PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0

SELF-SIMILARITY IN VISCOUS BOUSSINESQ EQUATION GRZEGORZ KARCH AND NICOLAS PRIOUX

Abstract. We study the existence and the asymptotic stability as the time variable escapes to infinity of self-similar solutions to the viscous Boussinesq equations posed in the whole three dimensional space.

1. Introduction In the natural or free convective transfer, the heat is transported between a solid surface and a fluid moving over it. Here, the fluid motion is entirely caused by the buoyancy forces arising from density changes that result from the temperature variations in the fluid. The fluid movement by the natural convection can be either laminar or turbulent. However, because of low velocities that usually exist in the natural convection, laminar flow occurs more frequently. By these reasons, we will focused ourselves on laminar natural convective flow. The reader is referred to the classical books [8, ch. 5, §56] as well to [1, ch. 4.2] for more details on models describing the heat transport in a fluid. Following [8] and putting all physical constants equal to 1, the initial value problem for the viscous Boussinesq equations, modeling the phenomena presented above, has the following form

(1.1)

vt + (v · ∇) v + ∇p = ∆v − βT + F, ∇ · v = 0, Tt + v · ∇T = ∆T, v(0, x) = v0 (x),

T (0, x) = T0 (x).

Here, the time variable t belongs to the interval ]0, ∞[, the space variable x lives in the whole space R3 , the vector v = (v1 (t, x), v2 (t, x), v3 (t, x)) is the unknown velocity of the fluid, and the scalar function T = T (t, x) is the unknown temperature. Moreover, F = F (t, x) is a given external force, the constant vector β ∈ R3 is proportional to the thermal expansion coefficient of the fluid and to the gravitational force. Finally, the functions v0 and T0 denote given initial conditions. Mathematical results on the existence and the uniqueness of solutions to system (1.1) can be found in [2, 6] and in the references therein. Here we note that (1.1) Received by the editors July 5, 2006. 2000 Mathematics Subject Classification. Primary 35Q30; Secondary 35B40; 76D05. Key words and phrases. Natural convection; Boussinesq system; self-similar solutions; large time asymptotics. The preparation of this paper by the first author was partially supported by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389. c

1997 American Mathematical Society

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GRZEGORZ KARCH AND NICOLAS PRIOUX

is invariant under the rescaling (1.2)

vλ (t, x) = λv(λ2 t, λx)

and Tλ (t, x) = λ3 T (λ2 t, λx)

for all λ > 0. Hence, the main goal of this paper is to construct self-similar solutions to system (1.1) that is solutions satisfying the scaling relation (1.3)

v(t, x) = λv(λ2 t, λx)

and T (t, x) = λ3 T (λ2 t, λx)

for all λ > 0, x ∈ R3 , and t > 0 as well as to study their role in the large time behavior of general solutions. We use the tools introduced in [4] where these questions were answered in the case of the Navier-Stokes equations posed in the whole space R3 and supplemented with small initial data belonging to the space PM2 , cf. (1.6) below. We begin our analysis by recalling the Leray projector P given formally by the formula Pv = v − ∇∆−1 (∇ · v). Hence, this is the pseudodifferential operator with −2 b the matrix symbol (P(ξ)) and we refer the reader to [3] for more j,k = δjk −ξj ξk |ξ| details and additional references. Using well-known properties of the projection P and the equalities (v · ∇) v = ∇·(v ⊗ v) and v ·∇T = ∇·(vT ) (being the immediate consequence of the equation ∇ · v = 0) we rewrite system (1.1) as follows vt − ∆v + P∇ · (v ⊗ v) = −βPT + PF,

Tt − ∆T + ∇ · (vT ) = 0.

Moreover, by the Duhamel principle, we obtain the following (formally equivalent to (1.1)) system of integral equations Z t Z t (1.4) v(t) = S(t)v0 + S(t − τ )PF (τ ) dτ − ∇S(t − τ )P(v ⊗ v)(τ ) dτ 0 0 Z t −β S(t − τ )PT (τ ) dτ 0 Z t T (t) = S(t)T0 − (1.5) ∇S(t − τ )(vT )(τ ) dτ. 0

Here, S(t) denotes the heat semigroup given as the convolution with the GaussWeierstrass kernel. In this paper, we study properties of solutions to the system of equations (1.4)-(1.5) in the spaces defined as follows. For a ≥ 0, we put ( ) 0 a 3 1 3 a b b (1.6) PM ≡ f ∈ S (R ), f ∈ L (R ), kf kPMa ≡ ess sup |ξ| f (ξ) < ∞ . loc

ξ∈R3

We construct solutions to system (1.4)-(1.5) (or system (1.1)) with the velocity v in the space
X = Cw [0, ∞[, PM2 supplemented with the norm kvkX = supτ >0 kv(τ )kPM2 and the temperature T belonging to  
0 a/2 Ya = T ∈ Cw [0, ∞[, PM , sup τ kT (τ )kPMa < ∞ τ >0

with the norm kT kYa = supτ >0 kT (τ )kPM0 + supτ >0 τ a/2 kT (τ )kPMa . We recall that Cw denotes the space of vector-valued functions which are weakly continuous in t (cf. [3]). In order to simplify notation, we denote the product of Banach spaces Z × Z × Z with the usual norm k(v1 , v2 , v3 )kZ×Z×Z = max{kv1 kZ , kv2 kZ , kv3 kZ } also by Z.

SELF-SIMILARITY IN VISCOUS BOUSSINESQ EQUATION

3

Our main result on the existence of solutions to system (1.1) reads as follows.
Theorem 1.1. Assume that v0 ∈ PM2 , T0 ∈ PM0 and F ∈ Cw [0, ∞[, PM0 . Fix 1 < a < 2. Suppose that |β| = max{|β1 |, |β2 |, |β3 |} < 1. There is γ > 0 such that if kv0 kPM2 + kT0 kPM0 + supτ >0 kF (τ )kPM0 < γ, then there exists a solution of system (1.4)-(1.5) in the space X × Ya corresponding to the initial datum (v0 , T0 ). Moreover, there exists another constant Γ > 0 such that this is the unique solution satisfying the condition kvkX + kT kYa < Γ. Remark 1.2. As we shall see below, γ = α1 (1 − |β|)2 (2α1 + α2 )−2 and Γ = (1 − |β|)(2α1 + α2 )−1 where the constants α1 and α2 are obtained in Lemmata 2.3 and 2.4, respectively. Now the standard procedure, based on the uniqueness of solutions from Theorem 1.1, allows us to deduce the existence of self-similar solutions to system (1.1). Corollary 1.3. Assume that v0 , T0 and F satisfy the assumptions of Theorem 1.1. Let, moreover, v0 be homogeneous of
degree -1, T0 homogeneous of degree -3, and F satisfies the relation λ3 F λ2 t, λx = F (t, x) for all λ > 0, x ∈ R3 , and t > 0. Then, the solution (v, T ) constructed in Theorem 1.1 is self-similar i.e. invariant under the rescaling (1.2). Remark 1.4. Note that the Dirac delta δ0 ∈ PM0 (R3 ) is a homogeneous distribution of degree −3. In our next theorem, we study the asymptotic stability of solutions in the sense proposed in [7] and developed in [4, 5]. Theorem 1.5. Assume that v0 , T0 , F and
v 0 , T 0 , F satisfy the assumptions of Theorem 1.1 and denote by (v, T ) and v, T the corresponding solutions of (1.4)(1.5). Suppose, moreover, that  lim kS(t)(v0 − v 0 )kPM2 + kS(t)(T0 − T 0 )kPM0 t→+∞ (1.7)  + kF (t) − F (t)kPM0 = 0. Then lim

t→∞







kv(t) − v(t)kPM2 + T (t) − T (t) PM0 + ta/2 T (t) − T (t) PMa = 0.

Remark 1.6. The PMa spaces seem to be exotic in the study of the nonlinear problem (1.1). However, it was shown in [5] that they appear in a very natural way in estimates of nonlinear terms like those in (1.4) and (1.5). Note, for example, that L1 (R3 ) (or more generally any finite measure on R3 ) is contained in PM0 . Moreover, repeating the interpolation argument from [5, Lemma 7.4] one can prove that for each a ∈ (3/2, 2), PM0 ∩ PMa ⊂ Lp (R3 ) for any p ∈ [2, 3/(3 − a)), which implies the corresponding regularity result for the temperature T = T (t, x). Finally, by the results from [5, Section 7], we immediately obtain that supt>0 t(1−3/q)/2 kv(t)kLq (R3 ) < ∞ for q > 3. Remark 1.7. We refer the reader to [7, Lemma 6.2] and [4, Remark following Theorem 5.1] for a discussion on the assumption (1.7). In particular, one can easily 1 3 ¯ ¯ Rshow that kS(t)(M δ0 − T0 )kPM0 → 0 as t → ∞ for any T0 ∈ L (R ) such that ¯ T (x) dx = M . R3 0

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GRZEGORZ KARCH AND NICOLAS PRIOUX

2. The existence of solutions The system of integral equations (1.4)-(1.5) can be written in the abstract form (2.1)

v = y1 + B1 (v, v) + LT, T = y2 + B2 (v, T ), Rt where y1 = S(t)v0 + 0 S(t − τ )PF (τ ) dτ , y2 = S(t)T0 , B1 (v, v) and LT are the third and fourth terms on the right hand side of equation (1.4), and B2 (v, T ) is the second term on the right hand side of equation (1.5). Solutions to the abstract system (2.1) can be constructed via the Banach fixed point algorithm and such problems were studied in [9]. Hence, we shall only sketch such a reasoning and we refer the reader to [9] for more details. Lemma 2.1. Given two Banach spaces X and Y, assume that the bilinear forms B1 : X × X → X and B2 : X × Y → Y satisfy the estimates kB1 (u, v)kX ≤ α1 kukX kvkX and kB2 (v, T )kY ≤ α2 kvkX kT kY with positive constants α1 and α2 , for all u, v ∈ X and T ∈ Y. Let the linear operator L : Y → Y be continuous and satisfy the inequality kLT kY ≤ ηkT kY with a constant η < 1. For every 2 (y1 , y2 ) ∈ X × Y such that ky1 kX + ky2 kY ≤ α1 (1 − η) /[2α1 + α2 ]2 , there exists a solution (v, T ) ∈ X × Y
to system (2.1). This is the unique solution satisfying the condition kvkX + kT kY < (1 − η)/(2α1 + α2 ). Proof. Let us choose arbitrary R ∈ ((1 − η)[2(α1 + α2 )]−1 , (1 − η)[2α1 + α2 ]−1 ). It suffices to show that the nonlinear operator Θ(v, T ) ≡ (y1 + B1 (v, v) + LT, y2 + B2 (v, T )) is the contraction on the closed ball B(0, R) = {(v, T ) ∈ X × Y : kvkX + kT kY ≤ R}. Here, we supplement the product space X × Y with the norm k(v, t)k ≡ kvkX + kT kY . First note that for every (v, T ) ∈ B(0, R) we have kΘ(v, T )k ≤ ky1 kX + α1 R2 + ηR + ky2 kY + α2 R2 . Now, observe that, by the smallness assumption on y1 and y2 , we obtain ky1 kX + ky1 kY ≤ (1 − η)R − (α1 + α2 )R2 . Indeed, the infimum of the right-hand side with 2 respect to R ∈ ((1−η)[2(α1 +α2 )]−1 , (1−η)[2α1 +α2 ]−1 ) equals α1 (1 − η) /[2α1 + α2 ]2 . Hence, kΘ(v, T )k ≤ R. To prove that Θ is the contraction in B(0, R), we proceed as follows. For any (v, T ), (e v , Te) ∈ B(0, R), we estimate kΘ(v, T ) − Θ(e v , Te)k

≤ 2α1 Rkv − vekX + ηkT − TekY +α2 R(kv − vekX + kT − TekY ) ≤ (R(2α1 + α2 ) + η)k(v, T ) − (e v , Te)k.

The proof is complete because R(2α1 + α2 ) + η < 1. Due to Lemma 2.1, we should estimate each term appearing on the right-hand side of equations (1.4) and (1.5).
Lemma 2.2. For every u0 ∈ PM2 , T0 ∈ PM0 , and F ∈ Cw [0, ∞[, PM0 we Rt have S(t)v0 ∈ X and 0 S(t − τ )PF (τ ) dτ ∈ X with the estimates (2.2)

Z t



kS(t)v0 kPM2 ≤ kv0 kPM2 and S(t − τ )PF (τ ) dτ kF (t)kPM0 .

2 ≤ sup t>0 0

PM

Moreover, for any a ≥ 0, it follows that S(·)T0 ∈ Ya and the following estimates hold true (2.3)

kS(t)T0 kPM0 ≤ kT0 kPM0 ,

and

ta/2 kS(t)T0 kPMa ≤ C kT0 kPM0 .

SELF-SIMILARITY IN VISCOUS BOUSSINESQ EQUATION

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Proof. For the proofs of (2.2), the reader is referred to [4, Lemmata 4.2 and 4.3]. The reasoning in the case of the first inequality in (2.3) is completely analogous, hence, we skip it. By the definition of the norm in PMa , we get
a/2 −t|ξ|2 e = C kT0 kPM0 , (2.4) ta/2 kS(t)T0 kPMa ≤ kT0 kPM0 sup t|ξ|2 ξ∈R3

which completes the proof that S(·)T0 ∈ Ya . The estimates of B1 , B2 and L required by Lemma 2.1 are shown in the following three lemmata. The bilinear form B1 is studied in the first of them. Lemma 2.3. There exists a constant α1 > 0 such that for every u, v ∈ X , we have

Z t   



≤ α sup ku(t)k S(t − τ )P∇ · (u ⊗ v)(τ ) dτ sup kv(t)k 1 PM2 PM2

2

t>0 t>0 0

PM

for all t > 0 This is the well-known estimate which appeared in the study of the Navier-Stokes system. Hence, the reader is referred to [4, Prop. 4.1] for the proof of Lemma 2.3. Now, we deal with the bilinear form B2 . Lemma 2.4. Assume that 1 < a < 2. There exists a constant α2 > 0 such that for every v ∈ X and T ∈ Ya the following estimate holds true

Z t

∇S(t − τ )(vT )(τ ) dτ

≤ α2 kvkX kT kYa

0

Ya

for all t > 0. Proof. This result is deduced from the equality | · |−2 ∗ | · |−a = C| · |−a+1 which holds true for every a ∈ (1, 3) in the space R3 . Indeed, using the definition of the Ya -norm we have Z t −(t−τ )|ξ|2 b)(τ, ξ) dτ iξe (b v ∗ T 0 Z t  2 (2.5) ≤ |ξ|e−(t−τ )|ξ| |ξ|−2 ∗ |ξ|−a τ −a/2 dτ kvkX kT kYa 0

Z = CkvkX kT kYa

t

2

|ξ|2−a e−(t−τ )|ξ| τ −a/2 dτ.

0

Hence, by the equality 2

sup |ξ|b e−(t−τ )|ξ| = C(t − τ )−b/2

(2.6)

ξ∈R3

with b = 2 − a, we may estimate the right-hand side of (2.5) by CkvkX kT kYa Rt because the integral 0 (t − τ )−1+a/2 τ −a/2 dτ is finite and independent of t for any a ∈ (0, 2). Consequently, we obtain the estimate of the first component of the Ya -norm

Z t



(2.7) sup ∇S(t − τ )(vT )(τ ) dτ

0 ≤ C1 kvkX kT kYa . t>0 0

PM

Now, let us deal with the second term of the Ya -norm. Note first the following two elementary inequalities: Z t Z t 2 2 −a/2 |ξ|2−a e−(t−τ )|ξ| τ −a/2 dτ ≤ (t/2) |ξ|2−a e−(t−τ )|ξ| dτ ≤ Ct−a/2 |ξ|−a t/2

t/2

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GRZEGORZ KARCH AND NICOLAS PRIOUX

and (by (2.6) with t − τ replaced by t/2 and with b = 2) Z t/2 Z t/2 −(t−τ )|ξ|2 −a/2 2−a −(t/2)|ξ|2 2−a τ −a/2 dτ = C|ξ|−a t−a/2 . e τ dτ ≤ |ξ| e |ξ| 0

0

Hence, proceeding as in (2.5), we obtain Z t −(t−τ )|ξ|2 b iξe (b v ∗ T )(τ, ξ) dτ 0 Z 2−a ≤ CkvkX kT kYa |ξ|

t/2

≤ C|ξ|

t

2

e−(t−τ )|ξ| τ −a/2 dτ

+ t/2

0 −a −a/2

!

t

Z

kvkX kT kYa

which immediately leads to the inequality

Z t

a/2 ∇S(t − τ )(vT )(τ ) dτ (2.8) sup t

t>0

≤ C2 kvkX kT kYa .

PMa

0

The proof of Lemma 2.4 is complete by (2.7) and (2.8) with α2 = C1 + C2 . Finally, we deal with the linear operator L. Lemma 2.5. Let 1 < a < 2. For every T ∈ Ya , the following estimate holds true

Z t



≤ sup kT (t)kPM0 S(t − τ )PT (τ ) dτ

0

PM2

t>0

for all t > 0. Note that the estimate from Lemma 2.5 is, in fact, presented as the second inequality in (2.2) with F replaced by T , hence here we skip the proof. Proof of Theorem 1.1. Lemmata 2.3, 2.4, and 2.5 provide the estimates required by Lemma 2.1. Hence Theorem 1.1 is proved under the assumption η = |β| < 1. Proof of Corollary 1.3. Let (v, T ) be the solution of (1.1) corresponding to the initial datum (v0 , T0 ) and the external force F constructed via Theorem 1.1. By the scaling invariance of system (1.1) we get that the functions given by (1.2) are also solutions with the same initial condition and the external force. Hence, by the uniqueness result of Theorem 1.1, the solution (v, T ) of (1.1) is self-similar because the norms in X and Ya are invariant under the rescaling (1.2). 3. Large time behavior of solutions We begin by preliminaries estimates. Lemma 3.1. There exists a constant C > 0 such that for all t > 0, 0 < τ < t, v ∈ PM2 and T ∈ Ya , the following estimates hold true (3.1)

kS(t − τ )PT (τ )kPM2 ≤ C(t − τ )−1 kT kYa ,

(3.2)

k∇S(t − τ )(vT )(τ )kPM0 ≤ C(t − τ )−1+a/2 τ −a/2 kvkX kT kYa ,

and (3.3)

k∇S(t − τ )(vT )(τ )kPMa ≤ C(t − τ )−1 τ −a/2 kvkX kT kYa .

SELF-SIMILARITY IN VISCOUS BOUSSINESQ EQUATION

7

Proof. By the definition of the PM2 -norm and (2.6) with b = 2, we immediately obtain 2

kS(t − τ )T (τ )kPM2 ≤ kT (τ )kPM0 sup |ξ|2 e−(t−τ )|ξ| ≤ C(t − τ )−1 kT kYa . ξ∈R3

0

Next, we deal with the PM -norm. By the calculations as in the proof of Lemma 2.4 (cf. (2.7)) we have 2 −(t−τ )|ξ|2 vb ∗ Tb(ξ) ≤ C|ξ|2−a e−(t−τ )|ξ| kv(τ )kPM2 kT (τ )kPMa iξe   ≤ C(t − τ )−1+a/2 τ −a/2 kv(τ )kPM2 τ a/2 kT (τ )kPMa . Next, we study the PMa -norm 2 2 |ξ|a iξe−(t−τ )|ξ| vb ∗ Tb(ξ) ≤ |ξ|2 e−(t−τ )|ξ| kv(τ )kPM2 kT (τ )kPMa   ≤ C(t − τ )−1 τ −a/2 kv(τ )kPM2 τ a/2 kT (τ )kPMa by (2.6) with b = 2. The proof of Lemma 3.1 is complete. Now we recall how to deal with the external force. The following lemma was shown in [4, Lemma 5.1], hence we skip its proof.
Lemma 3.2. Assume that F ∈ Cw [0, ∞[, PM0 satisfies lim kF (t)kPM0 = 0. t→∞ Then,

Z t

lim S(t − τ )F (τ ) dτ

2 = 0. t→∞ 0

PM

We are in a position to show our main result on the large time behavior of solutions to system (1.1). Proof of Theorem 1.5. Let (v, T ) and (v, T ) be two solutions of (1.1) constructed in Theorem 1.1. Recall that, by Lemma 2.1, one may choose R0 < (1−η)/(2α1 +α2 ) with the constants α1 , α2 and η = |β| obtained in Lemmata 2.3, 2.4, and 2.5, respectively, such that (3.4)

kvkX + kT kYa ≤ R0

and kvkX + kT kYa ≤ R0 .

We subtract the integral equation (1.4) for v and T from the analogous expression for v and T to get (3.5) Z t v(t) − v(t) = S(t)(v0 − v 0 ) + ∇S(t − τ )P ((v ⊗ v) (τ ) − (v ⊗ v) (τ )) dτ 0 Z t Z t

S(t − τ )P F (τ ) − F (τ ) dτ. +β S(t − τ )P T (τ ) − T (τ ) dτ + 0

0

We proceed in a same way with equation (1.5) Z t

∇S(t − τ ) (vT ) (τ ) − vT (τ ) dτ. (3.6) T (t) − T (t) = S(t)(T0 − T 0 ) + 0

Now, to shorter the notation, we introduce the auxiliary function

Z t




h(t) ≡ kS(t)(v0 − v 0 )kPM2 + S(t − τ )P F (τ ) − F (τ ) dτ

2 0 PM



+ S(t)(T0 − T 0 ) PM0 + ta/2 S(t)(T0 − T 0 ) PMa .

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GRZEGORZ KARCH AND NICOLAS PRIOUX

It follows immediately from Lemma 2.2 that h ∈ L∞ (0, ∞). Moreover, by assumption (1.7) (note the inequality ta/2 kS(t)T0 kPMa ≤ CkS(t/2)T0 kPM0 being the immediate consequence of (2.4)) and Lemma 3.2, we have limt→∞ h(t) = 0. By computing the PM2 -norm of (3.5) and the PM0 as well as PMa -norms multiplied by ta/2 of (3.6), and next adding the resulting inequalities we obtain



kv(t) − v(t)kPM2 + T (t) − T (t) PM0 + ta/2 T (t) − T (t) PMa

Z t

∇S(t − τ )P [(v ⊗ v) (τ ) − (v ⊗ v) (τ )] dτ ≤ h(t) +

2 0 PM

Z t

  S(t − τ )P T (τ ) − T (τ ) dτ +|β| (3.7)

2 0 PM

Z t




S(t − τ ) (vT ) (τ ) − vT (τ ) dτ +

0 0 PM

Z t


 a/2 . S(t − τ ) (vT ) (τ ) − vT (τ ) dτ +t

PMa

0

Given δ ∈ (0, 1) we decompose the second term on the right hand side of (3.7) R δt Rt as 0 ...dτ + δt ...dτ and we estimate each term separately. We use [4, estimate (5.5)] in the case of the integral over [0, δt] in order to show that (3.8) Z

δt

0

k∇S(t − τ )P [(v ⊗ v) (τ ) − (v ⊗ v) (τ )]kPM2 dτ Z

δt

(t − τ )−1 kv(τ ) − v(τ )kPM2 dτ 0   1 ≤ 4CR02 log . 1−δ ≤C



 sup kv(τ )kPM2 + sup kv(τ )kPM2 τ >0

τ >0

Rt The second term containing the integral term δt · · · dτ is treated directly by Lemma 2.3 (see also [5, estimate (5.6)]) to obtain

Z t

∇S(t − τ )P [(v ⊗ v) (τ ) − (v ⊗ v) (τ )] dτ

2

δt PM (3.9) ≤ 2α1 R0 sup kv(τ ) − v(τ )kPM2 . δt≤τ ≤t

We proceed in the analogous way with the third term on the right hand side of (3.7) using (3.1) and Lemma 2.5 to bound it as follows

Z t

 

|β| T (τ ) dτ S(t − τ )P T (τ ) −

2 0   PM  (3.10) 1 ≤ |β| CR0 log + sup kT (τ ) − T (τ )kPM0 . 1−δ δt≤τ ≤t In the similar way, using estimates (3.2)-(3.3) for the integrals on [0, δt] and Lemma 2.4 for τ ∈ [δt, t] we obtain the following inequalities for the forth and fifth

SELF-SIMILARITY IN VISCOUS BOUSSINESQ EQUATION

9

term on the right-hand side of (3.7) (3.11)

Z t




S(t − τ ) (vT ) (τ ) − vT (τ ) dτ

0 0 PM

Z t


 a/2 S(t − τ ) (vT ) (τ ) − vT (τ ) dτ +t

a 0 PM ! Z δ Z δ −1+a/2 −a/2 2 −1 −a/2 (1 − s) s ds + (1 − s) s ds ≤ CR0 0

0

 + α2 R0

sup kv(τ ) − v(τ )kPM2 + sup τ

δt≤τ ≤t

a/2

δt≤τ ≤t



T (τ ) − T (τ ) a PM

 .

Next, following [4] we put (3.12)  



A = lim sup kv(t) − v(t)kPM2 + T (t) − T (t) PM0 + ta/2 T (t) − T (t) PMa , t→∞

where “lim supt→∞ ” is understood as “limk∈N,k→∞ supt≥k ”. This number is finite due to the properties of the solutions (v, T ) and (v, T ) from Theorem 1.1, and our aim is to prove that A = 0. By this reason we compute “lim supt→∞ ” of the both sides of inequality (3.7). Obviously, we have 

lim sup sup kv(τ ) − v(τ )k 2 + sup T (τ ) − T (τ ) 0 t→∞

(3.13)

PM

δt≤τ ≤t

PM

δt≤τ ≤t

+ sup (τ )a/2 T (τ ) − T (τ ) PMa



≤ A.

δt≤τ ≤t

Recalling that limt→∞ h(t) = 0 we obtain from (3.7)- (3.13) the following inequality A ≤ g(δ) + (2α1 R0 + α2 R0 + η)A with g(δ) =

CR02

 log

1 1−δ



Z

δ −1+a/2 −a/2

(1 − s)

+

s

0

Z

!

δ −1 −a/2

(1 − s)

ds +

s

ds

0

which tends to 0 as δ & 0. Since δ ∈ (0, 1) can be arbitrary small and 2α1 R0 + α2 R0 + η < 1 by our assumption on R0 we obtain A = 0 and the proof of Theorem 1.5 is complete. References [1] Bejan, A; Convection Heat Transfer, 2nd edition, John Wiley and Sons Inc.,1995 [2] Cannon, J. R. & DiBenedetto, E., The initial value problem for the Boussinesq equations with data in Lp . Approximation methods for Navier-Stokes problems (Proc. Sympos., Univ. Paderborn, Paderborn, 1979), 129–144, Lecture Notes in Math., 771, Springer, Berlin, 1980. [3] Cannone, M., Harmonic analysis tools for solving the incompressible Navier-Stokes equations, Harmonic analysis tools for solving the incompressible Navier-Stokes equations. Handbook of mathematical fluid dynamics. Vol. III, 161–244, North-Holland, Amsterdam, 2004. [4] Cannone M. & Karch G., Smooth or singular solutions to the Navier-Stokes system ?, J. Differential Equations, 197 (2004), 247 – 274. [5] Cannone M. & Karch G., About regularized Navier-Stokes equations, J. Math. Fluid Mech. 7 (2005), 1–28. [6] Hishida, T., On a class of stable steady flows to the exterior convection problem, J. Differential Equations 141 (1997), 54–85.

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GRZEGORZ KARCH AND NICOLAS PRIOUX

[7] Karch G., Scaling in nolinear parabolic equations, J. Math. Anal. Appl., 234 (1999), 534 – 558. [8] Landau, L.D. & Lifchitz E.M., Theorical Physics: Fluid Mechanics, 2nd edition, Pergamon Press, 1987 [9] Prioux N., Asymptotic stability results for some nonlinear evolution equations, (2006), preprint http://prioux.n.free.fr/src/Download/Prioux-Asymptotic-2006.pdf, submitted. Instytut Mathematyczny, Uniwersytet Wroclawski pl. Wroclaw, Poland E-mail address: [email protected]

Grunwaldzki 2/4, 50-384

´matiques Applique ´es, Universite ´ de Marne-laLaboratoire d’Analyse et de Mathe ´e, Cite ´ Descartes-5, bd Descartes, Champs-sur-Marne, 77454 Marne-la-Valle ´e CeValle dex 2, France E-mail address: [email protected]