Existence of turbulent weak solutions to the generalized Navier-Stokes

Existence of turbulent weak solutions to the generalized Navier-Stokes equations in exterior domains and large time behaviour J¨org Wolf Abstract. Let Ω be an exterior domain in Rn (n = 2, 3, 4) , with boundary being not necessarily smooth. For any initial velocity u0 ∈ L2 (Ω)n such that ∇ · u0 = 0 (in sense of distribution) and external forces F ∈ L1 (0, ∞; L2 (Ω)n ) + L2 (0, ∞; W −1,2 (Ω)n ) we are able to construct a turbulent weak solution u ∈ Cw ([0, ∞); L2 (Ω)n ) ∩ L2 (0, ∞; W01,2 (Ω)n ) to the equations of motion of a non-Newtonian fluid. Simultaneously, we prove that this solution fulfils the non-uniform decay condition ku(t)kL2 (Ω) → 0 as t → ∞. Mathematics Subject Classification (2000). 35Q30, 35Q35, 35D05, 35B40. Keywords. General fluids, existence of weak solutions, asymptotic behaviour.

1. Introduction. Statement of the Main Result Let Ω ⊂ Rn (n = 2, 3, 4) be an exterior domain. We set Q := Ω × (0, ∞). In the present paper we consider the generalized Navier-Stokes equations ∇ · u = 0,

1

∂t u + ∇ · (u ⊗ u − S + pI) = f − ∇ · g

(1.1) in Q,

(1.2)

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J. Wolf

where p = pressure, u = {u1 , . . . , un } = velocity, S = {Sij | i, j = 1, . . . , n} = deviatoric stress tensor, f − ∇ · g = external force. Boundary and initial conditions. On the boundary of Ω we assume the following condition of adherence u = 0

on

∂Ω × (0, ∞).

(1.3)

At the initial time t = 0 we assume u(0) = u0

in Ω,

(1.4)

where u0 denotes a given initial velocity field, with ∇ · u0 = 0. Constitutive law. For the usual Navier-Stokes equations the stress S is given by S = S(D) = ν0 D, 2 where ν0 = const > 0 denotes the viscosity, while D = {Dij } denotes the ”rate of strain tensor”, which is defined by 1 D(u) := (∇u + ∇uT ). 2 However in many applications one often considers fluids where the viscosity is not constant but ranges between two positive constants. For instance considering a fluid where the mechanical energy is transferred into heat the viscosity may depend on the temperature. In this case the stress is given by the law S = S(x, t, D) = µ(θ(x, t))D,

(x, t) ∈ Q

(1.5)

with µ ∈ C(R),

0 < µ1 ≤ µ ≤ µ2 < ∞ in R, θ = temperature. (concerning the continuum mechanical background we refer to [2], [11]). Having in mind (1.5) as special case we impose the following conditions on the components of the deviatoric stress tensor S. (I)

1

2

2

S : Q × Mnsym → Mnsym 3

Here, ∇ · v := ∂xi v i , where ∂xi =

is a Carath´eodory function;

∂ (i = 1, . . . , n). Throughout repeated subscripts imply ∂xi

summation over 1 to n. Clearly, ∇ · D = ∆ on the space of all divergence free function. n2 3 Mn2 sym = vector space of all symmetric n × n matrices ξ = {ξij }. We equip Msym with scalar product ξ : η = ξij ηij and norm kξk := (ξ : ξ)1/2 . - By a · b we denote the usual scalar product in Rn and by |a| we denote the Euclidean norm.

2

Existence of turbulent solutions

3

growth condition: (II)

kS(x, t, ξ)k ≤ c0 kξk + κ1

2

∀ ξ ∈ Mnsym , f.a.a. (x, t) ∈ Q

(c0 > 0, κ1 ∈ L2 (Q), κ1 ≥ 0); coercivity: (III)

S(x, t, ξ) : ξ ≥ ν0 kξk2 − κ2

2

∀ ξ ∈ Mnsym , f.a.a. (x, t) ∈ Q

(ν0 > 0, κ2 ∈ L1 (Q), κ2 ≥ 0); strict monotonicity:  (S(x, t, ξ) − S(x, t, η)) : (ξ − η) > 0 (IV) ∀ ξ, η ∈ Mn2 (ξ 6= η), f.a.a. (x, t) ∈ Q. sym Weak solution to (1.1), (1.2). Before we introduce the notion of a weak solution to (1.1)-(1.4) let us provide some notations and function spaces which will be used in sequence of the paper. By W k, q (Ω), W0k, q (Ω) (k ∈ N; 1 ≤ q ≤ ∞) we denote the usual Sobolev spaces. Let (X, k · kX ) be a normed space. By Lq (0, T ; X) (0 < T ≤ ∞) we denote the space of all Bochner measurable functions ϕ : (0, T ) → X, such that  µZ T ¶ q1   q  0. Thirdly, we complete the proof of the strong energy inequality taking into account the monotonicity of S. Finally, by virtue of the global bound kuk 2 n+2 < ∞ verifying a similar estimate L n for u as we have found in the second step for um we get the non-uniform decay ku(t)kL2 → 0 as t → ∞.

2. Harmonic decomposition The purpose of this section is to state a few lemmas which form the base to construct an appropriate pressure function for weak solutions to the problem (1.1)(1.4).

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J. Wolf

During the whole section let G ⊂ Rn be a bounded domain with ∂G ∈ C 2 . By W02, r (G) (1 < r < ∞) we denote the closure of C0∞ (G) with respect to the norm µZ ¶ r1 kφkW 2, r (G) := |∆φ|r dx . 0

G

Clearly, by the well-known Calder´on-Zygmond’s estimate we have k∇2 φkLr (G) ≤ CCZ kφkW 2, r (G) , 0

(2.1)

where CCZ = const > 0 depending on n only. We begin our discussion by stating a variational estimate which is due to R. M¨ uller [17]. Lemma 2.1 (Variational estimate in W02, r (G)). Let 1 < r < ∞. Then there exists a constant Cr = Cr (r, n, G) > 0, such that R ∆u∆φ dx G kukW 2, r (G) ≤ Cr sup ∀ u ∈ W02, r (G). (2.2) 0 0 kφk 2, r 0 2, r W (G) 06=φ∈W (G) 0

0

¤ Next, we introduce the following subspaces of Lr (G) (1 < r < ∞), which will be used in the sequel Lr (G)

Ar (G) := {∆φ | φ ∈ C0∞ (G)} r

,

r

B (G) := {ϕ ∈ L (G) | ∆ϕ = 0 in G}. Remark 2.2. Owing to the reflexivity of W02, r (G) the space Ar (G) introduced above is given by (2.3) Ar (G) = {∆u | u ∈ W02, r (G)}. With help of Lemma 2.1 one obtains the following estimates which play an essential role in the proof of the main theorem (cf. Section 3). 2

Lemma 2.3. Let p ∈ Ar (G) and h = {hij } ∈ Lr (G)n (1 < r < ∞), such that Z Z p∆φ dx = h : ∇2 φ dx ∀ φ ∈ C0∞ (G). (2.4) G

G

Then kpkLr (G) ≤ Cr CCZ khkLr (G) . 0

(2.5)

Lemma 2.4. Let 1 < r < ∞. Then for every v ∗ ∈ (W02, r (G))∗ there exists a unique u ∈ W02, r (G) such that Z ∆u∆φ dx = hv ∗ , φi ∀ φ ∈ C0∞ (G). (2.6) G

Existence of turbulent solutions

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(For the proof of this lemma see in [26]). As an immediate consequence of Lemma 2.4 we get Corollary 2.5 (C. G. Simader). For every p ∈ Lr (G) (1 < r < ∞) there exist p0 ∈ Ar (G) and ph ∈ B r (G) such that p = p0 + ph .

(2.7)

In addition, there exists a constant Cr0 = Cr0 (r, n, G) such that kp0 kLr (G) + kph kLr (G) ≤ Cr0 kpkLr (G) , r

(2.8)

r

i.e. the sum A (G) + B (G) is direct. Proof. Let p ∈ Lr (G). According to Lemma 2.4 there exists u ∈ W02, r (G) satisfying Z Z p∆φ dx ∀ φ ∈ C0∞ (G). (2.9) ∆u∆φ dx = G

G

In particular, by (2.3) we have ∆u ∈ Ar (G). On the other hand, as one can easily check (2.9) is equivalent to Z (∆u − p)∆φ dx = 0 ∀ φ ∈ C0∞ (G). G

Using Weyl’s Lemma we have ∆(p − ∆u) = 0 in G. This shows that the function p − ∆u belongs to B r (G). Finally, setting p0 := ∆u and ph := p − ∆u gives (2.7). Now, it only remains to prove (2.8). For, let p = p0 + ph with p0 ∈ Ar (G) and ph ∈ B r (G). It is readily seen that Z Z p0 ∆φ dx = p∆φ dx ∀ φ ∈ C0∞ (G). G

G

Thus, applying Lemma 2.3 with h = p I shows that kp0 kLr (G) ≤ Cr CCZ kpkLr (G) . Whence, (2.8).

¤

Next, let us introduce the following subspace of B r (G) ½ ¾ ¯Z ¯ B˙ r (G) := p ∈ B r (G) ¯ p dx = 0 . G

Now we present the following result on the local pressure decomposition (cf. also in [26]). Theorem 2.6. Let u ∈ Cw ([0, ∞); L2 (G)n ) with ∇ · u = 0 (in the sense of distri2 butions) and let h ∈ L1loc ([0, ∞); Lr (G)n ) (1 < r ≤ 2). Suppose that Z ∞Z −u · ∂t ϕ + h : ∇ϕ dx dt = 0 (2.10) 0

G

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J. Wolf

for all ϕ ∈ C0∞ (G × (0, ∞))n with ∇ · ϕ = 0. Then there exist unique functions p0 ∈ L1loc ([0, ∞); Ar (G)), p˜h ∈ Cw ([0, ∞); B˙ r (G)) with such that Z ∞Z

Z

∞

Z

−u · ∂t ϕ + h : ∇ϕ dx dt = 0

G

p˜h (0) = 0,

p0 ∇ · ϕ + ∇˜ ph · ∂t ϕ dx dt 0

(2.11)

G

for all ϕ ∈ C0∞ (G × (0, ∞)). Furthermore, we have the a-priori estimates kp0 (t)kLr (G) ≤ c kh(t)kLr (G) , k˜ ph (t)kLr (G)) ≤ c ku(t) − u(0)kL2 (G)

° °Z t ° ° ° h(s) ds° + c° ° 0

(2.12) (2.13)

Lr (G)

for almost all t ∈ (0, ∞), where c = const > 0 depending only on r, n and G. Proof. Let ψ ∈ C0∞ (G)n with ∇·ψ = 0 and let η ∈ C0∞ (0, ∞). Into (2.10) inserting ϕ(x, t) = η(t)ψ(x) using Fubini’s theorem yields Z ∞ Z ∞ 0 − α η dt = β η dt, 0

where

0

Z α(t) :=

u(t) · ψ dx, G

Z

β(t) := −

h(t) : ∇ψ dx,

t ∈ (0, ∞).

G

By the assumptions of the theorem we have β ∈ L1loc ([0, ∞)). Therefore α ∈ W 1, 1 ([0, ∞)) with α0 = β. In particular, α is represented by an absolutely continuous function, which will be denoted also by α. Using integration by parts one calculates Z t

α(t) = α(0) +

β(s) ds

∀ t ∈ (0, ∞).

(2.14)

0

Define

Z

t

˜ := h(t)

h(s) ds,

t ∈ [0, ∞).

0

Let t ∈ (0, ∞) be fixed. Using Fubini’s theorem the identity (2.14) reads Z ˜ : ∇ψ dx = 0. (u(t) − u(0)) · ψ + h(t) G

Thus, according Z to [6] (Th. III. 3.1, Th. III. 5.2) there exists a unique function p˜(t) ∈ 0

Lr (G) with G

p˜(t) dx = 0 such that for all ψ ∈ W01, r (G)n

Z

Z ˜ : ∇ψ dx = (u(t) − u(0)) · ψ + h(t) G

p˜(t)∇ · ψ dx. G

(2.15)

Existence of turbulent solutions

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In addition, there exists a constant c > 0 depending only on r, n and G such that ˜ k˜ p(t)kLr (G) ≤ c (ku(t) − u(0)kLr (G) + kh(t)k Lr (G) ).

(2.16)

Next, we are going to prove that t 7→ p˜(t) is a Bochner measurable function. 0 For, let v ∈ Lr (G) be arbitrarily chosen. From [6] (Th. III. 3.4) we get the existence 0 of a function ψ ∈ W01, r (G)n such that ∇ · ψ = v − vG 5 (cf. also [23] (Lemma 2.1, p. 252)). Thus, from (2.15) we infer that Z Z p˜(t) (v − vG ) dx p˜(t) v dx = G

G

Z

˜ : ∇ψ dx. (u(t) − u(0)) · ψ + h(t)

= G

By the assumption of the theorem the function on the right is continuous, so also the function on the left. Consequently, p˜ ∈ Cw ([0, ∞); Lr (G)), and by a well-known theorem of Pettis this shows that p˜ is Bochner measurable. Let ϕ ∈ C0∞ (G × (0, ∞))n be a given test function. Into (2.15) putting ψ = ϕ(·, t) integrating both sides of this identity over the interval (0, ∞) yields Z ∞Z Z ∞Z ˜ : ∇ϕ dx dt = (u − u(0)) · ϕ + h p˜ ∇ · ϕ dx dt. (2.17) 0

G

0

G

Now, applying Corollary 2.5 one finds unique functions p˜0 ∈ Cw ([0, ∞); Ar (G)) and p˜h ∈ Cw ([0, ∞); B˙ r (G)) such that in G × (0, ∞). 6

p˜ = p˜0 + p˜h

As one may easily check (2.15) implies p˜(0) = 0. Thus, by Corollary 2.5 we deduce p˜h (0) = 0. Moreover, observing (2.8) using (2.16) gives (2.13). ph = 0 Into (2.17) inserting ψ = ∇φ for φ ∈ C0∞ (G) recalling ∇ · u = 0 and ∆˜ using integration by parts one obtains Z Z 2 ˜ h(t) : ∇ φ dx = p˜0 (t)∆φ dx ∀ φ ∈ C0∞ (G). G

G

Let t ∈ (0, ∞) and let 0 < ρ < 1. Then from the above identity we derive Z ˜ Z ˜ h(t + ρ) − h(t) p˜0 (t + ρ) − p˜0 (t) : ∇2 φ dx = ∆φ dx ρ ρ G G Z 1 v(x) dx. Here vG denotes the mean value Ln (G) G Z Z Z 6 Notice, that p˜0 (t) dx = 0 and p˜(t) dx = 0 implies p˜h (t) dx = 0. 5

G

G

G

(2.18)

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J. Wolf

for all φ ∈ C0∞ (G). With help of Lemma 2.3 one gets the estimate ° ° ° ° ˜ + ρ) − h(t) ˜ ° ° p˜0 (t + ρ) − p˜0 (t) ° ° h(t ° ° ° ° ≤ c° ° ° r ° r ρ ρ L (G) L (G) ° Z t+ρ ° °1 ° ° = c° h(s) ds° ° r . ρ t L (G) Then integrating both sides of the above estimate over the interval (0, T ) (0 < T < ∞) using Minkowski’s inequality and Fubini’s theorem gives ° ° ° p˜0 (· + ρ) − p˜0 ° ° ° ≤ c khkL1 (0,T +ρ;Lr (G)) , (2.19) ° ° 1 ρ L (0,T ;Lr (G)) where c = const > 0 depending only on r, n and G. This shows that p˜0 ∈ 1, 1 Wloc ([0, ∞); Ar (G)). Thus, setting p0 := ∂t p˜0 from (2.19) we deduce kp0 kL1 (0,T ;Lr (G)) ≤ c khkL1 (0,T ;Lr (G)) . Moreover, on both sides of (2.18) passing to the limit ρ → 0 using Riesz-Fischer’s theorem one gets Z Z h(t) : ∇2 φ dx = p0 (t)∆φ dx ∀ φ ∈ C0∞ (G) G

G

for almost all t ∈ (0, ∞). Then applying (2.5) (cf. Lemma 2.3) gives (2.12). On the other hand, the identity (2.11) easily follows from (2.17) replacing ϕ by ∂t ϕ therein and applying integration by parts. The uniqueness of p0 and p˜h follows directly from (2.12) and (2.13). ¤ Next, by the aid of Theorem 2.6 we will establish optimal estimates for the special case G = DR , where DR := {x ∈ Rn | R < |x| < 16R}. In the proof of the main result we will make extensive use of the following Corollary 2.7. Let 0 < R < ∞. Let u ∈ Cw ([0, ∞); L2 (DR )n ) with ∇ · u = 0 (in sense of distributions), such that Z ∞Z Z ∞Z − u · ∂t ϕ dx dt + (h1 + h2 ) : ∇ϕ − f · ϕ dx dt = 0 (2.20) 0

for all ϕ ∈

DR

C0∞ (DR

0

DR

n

× (0, ∞)) with ∇ · ϕ = 0, where h1 ∈ L

n+2 n

2

(DR × (0, ∞))n , 2

h2 ∈ L2 (DR × (0, ∞))n , f ∈ L1 (0, ∞; L2 (DR )n )

Existence of turbulent solutions

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are given functions. Then there exist unique functions p1 ∈ L

n+2 n

(0, ∞; A

n+2 n

(DR )),

p2 ∈ L2 (0, ∞; A2 (DR )), ∗

p3 ∈ L1 (0, ∞; A2 (DR )), n+2 p˜h ∈ Cw ([0, ∞); B˙ n (DR )),

such that Z ∞Z − 0

Z

DR

Z

= 0

∞

Z

u · ∂t ϕ dx dt + (h1 + h2 ) : ∇ϕ − f · ϕ dx dt 0 DR Z ∞Z Z ∞ (p1 + p2 + p3 )∇ · ϕ dx dt + ∇˜ ph · ∂t ϕ dx dt DR

0

for all ϕ ∈ C0∞ (DR × (0, ∞))n and Z Z p1 (t)∆φ dx = h1 (t) : ∇2 φ dx, D D Z R Z R p2 (t)∆φ dx = h2 (t) : ∇2 φ dx, DR DR Z Z p3 (t)∆φ dx = f (t) · ∇φ dx ∀ φ ∈ C0∞ (DR ) DR

(2.21)

DR

(2.22) (2.23) (2.24)

DR

for almost all t ∈ (0, ∞). In addition, we have the a-priori estimates kp1 (t)k

L

n+2 n

(DR )

≤ c kh1 (t)k

L

n+2 n

(DR )

,

(2.25)

kp2 (t)kL2 (DR ) ≤ c kh2 (t)kL2 (DR ) ,

(2.26)

kp3 (t)kL2 (DR ) ≤ c Rkf (t)kL2 (DR )

(2.27)

and k˜ ph (t)k2 n+2 L

n

n2 +4 n+2

ku(t) − u(0)k2L2 (DR ) + ckh1 k2 n+2 L n (DR ×(0,∞)) °Z t ° 2 ° n +4 ° n n−2 2 ° n+2 n+2 kh2 kL2 (DR ×(0,∞)) + c R (2.28) + cR f (s) ds° ° °

(DR )

≤ cR

0

L2 (DR )

for almost all t ∈ (0, ∞), where c = const > 0 depending only on n. Proof. First, let us prove the assertion for R = 1. To begin with we shall write 2 f = −∇ · h3 for an appropriate h3 : D1 → Rn . For this purpose with help of [6] (Th. III. 3.4) we introduce a linear operator B : L2 (D1 ) → W 1, 2 (D1 )n fulfilling the following properties (i) (ii)

∇ · Bg = g − gD1 ∀ g ∈ L2 (D1 ); kBgkW 1, 2 (D1 ) ≤ c kgkL2 (D1 ) ∀ g ∈ L2 (D1 ).

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J. Wolf

Then we set

1 (f (t))D1 ⊗ x, n Clearly, h3 ∈ L1 (0, ∞; W 1, 2 (D1 )). Next, defining h3 (t) := Bf (t) +

h3 := h3 − (h3 )D1

a.e. in

t ∈ (0, ∞).

D1 × (0, ∞)

it results ∇ · h3 = f

a.e. in

D1 × (0, ∞).

(2.29)

In addition, using Poincar´e’s inequality and H¨older’s inequality combined with the property (ii) of B one gets kh3 (t)kL2∗ (D1 ) ≤ c k∇h3 (t)kL2 (D1 ) ≤ c kf (t)kL2 (D1 )

(2.30)

for almost all t ∈ (0, ∞). n+2 Now, set r1 = , r2 = 2 and r3 = 2∗ . Let k ∈ {1, 2, 3}. According to n Lemma 2.4 for almost every t ∈ (0, ∞) there exist unique functions pk (t) ∈ Ark (D1 ), such that

Z

Z pk (t)∆φ dx = D1

D1

hk (t) : ∇2 φ dx ∀ φ ∈ C0∞ (D1 ).

(2.31) 0

We claim that t 7→ pk (t) is Bochner measurable. Indeed, letting v ∈ Lrk (D1 ) 2, r 0 be a arbitrarily chosen by Corollary 2.5 there exist unique functions φ ∈ W0 k (D1 ) 0 and p ∈ B rk (D1 ) such that v = ∆φ + p. Thus from (2.31) it follows that Z Z Z pk (t)v dx = pk (t)∆φ dx = hk (t) : ∇2 φ dx. (2.32) D1

D1

D1

Since t 7→ hk (t) is Bochner measurable the function on the left of (2.32) is Lebesgue measurable. Thus, by means of Petti’s theorem t 7→ pk (t) is Bochner measurable. n+2 n+2 Next, let p0 ∈ L1loc ([0, ∞); A n (D1 )) and p˜h ∈ Cw ([0, ∞); B˙ n (D1 )) denote the function obtained by Theorem 2.6. By an inspection of the proof therein one finds Z Z p0 (t)∆φ dx = (h1 (t) + h2 (t) + h3 (t)) : ∇2 φ dx ∀ φ ∈ C0∞ (D1 ) D1

D1

for almost all t ∈ (0, ∞). On the other hand, summing up (2.31) from k = 1 to 3 one sees that for almost all t ∈ (0, ∞) the function p1 (t) + p2 (t) + p3 (t) satisfies the above identity too. This implies p0 = p1 + p2 + p3

a.e. in

D1 × (0, ∞)

which proves (2.21), (2.22), (2.23) and (2.24). The a-priori estimate (2.25) ((2.26) resp.) immediately follows from (2.22) ((2.23) resp.) using Lemma 2.3, while (2.27) follows from (2.31) (with k = 3) using Lemma 2.3 along with (2.30) and H¨older’s inequality. On the other hand,

Existence of turbulent solutions

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the a-priori estimate (2.28) can verified easily from (2.13) applying Minkowski’s inequality. Next, let 0 < R < ∞ be arbitrarily chosen. Defining u(y, t) := Ru(Ry, t), h1 (y, t) := h1 (Ry, t), h2 (y, t) := h2 (Ry, t), f (y, t) := Rf (Ry, t),

(y, t) ∈ D1 × (0, ∞)

using the transformation formula of the Lebesgue integral the identity (2.20) turns into Z ∞Z Z ∞Z − u · ∂t ϕ dx dt + (h1 + h2 ) : ∇ϕ − f · ϕ dx dt = 0 (2.33) 0

D1 C0∞ (D1

for all ϕ ∈ get functions

0

D1

n

× (0, ∞)) with ∇ · ϕ = 0. From the first part of the proof we p1 ∈ L

n+2 n

(0, ∞; A

2

n+2 n

(D1 )),

2

p2 ∈ L (0, ∞; A (D1 )), ∗

p3 ∈ L1 (0, ∞; A2 (D1 )), p˜h ∈ Cw ([0, ∞); B˙

n+2 n

(D1 )),

satisfying the identities (2.21), (2.22), (2.23) and (2.24) and the a priori estimates (2.25), (2.26) , (2.27) and (2.28) (with R = 1). Finally setting pk (x, t) := pk (R−1 x, t) (k = 1, 2, 3) p˜h (x, t) := ph (R−1 x, t),

(x, t) := DR × (0, ∞)

once more applying the transformation formula of the Lebesgue integral shows that these functions fulfil (2.21), (2.22), (2.23) and (2.24) together with the a priori estimates (2.25), (2.26) , (2.27) and (2.28). This completes the proof of the corollary. ¤

3. Proof of the Main Theorem We divide the proof into four steps. At first we provide a sequence of weak solutions {um } to the corresponding approximate system, which converge to a weak solution u to (1.1)-(1.4). Secondly, for almost all 0 < T < ∞ we prove the strong convergence of um (T ) to u(T ) in L2 (Ω)n . Thirdly, based on the monotonicity condition of S and the L2 -convergence we have achieved in step 2 we deduce that u fulfils the strong energy inequality (1.7). Finally, in the fourth step we verify the decay condition (1.9).

14

J. Wolf Existence of a weak solution to the approximate system

Let Φ ∈ C ∞ ([0, ∞)) be a non-increasing function, such that Φ ≡ 1 on [0, 1], Φ ≡ 0 in [2, ∞) and 0 ≤ −Φ0 ≤ 2. For ε > 0 we set Φε (τ ) := Φ(ετ ),

τ ∈ [0, ∞). 2

Let u0 ∈ H, f ∈ L1 (0, ∞; L2 (Ω)n ) and g ∈ L2 (Q)n . Using the method developed in [26] one gets a unique weak solution uε ∈ L2 (0, ∞; V) ∩ BC(0, ∞; H) to the system ∇ · uε = 0

in

Q,

(3.1)

∂t uε + ∇ · (uε ⊗ uε Φε (|uε |) − S(x, t,D(uε )))

(3.2)

= −∇pε + f − ∇ · g uε

∂Ω×(0,T )

in Q,

= 0,

uε (0) = u0

(3.3) (3.4)

in

Ω,

(3.5)

i.e. the following identity Z −uε · ∂t ϕ + (S(x, t; D(uε ) − uε ⊗ uε Φε (|uε |) : ∇ϕ dx dt Q Z Z = f · ϕ + g : ∇ϕ dx dt + u0 · ϕ(0) dx Q

(3.6)

Ω

holds for every ϕ ∈ C ∞ (Q)n with ∇ · ϕ = 0 and supp(ϕ) ⊂⊂ Ω × [0, ∞). Furthermore, using integration by parts from (3.6) it follows Z tZ 1 kuε (t)k2H + S(x, τ, D(uε )) : D(uε ) dx dτ 2 s Ω Z tZ 1 = f · uε + g : ∇uε dx dτ + kuε (s)k2H (3.7) 2 s Ω for all 0 ≤ s < t < ∞. In what follows let K denote a positive constant, whose numerical value may vary from line to line and its dependence will be specified if necessary. From (3.7) observing (III) together with Korn’s inequality and Young’s inequality one easily gets kuε kL∞ (0,∞;H) + k∇uε kL2 (Q) ≤ K. (3.8) Furthermore observing (II) using (3.8) yields kS(·, D(uε )kL2 (Q) ≤ K.

(3.9)

By virtue of Sobolev’s imbedding theorem using multiplicative inequalities from (3.8) one deduces kuε kLr (0,∞;Lρ (Ω)) ≤ K (3.10)

Existence of turbulent solutions

15

for all ρ ∈ [2, 2∗ ] and all 1 ≤ r ≤ ∞ such that 2 n n + = . r ρ 2

(3.11)

By means of reflexivity of the spaces under consideration there exist a sequence of positive numbers {εm } with εm → 0 as m → ∞ and functions u ∈ L2 (0, ∞; V) ∩ ˜ ∈ L2 (Q)n2 such that L∞ (0, ∞; H) and S uεm → u ∇uεm → ∇u ˜ S(·, D(uεm )) → S

weakly in weakly in weakly in

L2

n+2 n

(Q)n ,

(3.12)

2

L2 (Q)n , 2

L2 (Q)n

(3.13)

as m → ∞.

(3.14)

As it has been proved in [26] we have u ∈ Cw ([0, ∞); H) ∩ L2 (0, ∞; V) and ˜ = S(x, t, D(u)) S

a.e. in

Q.

Thus, u is a weak solution to (1.1)-(1.4). Simultaneously, using Lions’ compactness ¯ and 0 < T < ∞ argument one has for every G ⊂⊂ Ω uεm

G×(0,T )

→u

G×(0,T )

strongly in

L2 (G × (0, T ))n

as m → ∞.

(3.15)

In addition, using Riesz-Fischer’s theorem, from (3.15) (eventually passing to a ¯ subsequence) one may assume that for almost all T ∈ (0, ∞) and for every G ⊂⊂ Ω there holds uεm (T ) → u(T ) in L2 (G)n as m → ∞. (3.16) G

G

Strong convergence of uεm (T ) in L2 (Ω) To begin with, let us define the following subsets of Rn G` := {x ∈ Rn | 2`−1 < |x| < 2`+3 }, b ` := {x ∈ Rn | 2` < |x| < 2`+2 }, G

` ∈ N.

Let N ∈ N, such that G` ⊂⊂ Ω for all ` ≥ N . Obviously, the identity (3.6) implies Z ∞Z − uεm · ∂t ϕ dx dt 0 G` Z ∞Z + (h1,m,` + h2,m,` ) : ∇ϕ − f · ϕ dx dt = 0 (3.17) 0

for all ϕ ∈

C0∞ (G`

G` n

× (0, ∞)) with ∇ · ϕ = 0, where

h1,m,` := −uεm ⊗ uεm Φεm (|uεm |), h2,m,` := S(·, D(uεm )) − g

a.e. in

G` × (0, ∞).

16

J. Wolf

Hence, we are in a position to apply Corollary 2.7 with h1 = h1,m,` and h2 = h2,m,` , which provides unique functions p1,m,` ∈ L

n+2 n

(0, ∞; A

2

n+2 n

(G` )),

2

p2,m,` ∈ L (0, ∞; A (G` )), ∗

p3,` ∈ L1 (0, ∞; A2 (G` )), n+2 p˜h,m,` ∈ Cw (0, ∞; B˙ n (G` ))

such that Z ∞Z − 0

G`

with

p˜h,m,` (0) = 0,

uεm · ∂t ϕ dx dt Z ∞Z + (h1,m,` + h2,m,` ) : ∇ϕ − f · ϕ dx dt 0 G` Z ∞Z = (p1,m,` + p2,m,` + p3,` )∇ · ϕ + ∇˜ ph,m,` ∂t ϕ dx dt 0

(3.18)

G`

for all ϕ ∈ C0∞ (G` × (0, ∞))n and Z Z p1,m,` (t)∆φ dx = h1,m,` (t) : ∇2 φ dx, G` G` Z Z p2,m,` (t)∆φ dx = h2,m,` (t) : ∇2 φ dx, G` G` Z Z p3,` (t)∆φ dx = f (t) · ∇φ dx, ∀ φ ∈ C0∞ (G` ) G`

(3.19) (3.20) (3.21)

G`

for almost all t ∈ (0, ∞). It is readily seen that (2.25), (2.26), (2.27) and (2.28) imply kp1,m,` (t)k

L

n+2 n

(G` )

≤ Kkuεm (t)k2 2 n+2 L

n

(G` )

(3.22)

,

kp2,m,` (t)kL2 (G` ) ≤ KkS(·, t, D(uεm (t))) − g(t)kL2 (G` ) , `

kp3,` (t)kL2 (G` ) ≤ K2 kf (t)kL2 (G` ) ,

(3.23) (3.24)

and k˜ ph,m,` (t)k2 n+2 L

n

(G` )

≤ K2`

n2 +4 n+2

kuεm (t) − u0 k2L2 (G` )

+ Kkh1,m,` k2 n+2 L

+ K2

`n n−2 n+2

n

(G` ×(0,∞))

kh2,m,` k2L2 (G` ×(0,∞)) °Z t °2 2 +4 ° ° ` nn+2 ° f (s) ds° + K2 ° ° 0

L2 (G` )

(3.25)

Existence of turbulent solutions

17

for almost all t ∈ (0, ∞) 7 . In particular, observing (II) using (3.8) and (3.10) from (3.22), (3.23) and (3.24) one deduces kp1,m,` (t)k

L

n+2 n

(G` )

+ kp2,m,` (t)kL2 (G` ) ≤ K,

(3.26)

kp3,` (t)kL2 (G` ) ≤ K2`

(3.27)

for almost all t ∈ (0, ∞), while from (3.25) one infers k˜ ph,m,` (t)k2 n+2 L

n

(G` )

2

+4 ` nn+2

≤ K2

³

f (t)k2L2 (G` ) + 2−2` kuεm (t) − u0 k2L2 (G` ) + k˜

´ (3.28)

for all t ∈ (0, ∞), where Z

t

˜f (t) :=

f (s) ds,

t ∈ (0, ∞).

0

We wish to mention that in all these estimates the constant K depends neither on m nor on ` and also not on t ∈ (0, ∞). Next setting vm,` := uεm + ∇˜ ph,m,`

in G` × (0, ∞),

from (3.18) we deduce 0 vm,` + ∇ · (uεm ⊗ uεm Φεm (|uεm |) − S(x, t, D(uεm )) + g)

= f − ∇(p1,m,` + p2,m,` + p3,` ) in

G` × (0, ∞).

(3.29)

0 Here vm,` ∈ L2 (0, ∞; W 1, 2 (G` )n ) + L1 (0, T ; X(G` )∗ ) stands for the distributive time derivative of v, where

X(G` ) := {η ∈ L2 (G` )n | ∇ · η ∈ L2 (G` )} 8 . To be more precise, for every T ∈ (0, ∞) we have the following identity

7 8

Recall that G` = D2`−1 (cf. Section 2). Here ∇ · η ∈ L2 (G` ) means there exists w ∈ L2 (G` ), such that Z

Z wφ dx = −

G`

η · ∇φ dx G`

∀ φ ∈ C0∞ (G` ).

18

J. Wolf

Z 0

T

Z 0 hvm,` (t), ϕ(t)i ds +

Z

0

Z

T

T

=

Z G`

S(x, t, D(uεm )) : ∇ϕ dx dt

(uεm ⊗ uεm Φεm (|uεm |) + g) : D(ϕ) dx dt

0

G` Z TZ

+

(p1,m,` + p2,m,` + p3,` )∇ · ϕ dx dt 0

G`

Z

T

Z

+

f · ϕ dx dt 0

(3.30)

G`

for all ϕ ∈ L2 (0, ∞; W01, 2 (G` )n ) ∩ L∞ (0, ∞; X(G` )). Clearly, for every ψ ∈ C0∞ (G` ) the function vm,` ψ appears to be an appropriate test function in (3.30). However having in mind a-priori estimates for v on the complement of a large ball we are going to define an appropriate partition of unity subordinated to the covering {G` }`≥N . That is a family of smooth functions {ψ` }`≥N , such that (1) (2)

b` , supp(ψ` ) ⊂ G ∞ P ψ` ≡ 1 in {x ∈ R3 | |x| > 2N +1 }, `=N

(3)

|∇ψ` | ≤ K2−` .

Obviously, as supp(ψ` ) ∩ supp(ψ`+2 ) = ∅ for all ` ≥ N besides (2) there holds (20 )

ν P

in {x ∈ R3 | 2k+1 < |x| < 2ν+1 }

ψ` ≡ 1

∀ N ≤ k < ν ≤ ∞.

`=k

Now, choose T ∈ (0, ∞) such that (3.16) is valid. Then into (3.30) inserting ϕ = vm,` ψ` using integration by parts yields Z Z 1 1 2 |vm,` (T )| ψ` dx ≤ |u0 |2 ψ` dx 2 Ω 2 Ω Z TZ − S(x, t, D(uεm )) : D(vm,` ψ` ) dx dt (3.31) 0

Ω

Z

T

Z

+ 0

Z

Ω

T

(uεm ⊗ uεm Φεm (|uεm |)) : ∇(vm,` ψ` ) dx dt

Z

(p1,m,` + p2,m,` + p3,` )vm,` · ∇ψ` dx dt

+ 0

Z

Ω

T

Z

+

Z

T

Z

f · vm,` ψ` dx dt + 0

Ω

g : ∇(vm,` ψ` ) dx dt. 0

Ω

Existence of turbulent solutions

19

Observing (III) using the product rule one estimates Z

T

Z

0

Ω

S(x, t, D(uεm )) : D(vm,` ψ` ) dx dt Z

Z

T

≥−

κ2 ψ` dx dt 0

Ω T Z

Z

+ 0

Z

Ω

T

S(x, t, D(uεm )) : (vm,` ⊗ ∇ψ` ) dx dt

Z

+ 0

Ω

S(x, t, D(uεm )) : ∇2 p˜h,m,` ψ` dx dt.

Thus, from (3.31) it follows that 1 2

Z

Z 1 |vm,` (T )| ψ` dx ≤ |u0 |2 ψ` dx. 2 Ω Ω Z TZ Z + κ2 ψ` dx dt + 2

0

Ω

0

T

Z f · vm,` ψ` dx dt Ω

+ Im,` + IIm,` + IIIm,` + IVm,` + Vm,` ,

(3.32)

where Z

T

Z

Im,` := − Z

0

Ω

T

Z

IIm,` := − Z

0

T

Z

T

Z

IIIm,` :=

S(x, t, D(uεm )) : (vm,` ⊗ ∇ψ` ) dx dt,

Ω

S(x, t, D(uεm )) : ∇2 p˜h,m,` ψ` dx dt,

g : ∇(vm,` ψ` ) dx dt, 0

Z

Ω

IVm,` := 0

Z

Ω

T

(uεm ⊗ uεm Φεm (|uεm |)) : ∇(vm,` ψ` ) dx dt,

Z

Vm,` :=

(p1,m,` + p2,m,` + p3,` )vm,` · ∇ψ` dx dt 0

Ω

(m, ` ∈ N; ` ≥ N ). In order to estimate integrals Im,` -Vm,` we first state the following Lemma, which for reader’s convenience will be proved in the appendix of this paper. Lemma 3.1. For every 1 < r < ∞, 1 ≤ q ≤ ∞ and d ∈ N ∪ {0} there exists a positive constant γ0 , which depends only on r, q, d and n such that k∇d wkLq (Gb` ) ≤ γ0 {2` }−d−n/r+n/q kwkLr (G` )

∀ w ∈ B r (G` ).

(3.33)

20

J. Wolf Proof of Theorem 1.3 continued. Combining (3.33) (with r =

(3.28) implies

n+2 ) and n

k∇d p˜h,m,` (t)kLq (Gb` ) ≤ K{2` }−d+1−n/2+n/q × × (kuεm (t) − u0 kL2 (G` ) + k˜f (t)kL2 (G` ) + 2−` )

(3.34)

for almost all t ∈ (0, ∞). Thus, integrating both sides of the last inequality over (0, T ) taking into account (3.8) applying Minkowski’s inequality shows that k∇d p˜h,m,` kL∞ (0,∞;Lq (Gb` )) ≤ K{2` }−d+1−n/2+n/q .

(3.35)

(i) Using H¨older’s inequality together with the property (3) of the family {ψ` }`≥N one finds √ |Im,` | ≤ K2−` T kS(·, D(uεm )kL2 (Q) kuεm kL∞ (0,∞;L2 (Ω)) √ ph,m,` kL∞ (0,T ;L2 (Gb` )) . + K2−` T kS(·, D(uεm )kL2 (Q) k∇˜ Making use of the a-priori estimates (3.9), (3.10) and applying (3.35) (with d = 1 and q = 2) yields √ |Im,` | ≤ K2−` T . (ii) Similarly, estimating √ |IIm,` | ≤ T kS(·, D(uεm ))kL2 (Q) k∇2 p˜h,m,` kL∞ (0,T ;L2 (Gb` )) using (3.35) (with d = 2 and q = 2) gives

√ |IIm,` | ≤ K2−` T .

(iii) Arguing as in (i) and (ii) one gets √

−`

|IIIm,` | ≤ K2

Z

T

Z

T + 0

Ω

|g| |∇uεm |ψ` dx dt.

(iv) With help of the product rule one easily calculates Z TZ IVm,` = (uεm ⊗ uεm Φεm (|uεm |)) : ∇uεm ψ` dx dt 0

Ω

Z

T

Z

+ 0

Z

Ω

T

Z

+ 0

(1)

= IVm,` +

(uεm ⊗ uεm Φεm (|uεm |)) : ∇2 p˜h,m,` ψ` dx dt

Ω (2) IVm,`

(uεm ⊗ uεm Φεm (|uεm |)) : vm,` ⊗ ∇ψ` dx dt (3)

+ IVm,` .

Firstly, applying integration by parts one finds Z TZ (1) IVm,` = − (uεm ⊗ uεm Φεm (|uεm |)) : uεm ⊗ ∇ψ` dx dt. 0

Ω

(3.36)

Existence of turbulent solutions

21

Then, observing property (3) of the covering {ψ` }`≥N applying H¨older’s inequality along with (3.10) gives (1)

|IVm,` | ≤ K2−` kuεm k3L3 (Q) ≤ K2−` T 1−n/4 . Secondly, by the aid of H¨older’s inequality one gets (2)

|IVm,` | ≤ KT 1/3 kuεm k2L3 (Q) k∇2 p˜h,m,` kL∞ (0,T ;L3 (Gb` )) . Applying (3.35) (with d = 2 and q = 3) using (3.10) implies (2)

|IVm,` | ≤ K2−`(1+n/6) T 1−n/6 . Thirdly, using H¨older’s inequality together with property (3) of the covering {ψ` }`≥N one estimates (3)

|IVm,` | ≤ K2−` (kuεm k3L3 (Q) + T 1/3 kuεm k2L3 (Q) k∇˜ ph,m,` kL∞ (0,T ;L3 (Gb` )) ). Once more appealing to (3.35) (with d = 1 and q = 3) taking into account (3.10) gives (3)

|IVm,` | ≤ K(2−` T 1−n/4 + 2−`(1+n/6) T 1−n/6 ). (1)

(2)

(3)

Thus, inserting the above estimates of IVm,` , IVm,` and IVm,` into (3.36) taking √ into account T 1−n/4 ≤ T + 1 for n = 2, 3, 4 shows that √ |IVm,` | ≤ K(2−` T + 1 + 2−`(1+n/6) T 1−n/6 ). (v) With help of H¨older’s inequality one easily estimates |Vm,` | ≤ c 2−` kp1,m,` k

L

n+2 n

(G` ×(0,T ))

kvm,` k

L

n+2 2

b` ×(0,T )) (G

−`

+ c 2 kp2,m,` kL2 (G` ×(0,T )) kvm,` kL2 (Gb` ×(0,T )) Z TZ + p3,` vm · ∇ψ` dx dt. 0

Ω

Using (3.22), (3.23) and the a priori estimate (3.10) along with (3.35) gives √ |Vm,` | ≤ K2−` T + 1 +

Z

T

Z p3,` vm · ∇ψ` dx dt.

0

Ω

Let k ∈ N with k ≥ N be arbitrarily chosen. Inserting the estimates for Im,` − Vm,` into (3.32) summing up the resulting inequalities from ` = k to ∞

22

J. Wolf

making use of the property (20 ) of the partition of unity {ψ` }`≥N gives ∞ Z X |vm,` (T )|2 ψ` dx `=k

Ω

≤ K(2 Z

√

−k

Z T +1 + 2

−k(1+n/6)

Z

T

Z

{|x|≥2k }

0

T

κ2 dx dt +

+

∞ Z X `=k

T

0

0

Z

T

1−n/6

)+ {|x|>2k }

Z

|u0 |2 dx

|g| |∇uεm | dx dt

{|x|≥2k }

f · vm,` ψ` dx dt Ω

+

∞ Z X `=k

T

Z p3,` vm,` · ∇ψ` dx dt. (3.37)

0

Ω

1. Estimation of the right hand side of (3.37) from above. a) With help of Cauchy-Schwarz’s inequality using (3.8) results Z

T

0

Z {|x|≥2k }

|g| |∇uεm | dx dt ≤ KkgkL2 (0,T ;L2 (Rn \B2k )) .

(3.38)

b) Next, we claim that the two sums on the right of (3.37) are convergent. To prove this fact we proceed as follows. From (3.34) with d = 1, q = 2 it follows Z |∇˜ ph,m,` (t)|2 dx ≤ K(kuεm (t) − u0 k2L2 (G` ) + k˜ f (t)k2L2 (G` ) + 2−2` ) (3.39) b` G

for almost all t ∈ (0, ∞). Taking the sum on both sides of (3.34) from ` = k to ∞ recalling that G` ∩ G`+4 = ∅ for every ` ≥ N using Minkowski’s inequality together with (3.8) yields ∞ Z X |∇˜ ph,m,` (t)|2 dx ≤ K (3.40) `=k

b` G

for almost all t ∈ (0, ∞), where K = const > 0 depending neither on m nor on t ∈ (0, ∞). Now, define ∞ X Ψk := ψ` in Rn . `=k

Recalling the definition of vm,` one calculates ∞ Z T Z X f · vm,` ψ` dx dt `=k

0

Ω

Z

T

Z

= 0

Ω

f · uεm Ψk dx dt +

∞ Z X `=k

T

Z f · ∇˜ ph,m,` ψ` dx dt. (3.41)

0

Ω

Existence of turbulent solutions

23

Firstly, by means of Cauchy-Schwarz’s inequality one estimates ¯Z ¯ ¯ ¯

Z

T

0

Ω

¯ ¯ f · uεm Ψk dx dt¯¯ ≤ Kkf kL1 (0,T ;L2 (Rn \B2k )) .

(3.42)

Secondly, applying Cauchy-Schwarz’s inequality for sequences in l2 and recalling that supp(ψ` ) ∩ supp(ψ`+2 ) = ∅ for all ` ≥ N using (3.40) yields ¯ ∞ ¯Z X ¯ ¯ ¯ f (t) · ∇˜ ph,m,` (t) ψ` dx¯¯ ¯ Ω

`=k

≤ 2kf (t)kL2 (Rn \B2k )

µX ∞ Z b` G

`=k

¶ 12 |∇˜ ph,m,` (t)|2 dx

≤ Kkf (t)kL2 (Rn \B2k ) for almost all t ∈ (0, ∞). Integrating both sides of the last estimate over the interval (0, T ) with respect to t one arrives at ¯ ∞ Z ¯X ¯ ¯

Z

T

0

`=k

Ω

¯ ¯ f · ∇˜ ph,m,` ψ` dx dt¯¯ ≤ Kkf kL1 (0,T ;L2 (Rn \B2k )) .

(3.43)

Estimating right hand side of (3.41) by (3.42) and (3.43) gives ¯ ∞ Z ¯X ¯ ¯ `=k

T

Z

0

Ω

¯ ¯ f · vm,` ψ` dx dt¯¯ ≤ Kkf kL1 (0,T ;L2 (Rn \B2k )) .

(3.44)

c) To estimate the second sum on the right hand side of (3.37) we argue as follows. Clearly, by the definition of vm,` we have ∞ Z X `=k

Z

T

p3,` vm,` · ∇ψ` dx dt

0

Ω

Z

T

Z

= 0

Ω

p3,k uεm · ∇Ψk dx dt +

∞ Z X `=k

0

T

Z p3,` ∇˜ ph,m,` · ∇ψ` dx dt. (3.45) Ω

b k applying Noticing that |∇Ψk | ≤ K2−k in Rn and |∇Ψk | ≡ 0 in Rn \ G Cauchy-Schwarz’s inequality together with the a-priori estimate (3.24) yields ¯Z ¯ ¯ ¯

0

T

Z Ω

p3,k uεm

¯ ¯ · ∇Ψk dx dt¯¯ ≤ Kkf k2L1 (0,∞;L2 (Gb

k ))

.

(3.46)

24

J. Wolf

Moreover, applying Cauchy-Schwarz’s inequality for sequences in l2 using (3.40) and (3.24) one finds ¯X ¯ Z Z ¯ ∞ T ¯ ¯ ¯ p (t)∇˜ p (t) · ∇ψ dx dt 3,` h,m,` ` ¯ ¯ `=k

0

Ω

Z

T

≤ K

µX ∞

0

−2`

2

kp3,` (t)k2L2 (G` )

¶ 12 dt.

`=k

≤ Kkf kL1 (0,T ;L2 (Rn \B2k−1 )) . Combining (3.45), (3.46) and (3.47) implies ∞ Z T Z X p3,` vm,` · ∇ψ` dx dt ≤ Kkf k2L1 (0,∞;L2 (Rn \B `=k

0

Ω

2k−1

)) .

(3.47)

(3.48)

2. Estimation of the left hand side of (3.37) from below. Calculating ∞ X

|vm,` (T )|2 ψ` = |uεm (T )|2 Ψk

`=k

+

∞ X

|∇˜ ph,m,` (T )|2 ψ` + 2

`=k

∞ X

uεm (T ) · ∇˜ ph,m,` (T )ψ`

`=k

almost everywhere in Ω making use of the property (20 ) of the family {ψ` }`≥N it follows that ∞ Z X |vm,` (T )|2 ψ` dx `=k

Ω

Z 2

≥ |x|≥2k+1

|uεm (T )| dx + 2

∞ Z X `=k

Ω

uεm (T ) · ∇˜ ph,m,` (T )ψ` dx.

(3.49)

Estimating the left hand side of (3.37) from below by (3.49) and the right hand side of (3.37) from above by (3.44) and (3.48) yields Z Z |uεm (T )|2 dx ≤ |u0 |2 dx {|x|>2k+1 }

{|x|>2k }

√

−k

+ K(2

T + 1 + 2−k(1+n/6) T 1−n/6 )

+ Kkκ2 kL2 (Rn \B2k−1 ×(0,∞)) + KkgkL2 (Rn \B2k−1 ×(0,∞)) + Kkf kL1 (0,T ;L2 (Rn \B2k−1 )) ∞ Z X −2 uεm (T ) · ∇˜ ph,m,` (T )ψ` dx. `=k

(3.50)

Ω

Now, it only remains to estimate the last term on the right of (3.50). For, take a sequence {η ν } in Dσ (Ω) which converges to uεm (T ) in L2 (Ω)n as ν → ∞.

Existence of turbulent solutions

25

Using (3.40) applying Cauchy-Schwarz’s inequality in l2 recalling G` ∩ G`+4 = ∅ for all ` ≥ N one easily deduces ° °X ° ∞ ° ° ph,m,` (T )ψ` ° ≤ Kkuεm (T ) − η ν kL2 (Ω) . (uεm (T ) − η ν ) · ∇˜ ° ° L2 (Ω)

`=k

Hence, ∞ X

η ν · ∇˜ ph,m,` (T )ψ` →

`=k

∞ X

uεm (T ) · ∇˜ ph,m,` (T )ψ`

in L2 (Ω)

as

ν → ∞.

`=k

Next, define Z

T

p˜1,m,` (x) :=

p1,m,` (x, t) dt, Z

0 T

p˜2,m,` (x) :=

p2,m,` (x, t) dt, Z

0 T

p˜3,` (x) :=

p3,` (x, t) dt, Z

0 T

˜ 1,m,` (x) := h

h1,m,` (x, t) dt, Z

0 T

˜ 2,m,` (x) := h

h2,m,` (x, t) dt,

x ∈ G`

` ≥ N.

0

As an immediate consequence of (3.19), (3.20) and (3.21) we have Z Z ˜ 1,m,` : ∇2 φ dx, p˜1,m,` ∆φ dx = h G` G` Z Z ˜ 2,m,` : ∇2 φ dx, p˜2,m,` ∆φ dx = h G` G` Z Z ˜ p˜3,` ∆φ dx = f (T ) · ∇φ dx ∀ φ ∈ C0∞ (G` ). G`

(3.51) (3.52) (3.53)

G`

With help of Minkowski’s inequality and H¨older’s inequality making use of (3.10) yields ˜ 1,m,` kL2 (G ) ≤ kh1,m,` kL1 (0,T ;L2 (G )) kh ` ` ≤ kuεm k2L2 (0,T ;L4 (G` )) ≤ T 1−n/4 kuεm k2L8/n (0,T ;L4 (G` )) ≤ KT 1−n/4 9 . Using H¨older’s inequality taking into account (3.9) gives ˜ 2,m,` kL2 (G ) ≤ kS(·, D(uε )) + gkL2 (G ×(0,T )) ≤ K. kh m ` ` 9

Notice that

n n 2n + = . 8 4 2

26

J. Wolf

Thus, owing to Lemma 2.3 it is readily seen that p˜1,m,` , p˜2,m,` , p˜3,` ∈ L2 (G` ). Furthermore, we have the following a-priori estimates √ k˜ p1,m,` kL2 (G` ) + k˜ p2,m,` kL2 (G` ) ≤ K T + 1,

(3.54)

k˜ p3,` kL2 (G` ) ≤ K2` kf kL1 (0,∞;L2 (G` )) .

(3.55)

By an inspection of the proof of Theorem 2.6 one gets a function p˜m ∈ L2loc (Ω\ B2N −1 ), such that for all ` ∈ N, ` ≥ N there holds p˜1,m,` + p˜2,m,` + p˜3,` + p˜h,m,` (T ) = p˜m − (˜ pm )G`

a.e. in

G` .

(3.56)

In addition, using (3.54), (3.55) and (3.34) (with d = 0, q = r = 2) we get the a-priori estimate k˜ pm − (˜ pm )Gk kL2 (Gk )

√ ≤ K2k (kuεm (T ) − u0 kL2 (Gk ) + kf kL1 (0,∞;L2 (Gk )) + 2−k T + 1).

(3.57)

Applying integration by parts using (3.56) one calculates ∞ Z X `=k

Z

Ω

η ν · ∇˜ ph,m,` (T )ψ` dx = − +

∞ Z X `=k

Ω

Ω

p˜m η ν ·

∞ X

∇ψ` dx

`=k

(˜ p1,m,` + p˜2,m,` + p˜3,` )η ν · ∇ψ` dx.

Thanks to (1) and (20 ) verifying that

∞ P

(3.58)

∇ψ` ≡ 0 in {|x| > 2k+1 }) yields

`=k

Z − Ω

p˜m η ν ·

∞ X

Z ∇ψ` dx = −

`=k

bk G

(˜ pm − (˜ pm )Gk )η ν · ∇(ψk + ψk+1 ) dx.

Observing property (3) of the partition of unity {ψ` }`≥N using Cauchy-Schwarz’s inequality and applying (3.57) one obtains ¯Z ¯ ∞ X ¯ ¯ ¯ ¯ p ˜ η · ∇ψ dx m ν ` ¯ ¯ Ω

`=k

√ ≤ Kkη ν kL2 (Gk ) (kuεm (T ) − u0 kL2 (Gk ) + kf kL1 (0,∞;L2 (Gk )) + 2−k T + 1).

Existence of turbulent solutions

27

On the other hand, using Cauchy-Schwarz’s inequality for sequences in l2 together with (3.54) and (3.55) it follows that ¯X ¯ Z ¯ ∞ ¯ ¯ (˜ p1,m,` + p˜2,m,` + p˜3,` )η ν · ∇ψ` dx¯¯ ¯ `=k

Ω

µX ∞

≤ kη ν k

L2 (Ω)

−2`

2

`=k √ −k

≤ Kkη ν kL2 (Ω) (2

k˜ p1,m,` + p˜2,m,` +

p˜3,` )k2L2 (G` )

¶ 12

T + 1 + kf kL1 (0,∞;L2 (Gk )) ).

Now, estimating the right hand side of (3.58) by the aid of the two estimates we have just obtained and afterwards carrying out the passage to the limit ν → ∞ leads to ¯X ¯ Z ¯ ∞ ¯ ¯ uεm (T ) · ∇˜ ph,m,` (T )ψ` dx¯¯ ¯ `=k

Ω

≤ K(kuεm (T )k2L2 (Gk ) + ku0 k2L2 (Gk ) + kf kL1 (0,T ;L2 (Rn \B2k−1 )) ) √ + K2−k T + 1. Hence using the estimate above from (3.50) it follows Z |uεm (T )|2 dx {|x|>2k+1 } Z Z ≤ K |uεm (T )|2 dx + K

{|x|>2k−1 }

Gk

|u0 |2 dx

(3.59)

+ Kkκ2 kL1 (Rn \B2k ×(0,∞)) + KkgkL2 (Rn \B2k ×(0,∞)) + Kkf kL1 (0,T ;L2 (Rn \B2k−1 )) √ + K(2−k T + 1 + 2−k(1+n/6) T 1−n/6 ). Next, let α > 0 be arbitrarily chosen. Clearly, one can select k ≥ N such that Z Z |u(T )|2 dx + |u0 |2 dx {|x|>2k−1 }

Gk

+ kκ2 kL1 (Rn \B2k ×(0,∞)) + kgkL2 (Rn \B2k ×(0,∞)) + kf kL1 (0,T ;L2 (Rn \B2k−1 )) √ α + (2−k T + 1 + 2−k(1+n/6) T 1−n/6 ) ≤ 4K and

Z |u(T )|2 dx ≤ {|x|>2k−1 }

α . 4(K + 1)

(3.60)

(3.61)

28

J. Wolf

By the aid of (3.16) one can choose m0 ∈ N such that Z Z α 2 , |uεm (T )| dx − |u(T )|2 dx ≤ 4K G Gk ¯Z k ¯ Z ¯ ¯ α 2 2 ¯ |uεm (T )| dx − |u(T )| dx¯¯ ≤ ¯ 4 Ω∩B2k+1 Ω∩B2k+1

(3.62) (3.63)

for all m ≥ m0 . Let m ≥ m0 . With help of the triangular inequality making use of (3.59) along with (3.60), (3.61), (3.62) and (3.63) one obtains ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯kuεm (T )k2L2 (Ω) − ku(T )k2L2 (Ω) ¯ ≤ ¯kuεm (T )k2L2 (Ω∩B k+1 ) − ku(T )k2L2 (Ω∩B k+1 ) ¯ 2

+ ≤

2

kuεm (T )k2L2 (R3 \B k+1 ) 2

+

ku(T )k2L2 (R3 \B k+1 ) 2

3 α + (K + 1)ku(T )k2L2 (R3 \B k−1 ) ≤ α. 2 4

Thus, kuεm (T )k2L2 (Ω) → ku(T )k2L2 (Ω)

as

m → ∞.

Moreover recalling that uεm (T ) converges weakly to u(T ) in L2 (Ω)n as m → ∞ one infers uεm (T ) → u(T )

strongly in L2 (Ω)n

as

m → ∞.

The Strong Energy Inequality Let us begin by defining the set J ⊂ [0, ∞) of all real numbers for which (3.16) is true. Clearly L1 ([0, ∞) \ J ) = 0. As it has been proved in the previous step there holds kuεm (t)kH → ku(t)kH

as

m→∞

∀t ∈ J.

(3.64)

Now, let s, t ∈ J with 0 < s < t < ∞ be fixed. Owing to (III) one can easily check that Z tZ S(·, D(uεm )) : D(uεm ) dx dτ s Ω Z tZ = S(·, D(u)) : D(uεm ) dx dτ s Ω Z tZ + (S(·, D(uεm )) − S(·, D(u))) : D(uεm ) dx dτ s Ω Z tZ S(·, D(u)) : D(uεm ) dx dτ ≥ s Ω Z tZ + (S(·, D(uεm )) − S(·, D(u))) : D(u) dx dτ. s

Ω

Existence of turbulent solutions

29

Thus, from (3.7) one deduces that Z tZ 1 S(·, D(u)) : D(uεm ) dx dτ kuεm (t)k2H + 2 s Ω Z tZ 1 ≤ f · uεm + g : ∇uεm dx dτ + kuεm (s)k2H 2 s Ω Z t (S(·, D(uεm )) − S(·, D(u))) : D(u) dx dτ. +

(3.65)

s

By means of (3.14), (3.13) and (3.64) we are in a position to carry out the passage to the limit m → ∞ on both sides of (3.65). This implies 1 ku(t)k2H + 2

Z tZ s

Ω

S(·, D(u)) : D(u) dx dτ Z tZ 1 ≤ f · u + g : ∇u dx dτ + ku(s)k2H . 2 s Ω

(3.66)

Whence, u is a turbulent solution. Decay for ku(T )kL2 (Ω) as T → ∞ First we define

¯ n o n+2 ¯ J˜ := T ∈ J ¯ u(T ) ∈ L2 n (Ω)n .

n+2 Recalling that u ∈ L2 n (Q)n shows that L1 ([0, ∞) \ J˜) = 0. Moreover, in J˜ there exists a sequence of numbers 1 < T1 < T2 < . . . < Tj < . . . with Tj → ∞ as j → ∞ such that Z n+2 1 |u(Tj )|2 n dx ≤ ∀ j ∈ N. (3.67) Tj ln(1 + Tj ) Ω

Let 0 < δ < 1 be a number which will be specified below. Let j ∈ N. Clearly, there exists a unique integer k = k(j) fulfilling p 2k−1 ≤ δ −1 Tj < 2k .

(3.68)

Without loss of generality we may assume that k − 1 > N , i.e. Gk ⊂ Ω. Let ` ≥ k. Arguing as in step 2 from Corollary 2.7 one gets unique functions p1,` ∈ L

n+2 n

2

(0, ∞; A

n+2 n

(G` )),

2

p2,` ∈ L (0, ∞; A (G` )), ∗

p3,` ∈ L1 (0, ∞; A2 (G` )), n+2 p˜h,` ∈ Cw (0, ∞; B˙ n (G` ))

with

p˜h,` (0) = 0,

30

J. Wolf

such that Z ∞Z − 0

G`

u · ∂t ϕ dx dt Z ∞Z + u ⊗ u + S(·, D(u(t))) − g) : ∇ϕ − f · ϕ dx dt 0 G` Z ∞Z = (p1,` + p2,` + p3,` )∇ · ϕ + ∇˜ ph,` · ∂t ϕ dx dt 0

(3.69)

G`

for all ϕ ∈ C0∞ (G` × (0, ∞))n Z p1,` (t)∆φ dx G Z ` p2,` (t)∆φ dx G Z ` p3,` (t)∆φ dx

and Z = (u(t) ⊗ u(t)) : ∇2 φ dx, G` Z = (S(·, D(u(t))) + g(t)) : ∇2 φ dx, G` Z = f (t) · ∇φ dx

G`

(3.70) (3.71) (3.72)

G`

for almost all t ∈ (0, ∞), for all φ ∈ C0∞ (G` ). Observing, (III) taking into account (3.16), (3.14) (3.12) and the fact b ` )) in Ln (0, T ; W 2, n (G

ph,m,` → ph,`

as

m→∞

(3.73)

carrying out the passage to the limit on both sides of (3.32) as m → ∞ (replacing T by Tj therein) one arrives at Z Z 1 1 2 |v` (Tj )| ψ` dx ≤ |u0 |2 ψ` dx. 2 Ω 2 Ω Z Tj Z Z Tj Z + κ2 ψ` dx dt + f · v` ψ` dx dt 0

Ω

0

Ω

+ I` + II` + III` + IV` + V` , where v` := u + ∇˜ ph,` and Z I` := − Z

Tj

Z S(x, t, D(u))) : (v` ⊗ ∇ψ` ) dx dt,

0

Ω

Tj

Z

S(x, t, D(u)) : ∇2 p˜h,` ψ` dx dt,

II` := − Z

0 Tj

Z

III` :=

Ω

g : ∇(v` ψ` ) dx dt, 0

Z

Ω

Tj

Z

(u ⊗ u) : ∇(v` ψ` ) dx dt,

IV` := 0

Z

Ω

Tj

Z

V` :=

(p1,` + p2,` + p3,` )v` · ∇ψ` dx dt 0

Ω

(3.74)

Existence of turbulent solutions

31

(` ∈ N; ` ≥ N ). By an analogous reasoning we have used to get (3.59) one proves the a-priori estimate Z |u(Tj )|2 dx Z Z ≤ K |u(Tj )|2 dx + K

{|x|>2k+1 }

{|x|>2k−1 }

B2k+3 ∩Ω

|u0 |2 dx

(3.75)

+ Kkκ2 kL1 (Rn \B2k ×(0,∞)) + KkgkL2 (Rn \B2k ×(0,∞)) + Kkf kL1 (0,∞;L2 (Rn \B2k−1 )) p 1−n/6 + K(2−k Tj + 2−2k(1−n/6) Tj ). Using H¨older’s inequality taking into account (3.68) and (3.67) one estimates Z

µ Z 2k |u(Tj )| dx ≤ K 2

2 n+2 n

2

|u(Tj )|

n ¶ n+2 dx

B2k+3 ∩Ω

B2k+3 ∩Ω

≤

K . + Tj ))2n/(n+2)

δ 2n/(n+2) (ln(1

(3.76)

Combining (3.75) and (3.76) one gets a positive constant K1 such that 2 2 2 ku(Tj )kL 2 (Ω) = ku(Tj )kL2 (Rn \B ) + ku(Tj )kL2 (B2k+1 ∩Ω) 2k+1 Z ≤ K1 |u0 |2 dx

(3.77)

{|x|>2k−1 }

+ K1 kκ2 kL1 (Rn \B2k ×(0,∞)) + K1 kgkL2 (Rn \B2k ×(0,∞))) + K1 kf kL1 (0,∞;L2 (Rn \B2k−1 )) +

K1 + K1 δ 2/3 . δ 2n/(n+2) (ln(1 + Tj ))2n/(n+2)

Let 0 < α < 1 be arbitrarily chosen. Let δ > 0 be fixed such that K1 δ 2/3 ≤

α . 4

(3.78)

32

J. Wolf

Clearly, there exists j1 ∈ N, such that Z |u0 |2 dx {|x|>2k(j)−1 }

+ kκ2 kL1 (0,∞;L2 (Rn \B2k(j) )) + kgkL2 (0,∞;L2 (Rn \B2k(j) ))

(3.79)

+ kf kL1 (0,∞;L2 (Rn \B2k(j)−1 )) +

1 α ≤ 4K1 δ 2n/(n+2) (ln(1 + Tj ))2n/(n+2)

for all j ≥ j1 . Hence, using (3.78) and (3.79) from (3.77) one gets α 2

ku(Tj )k2L2 (Ω) ≤

∀ j ∈ N, j ≥ j1 .

(3.80)

Owing to the strong energy inequality we have for all T ∈ J , T > Tj ku(T )k2L2 (Ω) ≤ ku(Tj )k2L2 (Ω) Z TZ −2 (S(x, t, D(u)) − g) : ∇u dx dt Tj

Ω

Z

T

Z

+2

(3.81)

f · u dx dt Tj

Ω

≤ ku(Tj )k2L2 (Ω) + Kk∇ukL2 (Ω×(Tj ,∞)) + Kkf kL1 (Tj ,∞;L2 (Ω)) . Finally choosing j2 ∈ N such that Kk∇ukL2 (Ω×(Tj ,∞)) + Kkf kL1 (Tj ,∞;L2 (Ω)) ≤

α 2

for all j ≥ j2 using (3.80) and (3.81) shows that ku(T )k2L2 (Ω) ≤ α

∀ T ≥ max{Tj1 , Tj2 }.

Hence the assertion of the theorem is completely proved.

¤

Appendix A. Caccioppoli Inequalities for Harmonic Functions Proof of Lemma 3.1. 1. L∞ -estimates for harmonic functions. Let V ⊂⊂ U be two open bounded sets with ρ := dist(V, ∂U ). Let p ∈ L1 (U ) being harmonic in U . Then Z 1 sup |w(x)| ≤ |w| dx. (A.1) Ln (B1 )ρn U x∈V

Existence of turbulent solutions

33

Indeed, let x ∈ V . Then Bρ (x) ⊂ U . By the mean value property of harmonic functions we have ¯Z ¯ Z ¯ ¯ 1 |w(x)| = ¯¯ − w(y) dy ¯¯ ≤ |w| dy. Ln (B1 )ρn Bρ (x)

U

Whence (A.1) 2. The case ` = 1. Let w ∈ B r (G1 ). We define the sets ¯7 n o ¯ G0 := x ∈ Rn ¯ < |x| < 5 , 8 ¯ n o 00 n¯ 5 G := x ∈ R ¯ < |x| < 7 . 8 1 b 1 , G0 ) = using (A.1) together with Caccioppli’s inequality yields Verifying dist(G 8 sup |∇d w(x)| ≤ Kk∇d wkL2 (G0 ) ≤ K sup |w(x)|. x∈G00

b1 x∈G

where K = const > 0 depending only on n and d. 1 Next, noticing that dist(G00 , G1 ) = once more appealing to (A.1) gives 8 Z 8n sup |w(x)| ≤ |w| dx. Ln (B1 ) G1 x∈G00 Thus, combining the last two estimates applying H¨older’s inequality implies µZ ¶ r1 µZ ¶ q1 r d q ≤ K − |w| dx . − |∇ w| dx b1 G

G1

This completes the proof of (3.33) for ` = 1. 3. The case ` ∈ N. Let w ∈ B r (G` ). Using an appropriate changing of coordinates applying the transformation formula of Lebesgue’s integral gives µZ ¶ q1 µZ ¶ r1 − |∇d w|q dx ≤ K2−`d − |w|r dx . b` G

G`

Whence, (3.33).

¤

References [1] R. A. Adams, Sobolev spaces. Academic Press, Boston 1978. [2] G. K. Batchelor, An introduction to fluid mechanics. Cambridge Univ. Press, Cambridge 1967. [3] W. Borchers and T. Miyakawa, L2 decay for the Navier-Stokes flows in unbounded domains with application to exterior stationary flows. Arch. Rat. Mech. Anal. 118 (1992), 273-295.

34

J. Wolf [4] A. P. Calder´ on and A. Zygmond, On the existence of certain singular integrals. Acta Math. 88 (1952), 85-139. [5] R. Farwig, H. Kozono and H. Sohr, An Lq -approach to Stokes and Navier-Stokes equations in general domains. (to appear). [6] G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I: Linearized steady problems. Springer-Verlag, New York 1994. [7] J. G. Heywood, The Navier-Stokes flow in unbounded domains: On the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29 (1980), 639-681. [8] H. Koch, V. A. Solonnikov and A. Vsevolod, Lq -estimates of the first order derivatives of solutions to the nonstationary Stokes problem. Birman, M. Sh. (Ed.) et al., Nonlinear problems in mathematical physics and related topics I. In honour of Professor O. A. Ladyzhenskaya, New York: Kluwer Academic/Plenum Publ. Int. Math. Ser., N.Y. 1 (2002), 203-218. [9] H. Kozono and T. Ogawa, Two dimensional Navier-Stokes flow in unbounded domains. Math. Ann. 297 (1993), 1-31. [10] H. Kozono, T. Ogawa and H. Sohr, Asymptotic behaviour in Lr for turbulent solutions of the Navier-Stokes equations in exterior domains. Manuscr. Math. 74 (1992) 253-275. [11] H. Lamb, Hydrodynamics. 6th ed., Cambridge Univ. Press 1945. [12] J. Leray, Sur le mouvement d’un liquide visqeux emplissant l’espace. Acta Math. 63 (1934), 193-248. [13] J. Lions, Quelques m´ethodes de r´esolution des probl`emes aux limites non lin´eaires. Dunod, Gauthier-Villars, Paris 1969. [14] P. Maremonti, Some results on the asymptotic behaviour of Hopf weak solutions to the Navier-Stokes equations in unbounded domains. Math. Z. 210 (1992), 1.-22. [15] K. Masuda, Weak solutions of the Navier-Stokes equations. Tohoku Math. J. 36 (1984), 623-646. [16] T. Miyakawa and H. Sohr, On energy inequality, smoothness and large time behaviour in L2 for weak solutions of the Navier-Stokes equations in exterior domains. Math. Z. 199 (1988), 455-478. [17] R. M¨ uller Das schwache Dirichletproblem in Lq f¨ ur den Bipotentialoperator in beschr¨ ankten und in Außengebieten. Bayreuther Mathematische Schriften 49 (1995), 115-211. [18] M. E. Schonbeck, L2 decay for weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 88 (1985), 209-222. [19] M. E. Schonbeck, Large time behavior of solutions of the Navier-Stokes equations. Comm. Partial Differential Equations. 11 (1986), 733-763. [20] J. Simon, On the existence of the pressure for solutions of the variational NavierStokes equations. Annali Mat. Pura Appl. 146 (1987), 65-96.

Existence of turbulent solutions

35

[21] G. C. Simader and H. Sohr The Dirichlet Problem for the Laplacian in bounded and unbounded domains. Pitman Research Notes in Mathematics Series, Vol. 360, Harlow: Addison Wesley Longman Ltd. (1996). [22] H. Sohr, W. von Wahl and M. Wiegner, Zur Asymptotik der Gleichungen von Navier-Stokes. Nachr. Akad. Wiss. G¨ ottingen, 3 (1986), 45-69. [23] V. A. Solonnikov, Stokes and Navier-Stokes equations in domains with noncompact boundaries. In ”Nonlinear partial differential equations and their applications”, Coll`ege de France Seminar, vol. IV, H. Br´ezis, J. L. Lions (editors), Pitman, Boston, London, 1983, 240-349. [24] S. Ukai, A solution formula for the Stokes equation in Rn + . Comm. Pure Appl. Math. 40 (1987), 611-621. [25] M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations in Rn . J. London Math. Soc. 35 (1987), 303-313. [26] J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity. J. Math. fluid Mech. 8 (2006), 1-35. J¨ org Wolf Humboldt-Universit¨ at zu Berlin Unter den Linden. 6 10099 Berlin Germany e-mail: [email protected]