≤ T0 .  Next, we turn our attention to the transport equation

 ∂t τ + u˜ .∇τ + bτ = g τ (0) = τ0

in (0, T ) × Ω , in Ω ,

(2.5)

where u˜ is a sufficiently smooth function. We show a priori estimates of the solution of (2.5) in the function spaces Xτ (T ) := L∞ (0, T ; Hq1 (Ω ))

and Yτ (T ) := L∞ (0, T ; Lq (Ω )).

Estimates of the solution of (2.5) in the weaker norm Yτ (T ) are used in the fixed point argument in Section 3. The proof may be considered as standard, yet a short sketch is carried out for the reader’s convenience. Proposition 2.3. Let 1 < p < ∞, n < q < ∞, T , T0 > 0, u˜ ∈ Hp1 (0, T ; Lq (Ω )) ∩ Lp (0, T ; Hq2 (Ω )) such that u˜ · ν = 0 on ∂ Ω , b ≥ 0 and Ω ⊂ Rn be a uniform C 2 -domain. (a) Then for each g ∈ L1 (0, T ; Hq1 (Ω )) ∩ L∞ (0, T ; Lq (Ω )) and τ0 ∈ Hq1 (Ω ) there exists a unique solution τ ∈ Xτ (T ) ∩ 1 W∞ (0, T ; Lq (Ω )) of (2.5). Moreover, there exists C3 > 0, independent of g , τ0 , u˜ and T , 0 < T ≤ T0 , such that

  C3 T 1−1/p ∥˜u∥ Lp (0,T ;Hq2 (Ω )) ∥τ ∥Xτ (T ) ≤ C3 ∥τ0 ∥Hq1 (Ω ) + ∥g ∥L1 (0,T ;Hq1 (Ω )) e . 1 (b) Let g ∈ L1 (0, T ; Lq (Ω )) and τ0 ∈ Lq (Ω ). Suppose there exists a solution τ ∈ Xτ ∩ W∞ (0, T ; Lq (Ω )) of (2.5). Then, there exists C4 > 0, independent of g , τ0 , u˜ and T , 0 < T ≤ T0 , such that

∥τ ∥Yτ (T ) ≤ C4 (∥τ0 ∥q + ∥g ∥T ,1,q )e

C ∥div u˜ ∥

L1 (0,T ;Hq1 (Ω ))

.

1 Proof. Assume that τ ∈ Xτ (T ) ∩ W∞ (0, T ; Lq (Ω )) is a solution of equation (2.5). We multiply equation (2.5) with q −2 |τ (t )| τ (t ) and integrate over Ω :

1 d q dt

∥τ (t )∥ ≤ ∥g (t )∥q ∥τ (t )∥ q q

q−1 q

       q −2 u˜ i (t )(∂i τjk (t ))|τ (t )| τjk (t ) +   i,j,k Ω 1

≤ ∥g (t )∥q ∥τ (t )∥qq−1 + ∥div u˜ (t )∥∞ ∥τ (t )∥qq . q

Thus, using the Sobolev embedding theorem, we obtain d dt

∥τ (t )∥q ≤ ∥g (t )∥q + C ∥div u˜ (t )∥Hq1 (Ω ) ∥τ (t )∥q .

and applying Gronwall’s Lemma, we obtain C ∥div u˜ ∥

∥τ ∥T ,∞,q ≤ C (∥τ0 ∥q + ∥g ∥T ,1,q )e

L1 (0,T ;Hq1 (Ω ))

.

(2.6)

This proves (b) and the uniqueness of the solution. We differentiate equation (2.5) with respect to xl , multiply the equation with |∂l τ (t )|q−2 ∂l τ (t ) and integrate over Ω . This yields d dt

  ∥∇τ (t )∥Lq (Ω ) ≤ C ∥g (t )∥Hq1 (Ω ) + ∥˜u(t )∥Hq2 (Ω ) ∥∇τ (t )∥Lq (Ω ) .

(2.7)

Applying Gronwall’s Lemma to (2.7) and adding equation (2.6), we obtain

  ∥˜u∥ 2 ∥τ ∥L∞ (0,T ;Hq1 (Ω )) ≤ C ∥τ0 ∥Hq1 (Ω ) + ∥g ∥L1 (0,T ;Hq1 (Ω )) e L1 (0,T ;Hq (Ω )) .

(2.8)

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Furthermore, we estimate the time derivative of τ using Proposition 2.1 by

∥∂t τ ∥T ,∞,q ≤ ∥˜u.∇τ ∥T ,∞,q + ∥g ∥T ,∞,q ≤ C (∥˜u∥T ,∞,∞ ∥∇τ ∥T ,∞,q + ∥g ∥T ,∞,q ).

(2.9)

Next we construct the solution of (2.5). We extend u˜ and g by 0 for t ̸∈ [0, T ] and then by an extension operator E, which is given by the Stein extension theorem [13, Theorem 5.24], to Rn . Similarly, τ0 is extended to Rn by E. Thus, we may assume w.l.o.g. u˜ ∈ Lp (R; Hq2 (Rn )) ∩ L∞ (R; L∞ (Rn )),

τ0 ∈ Hq1 (Rn ) and g ∈ L1 (R; Hq1 (Rn )).

For ε > 0 let ρε ∈ Cc∞ (R × Rn ) be a mollifier and define u˜ ϵ = u˜ ∗ ρε ,

gε = g ∗ ρε ,

and

τ0,ε = τ0 ∗ ρε .

Using the method of characteristic curves we construct a solution of



∂t τε + u˜ ε .∇τε + bτε = gε τε (0) = τ0,ε

in (0, T ) × Rn , in Rn .

Due to (2.8) and (2.9) τε is uniformly bounded in L∞ (0, T ; Hq1 (Rn )) and ∂t τε is uniformly bounded in L∞ (0, T ; Lq (Rn )). Thus

1 there exists a sequence (εn )n∈N and τ ∈ L∞ (0, T ; Hq1 (Rn )) ∩ W∞ (0, T ; Lq (Rn )) such that

∗

∗

τεn ⇀ τ in L∞ (0, T ; Hq1 (Rn )) and ∂t τεn ⇀ ∂t τ in L∞ (0, T ; Lq (Rn )). For n, m ∈ N with τn,m = τεn − τεm it holds that



∂t τn,m + u˜ εn .∇τn,m + bτn,m = gεn − gεm − (uεn − uεm ).∇τεm τn,m (0) = τ0,εn − τ0,εm

in (0, T ) × Rn , in Rn .



Hence, it follows from (2.6) that

 ∥τn,m ∥L∞ (0,T ;Lq (Rn )) ≤ C ∥τ0,εn − τ0,εm ∥q + ∥gεn − gεm ∥L1 (0,T ;Lq (Rn ))  C ∥˜uεn ∥L (0,T ;H 2 (Rn )) q 1 + ∥(uεn − uεm ).∇τεm ∥L1 (0,T ;Lq (Rn )) e  ≤ C ∥τ0,εn − τ0,εm ∥q + ∥gεn − gεm ∥L1 (0,T ;Lq (Rn ))  C ∥˜uε ∥ n L (0,T ;H 2 (Rn )) q 1 + ∥(uεn − uεm )∥L1 (0,T ;Hq1 (Rn )) ∥τεm ∥L∞ (0,T ;Hq1 (Rn )) e . Thus, τεn is a Cauchy sequence and we have τεn → τ in L∞ (0, T ; Lq (Ω )). The proposition follows now from standard arguments.  Finally, the following proposition is used for the fixed point argument. Proposition 2.4. Let X be a reflexive Banach space or let X have a separable pre-dual. Let K be a convex, closed and bounded subset of X and let X ↩→ Y , where Y is a Banach space. Let Φ : X → X map K into K and let η < 1, such that

∥Φ (x) − Φ (y)∥Y ≤ η∥x − y∥Y ,

x, y ∈ K .

Then there exists a unique fixed point of Φ in K . A proof can be found in [23]. 3. Proof of Theorem 1.1 The proof is based on a suitable linearization of (1.1) and a fixed point argument. However, it seems that the Banach fixed point theorem is not directly applicable, since the appearing term (˜u2 − u˜ 1 ).∇τ1 , see (3.6), cannot be controlled in L1 (0, T ; Hq1 (Ω )) due to lack of regularity of τ1 . Therefore, we employ a variant of Banach’s fixed point theorem, see Proposition 2.4. Although it seems to be tempting to deal with large initial data by linearizing around the generalizes Stokes operator investigated in [2], this is not possible at the moment due to the lack of the weak estimates given in Proposition 2.2 in this more general setting. Therefore, we restrict our results for large initial data to a constant viscosity function. Step 1. Linearization. Noting that (cf. (2.3))

A(u)u = −div Sv (Du) and A(0)u = −α1 (0, 0)1u,

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M. Geissert et al. / Nonlinear Analysis 75 (2012) 5015–5026

we rewrite (1.1) as

 ∂t u + A(0)u + ∇π = f + div (F (u) + µ(τ ))    div u = 0   ∂t τ + u.∇τ + bτ = g (∇ u, τ ) u|∂ Ω = 0     u(0) = u0 τ (0) = τ0

in (0, T0 ) × Ω , in (0, T0 ) × Ω , in (0, T0 ) × Ω , on (0, T0 ) × ∂ Ω , in Ω , in Ω ,

(3.1)

where F (u) = −u ⊗ u + (Sv (Du) − 2α1 (0, 0)Du). Then, we easily find that (1.1) can be rewritten as a fixed point problem of the map

u0 ,τ0 (u, f + div (F (u) + µ(τ )), g (∇ u, τ )), Φ (u, τ ) := Φ

(3.2)

where

˜ u0 ,τ0 : Xu (T0 ) × Lp (0, T0 ; Lq (Ω )) × L1 (0, T0 ; Hq1 (Ω )) ∩ L∞ (0, T0 ; Lq (Ω )) → Xu (T0 ) × Xτ (T0 ) Φ and

(˜u, f˜ , g˜ ) → (u, τ ) denotes the solution operator of following problem

 ∂t u + A(0)u + ∇π = f˜   div u = 0    ∂t τ + u˜ .∇τ + bτ = g˜ u| ∂ Ω = 0      u(0) = u0 τ (0) = τ0

in (0, T0 ) × Ω , in (0, T0 ) × Ω , in (0, T0 ) × Ω , in (0, T0 ) × ∂ Ω , in Ω , in Ω

(3.3)

2− 2

p ˜ u0 ,τ0 is for u0 ∈ (Lq,σ (Ω ), D(Aq ))1− 1 ,p = Bqp (Ω ) ∩ Hq1,0 (Ω ) ∩ Lq,σ (Ω ) (cf. Remark 1.2(f)) and τ0 ∈ Hq1 (Ω ). Note that Φ p well-defined by Propositions 2.2 and 2.3 since the equations for u and τ are now decoupled. Moreover, the following estimates hold:





∥u∥Xu (T0 ) ≤ C1 ∥f + div (F (˜u) + µ(τ˜ ))∥T0 ,p,q + ∥u0 ∥

2− 2 p Bqp (Ω )

,

  ∥τ ∥Xτ (T0 ) ≤ K1 (∥˜u∥Xu (T0 ) ) ∥τ0 ∥Hq1 (Ω ) + ∥g (∇ u˜ , τ˜ )∥L1 (0,T0 ;Hq1 (Ω )) ,

(3.4)

(3.5)

where K1 (·): R+ → R+ is a monotonically increasing function. Step 2. Fixed point argument for large time T0 > 0. The next lemma allows us to show that Φ maps K (T0 , R1 , R2 ) into K (T0 , R1 , R2 ) provided R1 , R2 and κ are small enough, where Ku (T0 , R1 ) := {˜u ∈ Xu (T0 ): u˜ (0) = u0 , ∥˜u∥Xu (T0 ) ≤ R1 }, Kτ (T0 , R2 ) := {τ˜ ∈ Xτ (T0 ): τ˜ (0) = τ0 , ∥τ˜ ∥Xτ (T0 ) ≤ R2 },

K (T0 , R1 , R2 ) := Ku (T0 , R1 ) × Kτ (T0 , R2 ).

Lemma 3.1. Fix T0 , R0 > 0 and p, q ∈ (1, ∞), such that

n + 2q < 12 . Then, there exist constants C , C5 > 0 and a function O: R+ → R+ with O(r ) → 0 for r → 0 such that for R1 , R2 ∈ (0, R0 ) and (˜u, τ˜ ) ∈ K (T0 , R1 , R2 ) 1 p

∥div F (˜u)∥T0 ,p,q ≤ CR21 + O(R1 )R1 , ∥g (∇ u˜ , τ˜ )∥L1 (0,T0 ;Hq1 (Ω )) ≤ (O(R1 + R2 ) + C κ) (R1 + R2 ), ∥g (∇ u˜ , τ˜ )∥T0 ,∞,q ≤ C , ∥div µ(τ˜ )∥T0 ,p,q ≤ C5 R2 hold. Proof. Proposition 2.1 and Hölder’s inequality yields

∥div (˜u ⊗ u˜ )∥T0 ,p,q ≤ C ∥˜u∥2Xu (T0 ) ≤ CR21 .

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Note, that Sv admits the same regularity as α1 and α2 , which can be seen in (2.2). Furthermore, since ∂(j,l) Sik (0) = 2α1 (0, 0)δji δlk ,

∥div (Sv (Du˜ ) − 2α1 (0, 0)Du˜ )∥T0 ,p,q ≤ C



∥∂(j,l) (Sik )(Du˜ ) − ∂(j,l) (Sik )(0)∥T0 ,∞,∞ ∥˜u∥Xu (T0 )

i,j,k,l

≤C

sup |η|≤∥∇ u˜ ∥T0 ,∞,∞

|(∇ Sv )(η) − (∇ Sv )(0)| ∥˜u∥Xu (T0 ) ≤ O(R1 )R1 ,

where O = O(r ) tends to zero for r → 0 since Sv is continuously differentiable. Similarly, since g (0, 0) = 0 and |∇ g (0, 0)| < κ , we obtain

∥g (∇ u˜ , τ˜ )∥T0 ,1,q ≤ ∥g (∇ u˜ , τ˜ ) − g (0, 0)∥T0 ,1,q ≤C sup |(∇ g )(η, ξ )|(∥∇ u˜ ∥T0 ,1,q + ∥τ˜ ∥T0 ,1,q ) |η|≤∥∇ u˜ ∥T ,∞,∞ 0 |ξ |≤∥τ˜ ∥T ,∞,∞ 0

≤C

|(∇ g )(η, ξ )|(R1 + R2 )

sup |η|≤∥∇ u˜ ∥T ,∞,∞ 0 |ξ |≤∥τ˜ ∥T ,∞,∞ 0





 ≤ C  |η|≤∥∇sup u˜ ∥

T0 ,∞,∞ |ξ |≤∥τ˜ ∥T ,∞,∞ 0

 |(∇ g )(η, ξ ) − (∇ g )(0, 0)| + |(∇ g )(0, 0)|  (R1 + R2 )

≤ (O(R1 + R2 ) + C κ)(R1 + R2 ). Once again, O(r ) → 0 for r → 0 due to the continuity of ∇ g. In a completely analogous way, |∇ g (0, 0)| ≤ κ yields

∥∇ g (∇ u˜ , τ˜ )∥T0 ,1,q ≤ (O(R1 + R2 ) + C κ)(R1 + R2 ). Clearly, by the same arguments, we obtain the corresponding estimate in the space L∞ (0, T0 ; Lq (Ω )). Finally, we calculate

∥div µ(τ˜ )∥T0 ,p,q ≤

sup |η|≤∥τ˜ ∥T0 ,∞,∞

|(∇µ)(η)| ∥∇ τ˜ ∥T0 ,p,q ≤ CR2 . 

Using estimates (3.4) and (3.5) together with Lemma 3.1, we obtain

 ∥u∥Xu (T0 ) ≤ C1 ∥f ∥T0 ,p,q + ∥div F (˜u)∥T0 ,p,q + ∥div µ(τ˜ )∥T0 ,p,q + ∥u0 ∥ ≤ C1 (κ + CR21 + O(R1 )R1 + C5 R2 ),

2− 2 p Bqp (Ω )



R1 , R2 ∈ (0, R0 )

and

  ∥τ ∥Xτ (T0 ) ≤ K1 (∥˜u∥Xu (T0 ) ) ∥τ0 ∥Hq1 (Ω ) + ∥g (∇ u˜ , ∇ τ˜ )∥L1 (0,T0 ,Hq1 (Ω )) ≤ C (κ + (O(R1 + R2 ) + C κ)(R1 + R2 )), R1 , R2 ∈ (0, R0 ). Hence, by first choosing R2 = 1/(4C1 C5 )R1 and then κ and R1 small enough, we find that Φ maps K (T0 , R1 , R2 ) into itself. Next, we show that Φ is a contraction in a suitable space. In order to do so, let (˜ui , τ˜i ) ∈ K (T0 , R1 , R2 ), i = 1, 2 and (ui , τi ) = Φ (˜ui , τ˜i ). Then, the difference (u, τ ) := Φ (˜u2 , τ˜2 ) − Φ (˜u1 , τ˜1 ) with difference in pressure φ = π2 − π1 fulfills the equation:

∂ u + A(0)u + ∇φ = div (F (˜u ) − F (˜u ) + µ(τ˜ ) − µ(τ˜ )) t 2 1 2 1   div u = 0   ∂t τ + u˜ 2 .∇τ + bτ = g (∇ u˜ 2 , τ˜2 ) − g (∇ u˜ 1 , τ˜1 ) − u˜ .∇τ1 u|∂ Ω = 0    u(0) = 0 τ (0) = 0

in (0, T0 ) × Ω , in (0, T0 ) × Ω , in (0, T0 ) × ∂ Ω , on (0, T0 ) × ∂ Ω , in Ω , in Ω .

(3.6)

Moreover, the terms on the right hand side of the latter equation can be estimated as follows. n + 2q < 12 . Then, there exist constants C , C6 > 0 and a function O: R+ → R+ with O(r ) → 0 for r → 0 such that for R1 , R2 ∈ (0, R0 ) and (ui , τi ), (˜ui , τ˜i ) ∈ K (T0 , R1 , R2 ) for i = 1, 2

Lemma 3.2. Fix T0 , R0 > 0 and p, q ∈ (1, ∞) such that

1 p

∥F (˜u2 ) − F (˜u1 )∥T0 ,p,q ≤ CR1 ∥˜u2 − u˜ 1 ∥Yu (T0 ) , ∥g (∇ u˜ 2 , τ˜2 ) − g (∇ u˜ 1 , τ˜1 )∥T0 ,1,q ≤ C (O(R1 + R2 ) + κ)(∥˜u2 − u˜ 1 ∥Yu (T0 ) + ∥τ˜2 − τ˜1 ∥Yτ (T0 ) ), ∥(˜u2 − u˜ 1 ).∇τ1 ∥T0 ,1,q ≤ CR2 ∥˜u2 − u˜ 1 ∥Yu (T0 ) , ∥µ(τ˜2 ) − µ(τ˜1 )∥T0 ,p,q ≤ C6 ∥τ˜2 − τ˜1 ∥Yτ (T0 ) hold.

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Proof of Lemma 3.2. Thanks to Proposition 2.1, we obtain

∥˜u2 ⊗ u˜ 2 − u˜ 1 ⊗ u˜ 1 ∥T0 ,p,q = ∥(˜u2 + u˜ 1 ) ⊗ (˜u2 − u˜ 1 )∥T0 ,p,q ≤ (∥˜u1 + u˜ 2 ∥T0 ,∞,∞ )∥˜u2 − u˜ 1 ∥T0 ,p,q ≤ CR1 ∥˜u2 − u˜ 1 ∥T0 ,p,q . Defining the function S˜ by S˜ (X ) = 2α1 (tr(X 2 ), tr(X 3 )) + 2α2 (tr(X 2 ), tr(X 3 ))X ,

X ∈ Rn×n ,

we easily see that S˜ (X )X = Sv (X ) and S˜ ∈ C 1 (Rn×n , Rn×n ). Therefore, we may write Sv (Dv˜ ) − 2α1 (0, 0)Dv˜ = (S˜ (Dv˜ ) − S˜ (0))Dv˜ ,

v˜ ∈ Xu (T )

and obtain the equality Sv (Du˜ 2 ) − 2α1 (0, 0)Du˜ 2 − (Sv (Du˜ 1 ) − 2α1 (0, 0)Du˜ 1 ) = (S˜ (Du˜ 2 ) − S˜ (0))D(˜u2 − u˜ 1 ) + (S˜ (Du˜ 2 ) − S˜ (Du˜ 1 ))Du˜ 1

=: Q1 D(˜u2 − u˜ 1 ) + Q2 Du˜ 1 . We estimate both terms

∥Q1 ∥T0 ,∞,∞ ≤ ∥Q2 ∥T0 ,p,q ≤

sup |η|≤∥∇ u˜ 2 ∥T0 ,∞,∞

|(∇ S˜ )(η)| ∥˜u2 ∥L∞ (0,T0 ;W∞ 1 (Ω )) ≤ CR1 ,

sup |η|≤max{∥∇ u˜ 1 ∥T0 ,∞,∞ ,∥∇ u˜ 2 ∥T0 ,∞,∞ }

|(∇ S˜ )(η)| ∥˜u2 − u˜ 1 ∥Lp (0,T0 ;Hq1 (Ω ))

≤ C ∥˜u2 − u˜ 1 ∥Yu (T0 ) . Therefore,

∥(Sv (Du˜ 1 ) − 2α1 (0, 0)Du˜ 2 ) − (Sv (Du˜ 1 ) − 2α1 (0, 0)Du˜ 1 )∥T0 ,p,q ≤ CR1 ∥˜u2 − u˜ 1 ∥Yu (T0 ) . Furthermore we have

∥µ(τ˜2 ) − µ(τ˜1 )∥T0 ,p,q ≤

sup |ξ |≤max{∥τ˜1 ∥T0 ,∞,∞ ,∥τ˜2 ∥T0 ,∞,∞ }

|(∇µ)(ξ )|∥τ˜2 − τ˜1 ∥T0 ,p,q

≤ C6 ∥τ˜2 − τ˜1 ∥Yτ (T0 ) . Since |∇ g (0, 0)| < κ , similarly to the proof of Lemma 3.1, we obtain

∥g (∇ u˜ 2 , τ˜2 ) − g (∇ u˜ 1 , τ˜1 )∥T0 ,1,q ≤ C

sup |η|≤ max {∥∇ u˜ i ∥T ,∞,∞ } 0 i=1,2 |ξ |≤ max {∥τ˜i ∥T ,∞,∞ } 0 i=1,2

|(∇ g )(η, ξ )|(∥∇(˜u2 − u˜ 1 )∥T0 ,1,q + ∥τ˜2 − τ˜1 ∥T0 ,1,q )

≤ C (O(R1 + R2 ) + κ)(∥˜u2 − u˜ 1 ∥Yu (T0 ) + ∥τ˜2 − τ˜1 ∥Yτ (T0 ) ), with O(r ) → 0 for r → 0. Finally, we conclude the proof with the calculation

∥(˜u2 − u˜ 1 ).∇τ1 ∥T0 ,1,q ≤ CR2 ∥˜u2 − u˜ 1 ∥Yu (T0 ) .  Using Propositions 2.2 and 2.3, we obtain

∥u∥Yu (T0 ) ≤ C2 (∥F (˜u2 ) − F (˜u1 )∥T0 ,p,q + ∥µ(τ˜2 ) − µ(τ˜1 )∥T0 ,p,q ) ≤ CR1 ∥˜u∥Yu (T0 ) + C2 C6 ∥τ˜ ∥Yτ (T0 ) and with R2 = 1/(4C1 C5 )R1

∥τ ∥Yτ (T0 ) ≤ C4 (∥g (∇ u˜ 2 , τ˜2 ) − g (∇ u˜ 1 , τ˜1 ) + u˜ .∇τ1 ∥T0 ,q,q ) ≤ C (κ + O(R1 ))(∥˜u∥Yu (T0 ) + ∥τ˜ ∥Yτ (T0 ) ). Choosing now R1 , and κ sufficient small and multiplying the estimate for τ with 4C2 C6 , we deduce

∥u∥Yu (T0 ) + 4C2 C6 ∥τ ∥Yτ (T0 ) ≤

1 2

(∥˜u∥Yu (T0 ) + 4C2 C6 ∥τ˜ ∥Yτ (T0 ) ).

Hence, the mapping Φ is a contraction with respect to the norm ∥u∥Yu (T0 ) + 4C2 C6 ∥τ ∥Yτ (T0 ) . Now, Theorem 1.1(a) follows 1 from Proposition 2.4. The time regularity of τ ∈ W∞ (0, T0 , Lq (Ω )) follows from Proposition 2.3 and the regularity of g (∇ u, τ ) ∈ L∞ (0, T0 ; Lq (Ω )) (cf. Lemma 3.1).

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Step 3. Fixed point argument for large initial data. Let us now consider the situation of Theorem 1.1(b), i.e. let α1 > 0 be constant, α2 = 0 and 0 < T ≤ T0 . For fixed T0 > 0, u0 and τ0 , we choose R0 > 0 large enough such that K (T , R0 , R0 ) is not empty. In order to apply Proposition 2.4, we show

Φ (K (T , R0 , R0 )) ⊂ K (T , R0 , R0 ) and

∥Φ (˜u2 , τ˜2 ) − Φ (˜u1 , τ˜1 )∥Yu (T )×Yτ (T ) ≤

1 2

(∥˜u2 − u˜ 1 ∥Yu (T ) + ∥τ˜2 − τ˜2 ∥Yτ (T ) ),

provided 0 < T ≤ T0 is small enough. Here, Φ is defined by (3.2). However, note that in this case F (u) = −u ⊗ u. Lemma 3.3. Fix T0 , R0 > 0 and p, q ∈ (1, ∞) such that

n + 2q < 21 . Then, there exist a constant C > 0 and a function O: R+ → R+ with O(t ) → 0 for t → 0 such that for all T ∈ (0, T0 ) and (˜ui , τ˜i ), (˜u, τ˜ ), (ui , τi ) ∈ K (T , R0 , R0 ), i = 1, 2, 1 p

∥˜u.∇ u˜ ∥T ,p,q ≤ O(T ), ∥g (∇ u˜ , τ˜ )∥L1 (0,T ;Hq1 (Ω )) ≤ O(T ), ∥g (∇ u˜ , τ˜ )∥T ,∞,q ≤ C , ∥div µ(τ˜ )∥T ,p,q ≤ O(T ), ∥˜u2 ⊗ u˜ 2 − u˜ 1 ⊗ u˜ 1 ∥T ,p,q ≤ O(T )∥˜u2 − u˜ 1 ∥Yu (T ) , ∥g (∇ u˜ 2 , τ˜2 ) − g (∇ u˜ 1 , τ˜1 )∥T ,1,q ≤ O(T )(∥˜u2 − u˜ 1 ∥Yu (T ) + ∥τ˜2 − τ˜1 ∥Yτ (T ) ), ∥µ(τ˜2 ) − µ(τ˜1 )∥T ,p,q ≤ O(T )∥τ˜2 − τ˜1 ∥Yτ (T ) , ∥(˜u2 − u˜ 1 ).∇τ1 ∥T ,1,q ≤ O(T )∥˜u2 − u˜ 1 ∥Yu (T ) holds. Proof. Note first that for u∗ (t ) := e−Aq t u0 and w ˜ = u˜ − u∗ u∗ ∈ Xu (T ) and w( ˜ 0) = 0 holds. Therefore, Proposition 2.1 yields ∗ ∥˜u∥L∞ (0,T ;W∞ ˜ L∞ (0,T ;W∞ 1 (Ω )) ≤ ∥w∥ 1 (Ω )) + ∥u ∥L (0,T ;W 1 (Ω )) ≤ CR0 + C , ∞ ∞

∥˜u∥L∞ (0,T ;Lq (Ω )) ≤ CR0 + C .

(3.7)

It holds u˜ .∇ u˜ = w.∇ ˜ w ˜ + u∗ .∇ w ˜ + w.∇ ˜ u∗ + u∗ .∇ u∗ and

∥w.∇ ˜ u∗ ∥T ,p,q ≤ ∥w∥ ˜ T ,∞,∞ ∥u∗ ∥Lp (0,T ;Hq1 (Ω )) ≤ O(T )R0 . Similarly ∗ ∥u∗ .∇ w∥ ˜ T ,p,q ≤ ∥w∥ ˜ L∞ (0,T ;W∞ 1 (Ω )) ∥u ∥T ,p,q ≤ O(T )R0 .

Furthermore, we estimate with similar arguments as above 1/p ∥w.∇ ˜ w∥ ˜ T ,p,q ≤ C ∥w∥ ˜ T ,∞,q ∥w∥ ˜ L∞ (0,T ;W∞ ≤ O(T )R20 . 1 (Ω )) T ′

Since u∗ .∇ u∗ ∈ Lp (0, T ; Lq (Ω )) the first conclusion follows. By (3.7),

∥˜u∥L1 (0,T ;Hp2 (Ω )) ≤ O(T ) and ∥τ˜ ∥Lp (0,T ;Hp1 (Ω )) ≤ O(T ) it easily follows that

∥div µ(τ˜ )∥T ,p,q ≤

sup |η|≤∥τ˜ ∥T ,∞,∞

|(∇µ)(η)| ∥∇ τ˜ ∥T ,p,q ≤ C ∥∇ τ˜ ∥T ,p,q ≤ O(T ).

Moreover, (3.8) and similar calculations show, as in the proof of Lemma 3.1, that

∥g (∇ u˜ , τ˜ )∥L1 (0,T ;Hq1 (Ω )) ≤ O(T ) and ∥g (∇ u˜ , τ˜ )∥T ,∞,q ≤ C .

(3.8)

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M. Geissert et al. / Nonlinear Analysis 75 (2012) 5015–5026

Furthermore, we estimate

∥˜u2 ⊗ u˜ 2 − u˜ 1 ⊗ u˜ 1 ∥T ,p,q = ∥(˜u2 + u˜ 1 ) ⊗ (˜u2 − u˜ 1 )∥T ,p,q = ∥˜u2 + u˜ 1 ∥T ,p,q ∥˜u2 − u˜ 1 ∥T ,∞,∞   = CT 1/p ∥˜u2 + u˜ 1 − 2u∗ ∥T ,∞,q + 2∥u∗ ∥T ,p,q ∥˜u2 − u˜ 1 ∥Yu (T ) ≤ O(T )∥˜u2 − u˜ 1 ∥Yu (T ) . By (3.7) and (3.8), we calculate

∥g (∇ u˜ 2 , τ˜2 ) − g (∇ u˜ 1 , τ˜1 )∥T ,1,q ≤ C

sup |η|≤ max {∥∇ u˜ i ∥T ,∞,∞ } i=1,2 |ξ |≤ max {∥τ˜i ∥T ,∞,∞ } i=1,2

|(∇ g )(η, ξ )|(∥∇(˜u2 − u˜ 1 )∥T ,1,q + ∥τ˜2 − τ˜1 ∥T ,1,q )

≤ C (∥∇(˜u2 − u˜ 1 )∥T ,1,q + ∥τ˜2 − τ˜1 ∥T ,1,q ) ≤ O(T )(∥˜u2 − u˜ 1 ∥Yu (T ) + ∥τ˜2 − τ˜1 ∥Yτ (T ) ), and similarly to the proof of Lemma 3.2 we obtain using (3.8)

∥µ(τ˜2 ) − µ(τ˜1 )∥T ,p,q ≤ O(T )∥τ˜2 − τ˜1 ∥Yτ (T ) , ∥(˜u2 − u˜ 1 ).∇τ1 ∥T ,1,q ≤ O(T )∥˜u2 − u˜ 1 ∥Yu (T ) .  Using Proposition 2.2, Proposition 2.3 and Lemma 3.3 and choosing T sufficiently small leads to

∥Φ (u, τ )∥Xu (T )×Xτ (T ) ≤ R0 ,

(u, τ ) ∈ K (T , R0 , R0 ),

and 1 (∥˜u2 − u˜ 1 ∥Yu (T ) + ∥τ˜2 − τ˜2 ∥Yτ (T ) ), 2 and Theorem 1.1(b) follows from Proposition 2.4.

∥Φ (˜u2 , τ˜2 ) − Φ (˜u1 , τ˜1 )∥Yu (T )×Yτ (T ) ≤

(˜u1 , τ˜1 ), (˜u2 , τ˜2 ) ∈ K (T , R0 , R0 ).

Acknowledgments The second and third authors were supported by the DFG International Research Training Group 1529 on Mathematical Fluid Dynamics at TU Darmstadt. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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