LORENZO BRANDOLESE AND MARIA E. SCHONBEK

Transcription

1 LARGE TIME DECAY AND GROWTH FOR SOLUTIONS OF A VISCOUS BOUSSINESQ SYSTEM LORENZO BRANDOLESE AND MARIA E. SCHONBEK Abstract. In this aer we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow u as t in the sense that the energy and the L -norms of the velocity field grow to infinity for large time for 1 < 3. In the case of strong solutions we rovide shar estimates both from above and from below and exlicit asymtotic rofiles. We also show that solutions arising from (u, θ ) with zero-mean for the initial temerature θ have a secial behavior as x or t tends to infinity: contrarily to the generic case, their energy dissiates to zero for large time. 1. Introduction In this aer we address the roblem of the heat transfer inside viscous incomressible flows in the whole sace R 3. Accordingly with the Boussinesq aroximation, we neglect the variations of the density in the continuity equation and the local heat source due to the viscous dissiation. We rather take into account the variations of the temerature by utting an additional vertical buoyancy force term in the equation of the fluid motion. This leads us to the Cauchy roblem for the Boussinesq system t θ + u θ = κ θ t u + u u + = ν u + βθe 3 (1.1) x R 3, t R + u = u t= = u, θ t= = θ. Here u: R 3 R + R 3 is the velocity field. The scalar fields : R 3 R + R and θ : R 3 R + R denote resectively the ressure and the temerature of the fluid. Moreover, e 3 = (,, 1), and β R is a hysical constant. For the decay questions that we address in this aer, it will be imortant to have strictly ositive viscosities in both equations: ν, κ >. By rescaling the unknowns, we can and do assume, without loss of generality, that ν = 1 and β = 1. To simlify the notation, from now on we take the thermal diffusion coefficient κ > such that κ = 1. Date: Setember 17, Mathematics Subject Classification. Primary 76D5; Secondary 35Q3, 35B4. Key words and hrases. Boussinesq, energy, heat convection, fluid, dissiation, Navier Stokes, long time behaviour, blow u at infinity. The work of L. Brandolese and M. Schonbek were artially suorted by FBF Grant SC The work of M. Schonbek was also artially suorted by NSF Grants DMS-999 and grant FRG

2 2 LORENZO BRANDOLESE AND MARIA E. SCHONBEK Like for the Navier Stokes equations, obtained as a articular case from (1.1) utting θ, weak solutions to (1.1) do exist, but their uniqueness is not known. The global existence of weak solutions, or strong solutions in the case of small data has been studied by several authors. See, e.g. [1], [11], [17], [18], [34]. Conditional regularity results for weak solutions (of Serrin tye) can be found in [9]. The smoothness of solutions arising from large axisymmetric data is addressed in [2] and [24]. Further regularity issues on the solutions have been discussed also [2, 15]. The goal of this aer is to study in which way the variations of the temerature affect the asymtotic behavior of the velocity field. We oint out that several different models are known in the literature under the name of viscous (or dissiative) Boussinesq system. The asymtotic behaviour of viscous Boussinesq systems of different nature have been recently addressed, e.g., in [3, 12]. But the results therein cannot be comared with ours. Only few works are devoted to the study of the large time behavior of solutions to (1.1). See [21, 26]. These two aers deal with self-similarity issues and stability results for solutions in critical saces (with resect to the scaling). On the other hand, we will be mainly concerned with instability results for the energy norm, or for other subcritical saces, such as L, with < 3. A simle energy argument shows that weak solutions arising from data θ L 1 L 2 and u L 2 σ satisfy the estimates u(t) 2 C(1 + t) 1/4 and θ(t) 2 C(1 + t) 3/4. The above estimate for the temerature looks otimal, since the decay agrees with that of the heat kernel. On the other hand the otimality of the estimate for the velocity field is not so clear. For examle, in the articular case θ =, the system boils down to the Navier Stokes equations and in this simler case one can imrove the bound for the velocity into u(t) 2 u 2. In fact, u(t) 2 for large time by a result of Masuda [28]. Moreover, in the case of Navier Stokes the decay of u(t) 2 agrees with the L 2 -decay of the solution of the heat equation. See [35, 25, 39] for a more recise statement. The goal of the aer will be to show that the estimate of weak solutions u(t) 2 C(1 + t) 1/4 can be imroved if and only if the initial temerature has zero mean. To achieve this, we will establish the validity of the corresonding lower bounds for a class of strong solutions. In articular, this means that very nice data (say, data that are smooth, fast decaying and small in some strong norm) give rise to solutions that become large as t : our results imly the growth of the energy for strong solutions : (1.2) c(1 + t) 1/4 u(t) 2 C(1 + t) 1/4, t > 1.

3 DECAY AND GROWTH OF SOLUTIONS OF BOUSSINESQ (Setember 17, 211) 3 The validity of the lower bound in (1.2) (namely, the condition c > ), will be ensured whenever the initial temerature is sufficiently decaying but θ. We feel that is imortant to oint out here an erratum to the aer [23]. Unfortunately, the lower bound in (1.2) contredicts a result in [23, Theorem 2.3], where the authors claimed that u(t) 2 under too general assumtions, weaker than those leading to our growth estimate. The roof of their theorem (in articular, of inequalities (5.6) and (5.8) in [23]) can be fixed by utting different conditions on the data, including θ =. This is essentially what we will do in art (b) of our Theorem 2.2 below. Similarity, the statement of Theorem 2.4 in [23] contredicts our lower bound (1.6) below (inequalities (5.18) (5.2) in their roof do not look correct). This will be also corrected by our Theorem 2.2. We would like to give credit to the aer [23] (desite the above mentioned errata), because we got from there insiration for our results of Sections 3 and 4. Our main tool for establishing the the lower bound will be the derivation of exact ointwise asymtotic rofiles of solutions in the arabolic region x > t. This will require a careful choice of several function saces in order to obtain as much information as ossible on the ointwise behavior of the velocity and the temerature. A similar method has been alied before by the first author in [6] in the case of the Navier Stokes equations, although the relevant estimates were erformed there in a different functional setting. Even though several other methods develoed for Navier Stokes could be effective for obtaining estimates from below (see, e.g., [13, 22, 3]), our analysis has the advantage of utting in evidence some features that are secific of the Boussinesq system: in articular, the different behavior of the flow when x 3 or when x x2 2, due to the verticality of the bouyancy forcing term θe 3 (see Theorem 2.6 below). Moreover, the analysis of solutions in the region x > t and our use of weighted saces comletely exlains the henomenon of the energy growth: the variations of the temerature ush the fluid articles in the far field; even though in any bounded region the fluid articles slow down as t (this effect is measured e.g. by the decay of the L -norm established in Proosition 2.5), large ortions of fluid globally carry an increasing energy during the evolution. Our result thus illustrates the hysical limitations of the Boussinesq aroximation, at least for the study of heat convection inside fluids filling domains where Poincaré s inequality is not available, such as the whole sace. In fact, our method alies also to weighted L -saces, so let us introduce the weighted norm ( 1/ f L r = f(x) (1 + x ) dx) r. Then we will show that strong solutions starting from suitably small and well decaying data satisfy, for t > large enough, (1.3) c(1 + t) 1 2 (r+ 3 1) u(t) L r C(1 + t) 1 2 (r+ 3 1),

4 4 LORENZO BRANDOLESE AND MARIA E. SCHONBEK for all r, 1 < <, r + 3 < 3. As before in (1.2), these lower bounds hold true with a constant c > as soon as the initial temerature has non-zero mean. Notice that in this case the L -norms asymtotically blow u for large time if and only if < 3. The above restriction r + 3 < 3 on the arameters is otimal. Indeed, under the same conditions yielding to (1.3) we will also rove that when r, 1 < and r + 3 3, then one has u(t) L =, for all t >. The fact that the L r r-norm becomes infinite instanteneously for this range of the arameters is related to that the velocity field immediately satially sreads out and cannot decay faster than x 3 as x for t >, and this even if u C (R3 ). As we observed before, the lower bound in (1.3) brakes down when θ =. In such case, our decay estimates can be imroved. We will establish the uer bound for weak solutions (1.4) u(t) 2 C (1 + t) 1/4 by the Fourier slitting method. This method was first introduced in [35]. We will need no smallness assumtion on u to rove (1.4). We have to ut however a smallness condition of the form (1.5) θ 1 < ε. We do not know if it is ossible to get rid of (1.5) to establish (1.4). Such smallness condition however looks natural as it resects the natural scaling invariance of the system (1.1). As before, the decay estimate (1.4) is otimal for generic solutions satisfying θ =. Indeed, when we start from localized and small velocity, we can establish the uer-lower bounds for strong solutions (1.6) c (1 + t) 1 2 (r+ 3 2) u(t) L r C (1 + t) 1 2 (r+ 3 2), for all r, 1 < <, r + 3 < 4. Similarily as before, the validity of the lower bound (the condition c > ) now requires a non vanishing condition in the first moments of θ Notations. We denote by C the sace of smooth functions with comact suort. The L -will be denoted by and in the case = 2 we will simly write. Moreover, V = {φ C φ = }, L σ denotes the comletion of V under the norm, and V the closure of V in H 1 We will adot the following convention for the Fourier transform of integrable functions: Ff(ξ) = f(ξ) = f(x)e 2πix ξ dx. Here and throughout the aer all integral without integration limits are over the whole R 3. The notations L,q and L r have a different meaning. For 1 < < and 1 q, L,q denotes the classical Lorentz sace. For the definition and the basic inequalities

5 DECAY AND GROWTH OF SOLUTIONS OF BOUSSINESQ (Setember 17, 211) 5 concerning Lorentz saces (namely the generalisation of the straightforward L -L q Hölder and Young convolution inequalities) the reader can refer to [27, Chater 2]. Here we just recall that L, agrees with the usual Lebesgue sace L, and L, agrees with the weak Lebesgue sace L w (or Marcinkiewicz sace) The quasi-norm L w = {f : R n C, measurable, f L w < }. f L w = su t[λ f (t)] 1 t> is equivalent to the natural norm on L,, for 1 < <. Here as it is usual we defined λ f (s) = λ{x : f(x) > s} where λ denotes the Lebesgue measure. On the other hand, for 1 and r, L r is the weighted Lebesgue sace, consisting of the functions f such that (1+ x ) q f L. Notice that the bold subscrit σ introduced above is not a real arameter: in the notation L σ, this subscrit simly stands for solenoidal. To denote general constants we use C which may change from line to line. In certain cases we will write C(α) to emhasize the constants deendence on α. We denote by e t the heat semigrou. Thus, e t u = g t (x y)u (y) dy, where g t (x) = (4π) 3/2 e x 2 /(4t) is the heat kernel. We denote by E(x) the fundamental solution of in R 3. The artial derivatives of E are denoted by E xj, E xj,x k, etc Organization of the aer. All the main results are stated without roof in Section 2. The statement of our theorems are slitted into two arts: art (a) is devoted to the roerties of solutions in the general case (where the integral θ is not necessarily zero). In Part (b) of our theorems we are concerned with the secial case θ =. The rest of the aer is organized as follows. Sections 3-4 are devoted to the roof of our results on weak solutions and Sections 5-6 to strong solutions. In Section 7 we collect a few technical remarks. 2. Statement of the main results 2.1. Results on weak solutions. We will consider weak solutions to the viscous Boussinesq equations and establish existence and natural decay estimates in L saces, 1 <, for the temerature, together with bounds for the growth of the velocities. Such estimates rely on the fact that in both equations of the system (1.1) we have a diffusion term. They comlete those of obtained in [16] where there was no temerature diffusion. We start recalling the basic existence result of weak solutions to the system (1.1). See, e.g., [9, 16]. Proosition 2.1. Let (θ, u ) L 2 L 2 σ. There exists a weak solution (θ, u) of the Boussinesq system (1.1), continuous from R + to L 2 with the weak toology, with data (u, θ ) such that, for any T >, θ L 2 (, T ; H 1 ) L (, T ; L 2 σ), u L 2 (, T ; V ) L (, T ; L 2 σ).

6 6 LORENZO BRANDOLESE AND MARIA E. SCHONBEK Such solution satisfies, for all t [, T ], the energy inequalities (2.1) θ(t) and (2.2) u(t) t t for all t, and some absolute constant C >. θ(s) 2 ds θ 2. u(s) 2 ds +C ( u + t 2 θ 2) One can imrove the growth estimate on the velocity as soon as θ belongs to some L sace, with < 2. For simlicity we will consider only the case θ L 1 L 2. Moreover, it is natural to ask under which sulementary conditions on the initial data one can insure that the energy of the fluid u(t) 2 remains uniformly bounded. Theorem 2.2 rovides an answer. Beside a smallness assumtion on θ 1, we need to assume that θ =. In addition, it is ossible to rove that u(t) not only remains uniformly bounded, but actually decays at infinity (without any rate). Exlicit decay rates for u(t) can also be rescribed rovided the linear art e u decays at the aroriate rate. Theorem 2.2. (a) Let (θ, u ) L 2 L 2 σ. Under the additional condition θ L 1 the estimates on the weak solution constructed in Proosition 2.1 can be imroved into (2.3) θ(t) 2 C(t + 1) 3 2, u(t) 2 C(t + 1) 1 2, Moreover, if θ L 1 L, for some 1 <, then θ(t) C()(t + 1) 3 2 (1 1 ). (b) (The θ = case) In this art we additionally assume θ L 1 1 and θ =. Then there exists an absolute constant ε > such that if (2.4) θ 1 < ε then the weak solution of the Boussinesq system (1.1) constructed in Proosition 2.1 satisfies, for some constant C > and all t R +, (2.5) θ(t) 2 C(1 + t) 5 2 and (2.6) u(t) 2 as t. Moreover under the additional condition u L 3/2 L 2 σ, we have (2.7) u(t) 2 C(1 + t) 1/2.

7 DECAY AND GROWTH OF SOLUTIONS OF BOUSSINESQ (Setember 17, 211) 7 The roof of art (a) of Theorem 2.2 is straightforward. Part (b) is more subtle: its roof relies on an inductive argument: assuming that the aroximate velocity u n 1 grows in L 2 at most like t 1/8 (which is actually better than rovided by Proosition 2.1) we can rove that the same growth estimate remains valid for u n. A smallness condition on θ is needed to insure that in the inequality u n (t) C(1 + t) 1/8 the constant can by taken indeendently on n. With this imroved control on the growth of the velocity, alying the Fourier slitting method (introduced in [35]) and a few boot-straing we can imrove our estimates u to the rates given in (2.5) and (2.7). Remark 2.3. Solutions of the Boussinesq system (1.1) with energy decaying faster than t 1/2 might exist, but they are likely to be highly non-generic. Indeed, it seems difficult to construct such solutions assuming only that the data belong to suitable function saces (ossibly with small norms). The main obstruction is that one would need stringent cancellations roerties on the data, which however turn out to be non-invariant under the Boussinesq flow. A ossible way to obtain such fast decaying solutions would be to start with data satisfying some secial rotational symmetries, as those described in [5] Results on strong solutions. The best way to rove the otimality of the estimates contained in Theorem 2.2 is to establish the corresonding lower bound estimates at least for a subclass of solutions. For the study of the estimates from below we will limit our considerations to a class of strong solutions. This is not a real restriction as lower bound estimates established for solutions emanating from well localized, smooth and small data are exected to remain valid in the larger class of weak solutions. Studying strong solutions has also the advantage of better utting in evidence some interesting roerties secific of the Boussinesq system, such as the influence of the vertical buoyancy force on the ointwise behavior of the fluid in the far-field. The existence of strong solutions to the system (1.1) will be insured by a fixed oint theorem in function saces invariant under the natural scaling of the equation. Thus, if u X, where X is a Banach sace to be determined, we want to have, for all λ >, u λ X = u X, where u λ (x, t) = λu(λx, λ 2 t) is the rescaled velocity. A suitable choice for norm of the sace X, insired by [1], is (2.8) u X = ess su x R 3, t> ( t + x ) u(x, t). This choice for X is quite natural. Indeed, whenever u (x) C x 1, the linear evolution e t u belongs to X and this aves the way for the alication of the fixed oint theorem in such sace. More recisely, we define X as the Banach sace of all locally integrable divergence-free vector fields u such that u X <, and continuous with resect to t in the following usual sense: u(t) u() in the distributional sense as t and ess su x R 3 x u(x, t) u(x, t ) as t t if t >. In the same way, if θ belongs to a Banach sace Y, we want to have θ λ Y = θ Y, where θ λ (x, t) = λ 3 θ(λx, λ 2 t)

8 8 LORENZO BRANDOLESE AND MARIA E. SCHONBEK is the rescaled temerature. We then define a Banach sace Y of scalar functions through the norm (2.9) θ Y = θ L t (L 1 ) + ess su x R 3, t> ( t + x ) 3 θ(x, t) and the natural continuity condition on the time variable as before. The starting oint of our analysis will be the following roosition roviding a simle construction of mild solutions (u, θ) X Y. We will refer to them also as strong solutions. Indeed, one could rove that such solutions turn out to be smooth, as one could check by adating to the system (1.1) classical regularity criteria for the Navier Stokes equations like that of Serrin [37]. See the aer [9]. The smoothness of these solutions, however, lays no secial role in our arguments. Proosition 2.4. There exists an absolute constant ɛ > such that if (2.1) θ 1 < ɛ, ess su x R 3 x 3 θ (x) < ɛ, ess su x R 3 x u (x) < ɛ where u is a divergence-free vector field, then there is a constant C > and a (mild) solution (u, θ) X Y of (1.1), such that (2.11) u X Cɛ and θ Y Cɛ. Moreover, these conditions define u and θ uniquely. Next Proosition shows it is ossible to obtain better sace-time decay estimates rovided one starts with suitably decaying data. Proosition 2.5. (a) Let u and θ as in Proosition 2.4, and satisfying the additional decay estimates, for some 1 a < 3, b 3, and a constant C >, (2.12) u (x) C(1 + x ) a, θ (x) C(1 + x ) b. Then the solution constructed in Proosition 2.4 satisfies, for another constant C > indeendent on x and t, (2.13) u(x, t) C inf η a x η (1 + t) (η 1)/2 and (2.14) θ(x, t) C inf η b x η (1 + t) (η 3)/2. (b) (the θ = case) Assume now 2 a < 4, a 3, and b 4 and let u and θ satisfying the revious assumtions. If, in addition, (2.15) θ = and θ L 1 1, (2.16) then the decay of u and θ is imroved as follows: u(x, t) C θ(x, t) C inf (1 + η a x ) η (1 + t) (η 2)/2, inf (1 + η b x ) η (1 + t) (η 4)/2.

9 DECAY AND GROWTH OF SOLUTIONS OF BOUSSINESQ (Setember 17, 211) 9 Recall that the fundamental solution of in R 3 is E(x) = c x 1. Thus, in the following asymtotic exansions, E x3 and E xh,x 3 are vectors whose comonents are homogeneous functions of degree 3 and 4 resectively. In articular E x3 (x) C x 3 and E xj,x 3 (x) C x 4. We are now in the osition of stating our main results on strong solutions. The first theorem describes the asymtotic rofiles of solutions in the arabolic region x > t. Roughly, it states that all sufficiently decaying solutions (u, θ) of (1.1) behave in such region like a otential flow. Theorem 2.6. (a) Let a > 3 2 and b > 3. Let (u, θ) be a (mild) solution of (1.1) satisfying the decay estimates (2.13)-(2.14). Then the following rofile for u holds: ( ) (2.17) u(x, t) = e t u (x) + θ t ( ) E x3 (x) + R(x, t) where R(x, t) is a lower order term with resect to t E x3 (x) for x > t, namely, (2.18) lim x t R(x, t) =. t x 3 (b) (the θ = case) Assume now a > 2 and b > 4. Assume also that θ =. Let (u, θ) be a solution satisfying the decay condition (2.16). Then the following rofiles for u j (j = 1, 2, 3) hold: ( t ) (2.19) u j (x, t) = e t u (x) E xj x 3 (x) y θ(y, s) dy ds + R(x, t) where R is a lower order term for x > t > 1, namely (2.2) lim t, x t R(x, t) =. t x 4 The following remark should give a better understanding of the theorem. Remark 2.7. (a) (The case θ ). We deduce from the asymtotic rofile (2.17) the following: when e t u (x) < t x 3 (this haens, e.g., when we assume also u (x) C x 3 and x > t > 1) and θ then ( ) (2.21) u(x, t) θ t ( ) E x3 (x), for x > t. (The exact meaning of our notation and of statements (2.21) and (2.22) below is made recise in the roof). (b) (The θ = case) We deduce from the rofile (2.19) the following: when e t u (x) < t x 4 (this haens, e.g., when we assume also u (x) C x 4 and x > t > 1) then ( t ) (2.22) u j (x, t) E xj x 3 (x) y θ(y, s) dy ds for x > t > 1.

10 1 LORENZO BRANDOLESE AND MARIA E. SCHONBEK A remarkable consequence of the revious theorem is the following. Corollary 2.8. (a) Let a > 3 2, b > 3 and let (u, θ) be a solution as in Part (a) of Theorem 2.6. Then for all r, such that r, 1 < <, r + 3 < min{a, 3}, there exists t > such that the solution satisfies the uer and lower estimates in the weighted-l -norm (2.23) φ( m ) ( 1 + t ) 1 2 (r+ 3 1) u(t) L r C ( 1 + t ) 1 2 (r+ 3 1) for all t t. Here, m = θ and φ: R + R is some continuous function such that φ() = and φ(σ) > if σ >. (b) (the θ = case) Under the assumtions of the revious item, with the stronger conditions a > 2, b > 4 and the additional zero mean condition m =, let us set 1 m = lim inf t t t. yθ(y, s) dy ds Then, for all r, such that r, 1 < <, r + 3 < min{a, 4}, we have (2.24) φ ( m ) ( 1 + t ) 1 2 (r+ 3 2) u(t) L r C ( 1 + t ) 1 2 (r+ 3 2) for another suitable continuous function φ: R + R such that φ() = and φ(σ) > for σ >. Remark 2.9. When θ, we thus get by (2.23) the shar large time behavior u(t) L r t 1 2 (r+ 3 1). When θ = and m we have the faster shar decay u(t) L r t 1 2 (r+ 3 2). The condition m is satisfied for generic solutions. It revents θ to have oscillations at large times. 3. The mollified Boussinesq system and existence of weak solutions The existence of weak solutions to the Boussinesq system is well known, see [9]. Their uniqueness, however, is an oen roblem. Moreover, we do not know if any weak solutions satisfy the energy inequality and the decay estimates stated in Proosition 2.1. For this reason, we now briefly outline another construction of weak solutions, which is well suited for obtaining all our estimates. We begin by introducing a mollified Boussinesq system. As the construction below is a straightworward adatation of that of Caffarelli, Kohn and Nirenberg, [8], we will be rather sketchy. For comleteness we recall the definition of the retarded mollifier as given in [8]. Let ψ(x, t) C such that ψ, ψdx dt = 1, su ψ {(x, t) : x 2 < t, 1 < t < 2}.

11 DECAY AND GROWTH OF SOLUTIONS OF BOUSSINESQ (Setember 17, 211) 11 For T > and u L 2 (, T ; L 2 σ), let ũ : R 3 R R 3 be { u(x, t) if (x, t) R 3 (, T ), ũ = otherwise. Let δ = T/n. We set Ψ δ (u)(x, t) = δ 4 R 4 ψ ( y δ, τ δ ) ũ(x y, t τ) dydτ. Consider, for n = 1, 2,... and δ = T/n, the mollified Cauchy roblem t θ n + Ψ δ (u n 1 ) θ n = θ n (3.1) t u n + (Ψ δ (u n ) u n ) + n = u n + θ n e 3 x R 3, t R + u n =. with data (3.2) θ n t= = θ and u n t= = u. The iteration scheme starts with u =. Note that since div u =, we also have div (Ψ δ (u n )) =, for t R +. At each ste n, one solves recursively n + 1 linear equations: first one solves the transort-diffusion equation (with smooth convective velocity) for the temerature; after θ n is comuted, solving the second of (3.1) amounts to solving a linear equation on each stri R 3 (mδ, (m + 1)δ), for m =, 1,..., n 1. For solutions to (3.1) we have the following existence and uniqueness result, Proosition 3.1. Let (θ, u ) L 2 L 2 σ. For each n {1, 2,... } there exists a unique weak solution (θ n, u n, n ) of the aroximating equations with data (3.2) such that, for any T >, and θ n L 2 (, T ; H 1 ) L (, T ; L 2 ), u n L 2 (, T ; V ) L (, T ; L 2 σ) n L 5/3 (, T ; L 5/3 ) + L (, T ; L 6 ) Moreover, for all t >, θ n and u n satisfy the energy inequalities as in (2.1) and (2.2). In articular, the sequences θ n, u n and n, n = 1, 2,... are bounded in their resective saces. Proof. This can be roved using the Faedo-Galerkin method. As the the argument is standard (see, e.g. [38, Theorem 1.1, Chater III], or [8, Aendix]), we ski the details. We only rove the condition on the ressure since this is the only change that we have to make in [8]. Taking the divergence of the second equation in (3.1) we get = n 1 + n 2, where n 1 = i,j i j (u n i u n j ) and n 2 = x3 θ n.

12 12 LORENZO BRANDOLESE AND MARIA E. SCHONBEK Thus, n 1 L5/3 (, T ; L 5/3 ) uniformly with resect to n, by the energy inequality for u n, interolation, and the Calderon-Zygmund theorem, as roved in [8]. On the other hand, n 2 = E x 3 θ, where E(x) is the fundamental solution of. Thus, E x3 (x) = c x 3 x 3 belongs to the Lorentz sace L 3/2, (R 3 ). But θ n L (, T ; L 2 ) uniformly with resect to n, hence Young convolution inequality in Lorentz saces (see [27, Chater 2]) yields n 2 L (, T ; L 6 ) uniformly with resect to n. It follows from Proosition 3.1 that, extracting suitable subsequences, n = n 1 + n 2, where ( n 1 ) converges weakly in L5/3 (, T ; L 5/3 ) and n 2 converges in L (, T ; L 6 ) in the weak-* toology. Moreover, (u n ) is convergent with resect to the toologies listed in [8, ]. On the other hand (θ n ) will be convergent with resect to the same toologies, because all estimates available for u n hold also for θ n. No additional difficulty in the assage to the limit in the nonliner terms arises, in the equation of the temerature other those already existing for the Navier Stokes equations. Hence, the distributional limit (θ, u, ) of a convergent subsequence of (θ n, u n, n ) is a weak solutions of the Boussinesq system. This establishes Proosition 2.1. We finish this section by establishing the natural L -estimates for the aroximating temeratures: Lemma 3.2. Let (θ, u ) L 2 L 2 σ and let (θ n, u n, n ) be the solution of the mollified Boussinesq system (3.1) for some n {1, 2,...}. Let also 1 <. If θ L 1 L, then ( c t ) 3 (3.3) θ n (t) θ 1 + A 2 (1 1 ), where A = A(, θ 1, θ ) and c > is an absolute constant. Proof. First notice that, for each n, θ n is the solution of a linear transort-diffusion equation with smooth and divergence-free velocity Ψ(u n 1 ). The L decay estimates for these equations are well known. We reroduce the same roof as in [14, 19]) exliciting better the constants, as we will need the exressions of such constants later on. A basic estimate (valid for 1 ) is (3.4) θ n (t) θ. See [14, Corollary 2.6] for a nice roof of (3.4) that remains valid in the much more general case of trasort equations with (or without) fractional diffusion. We start with the case 2 <. Multilying the equation for θ n by θ n 2 θ n and integrating we get d dt θn (t) + 4( 1) ( θ n /2 )(t) 2. By the Sobolev embedding theorem, Ḣ 1 L 6, hence θ n (t) 3 C ( θn /2 )(t) 2.

13 DECAY AND GROWTH OF SOLUTIONS OF BOUSSINESQ (Setember 17, 211) 13 The interolation inequality yields θ n (t) θ n (t) 2/(3 1) 1 θ n (t) 3( 1)/(3 1) 3. Combining these two inequalities with the basic estimate θ n (t) 1 θ 1, we obtain the differential inequality d dt θn (t) 4( 1) C θ 2/(3 3) 1 Integrating this we get ( θ n (t) 8 t 3C θ (2)/(3 3) 1 and estimate (3.3) follows with + ( θ n (t) ) θ (2)/(3 3) ) 3( 1)/2. ( ) θ (2)/(3 3) 1 A =. θ The case 1 < 2 is deduced by interolation. In the = 2 case, we obtain the following Lemma 3.3. Let θ L 1 L 2 and u L 2 σ. Let (θ n, u n, n ) be the solution of the mollified Boussinesq system (3.1) for some n {1, 2,...}. Then, (3.5) θ n (t) θ 1 ( C t + A ) 3/4, u n (t) u + C θ 1 t 1/4, for two absolute constants C, C > and A = ( θ 1 / θ 2 ) 4/3. Proof. We only have to estimate the L 2 norm of the velocity. We make use of the identity d dt un (t) 2 2 = u n t u n, that can be justified exactly as for the mollified Navier-Stokes equations, see [8] and [35]. Multilying the velocity equation in the Boussinesq system (3.1) by u n and integrating we get (3.6) d dt un (t) u n (t) 2 2 u n (t) θ n (t). Dividing by u n (t), d dt un (t) θ n (t). Now we use the decay of θ n (t) obtained in Lemma 3.2. Integrating we obtain the second of (3.5).

14 14 LORENZO BRANDOLESE AND MARIA E. SCHONBEK 4. Imroved bounds for weak solution in the case θ = The estimates obtained in Lemma 3.3 can be considerably imroved rovided we additionally assume θ = and the moment condition θ L 1 1. First of all, from an elementary heat kernel estimate one easily checks that in this case (4.1) e t θ 2 A 1 (t + 1) 5/2, where A 1 > deends only on the data, through its L 2 -norm and the x θ (x) dx integral. We will now see that in this case the aroximated temerature θ n (t) also decays at the faster rate (t + 1) 5/4 in the L 2 -norm. Using this new decay rate for θ n it is ossible to show that the velocities are uniformly bounded in L 2. Once we have a solution such that u n (t) remains bounded as t one can go further and rove that u n actually decays at infinity in L 2 at some algebraic decay rate, deending on the decay of the linear evolution e t u. In view of the assage to the limit from the mollified system (3.1) to the Boussinesq system (1.1), all the estimates must be indeendent on n. Proosition 4.1. Let (θ, u ) (L 1 1 L2 ) L 2 σ and assume θ =. There exists an absolute constant ε > such that if (4.2) θ 1 < ε then the solution of the mollified Boussinesq system (3.1), with data (u, θ ) satisfies (4.3) θ n (t) 2 A(1 + t) 5 2 and (4.4) u n (t) 2 A for all n N and t R +. Here A > is some constant deending on the data u and θ, and indeendent on n, and t. Proof. We denote by C a ositive absolute constant, which may change from line to line. We also denote by A 1, A 2,... ositive constants that deend only on the data. More recisely, A j = A j ( θ, θ L 1 1, u ). Ste 1: An auxiliary estimate. We make use of the Fourier slitting method introduced in [35]. The first ste consists in multilying the temerature equation by θ n and to integrate by arts. Using the Plancherel theorem in the energy inequality for θ n, we get 1 d θ 2 dt n (ξ, t) 2 dξ ξ 2 θ n (ξ, t) 2 dξ Now slit the integral on the right hand side into S S c, where { ( ) } k 1/2 (4.5) S = ξ : ξ, 2(t + 1)

15 DECAY AND GROWTH OF SOLUTIONS OF BOUSSINESQ (Setember 17, 211) 15 and k is a constant to be determined below. Noting that for ξ S c one has ξ 2 k 2(t+1), it follows that d θ dt n (ξ, t) 2 dξ k θ t + 1 n (ξ, t) 2 S c = k θ t + 1 n (ξ, t) 2 dξ + k θ t + 1 n (ξ, t) 2 dξ. S Multilying by (1 + t) k we obtain ] d (4.6) [(1 + t) k θ dt n (ξ, t) 2 dξ k(t + 1) k 1 S θ n (ξ, t) 2 dξ, where S is as in (4.5). Taking the Fourier transform in the equation for θ n in (3.1) we get ( t 2 (4.7) θ n (ξ, t) 2 2 e t ξ 2 θ ξ 2 u n 1 (s) θ (s) ds) n. Hence S θ n (ξ, t) 2 dξ C [ e ξ 2t θ 2 + (1 + t) 5/2 ( t ) 2 ] u n 1 (s) θ n (s) ds. Relacing this in (4.6) and alying the Plancherel theorem we get [ d [ (1 + t) k θ n (t) 2] ( t ) 2 ] C e t θ 2 (1 + t) k 1 + (1 + t) k 7/2 u n 1 θ n ds. dt From where it follows, letting k = 7/2, [( t θ n (t) 2 (1 + t) { θ 7/2 2 + C (4.8) + t ) e s θ 2 (1 + s) 5/2 ds ( s 2 ]} u n 1 (r) θ (r) dr) n ds. Recalling the estimate (4.1) for the linear evolution we get, for all n N, [ t ( ] s 2 (4.9) θ n (t) 2 C A 1 (1 + t) 5/2 + (1 + t) 7/2 u n 1 (r) θ (r) dr) n ds. We now use the following inequality, deduced from estimate (3.5), (4.1) θ n (t) C θ 1 t 3/4. Putting this inside (4.9) we obtain a new bound for θ n 2, namely (4.11) [ t ( ] s 2 θ n (t) 2 C A 1 (1 + t) 5/2 + θ 2 1(1 + t) 7/2 u n 1 (r) r dr) 3/4 ds. Ste 2: The inductive argument. We will now rove by induction that, for all ositive integer n we have (4.12) u n 1 (t) u + Mt 1/8

16 16 LORENZO BRANDOLESE AND MARIA E. SCHONBEK where M > is some constant indeendent on n (but ossibly deendent on the data θ, u ) to be determined. Notice that estimate (4.12) is actually better than what we have so far (comare with the second of (3.5)). For n = 1 the inductive condition (4.12) is immediate since u =. Let us now rove that u n u + Mt 1/8 assuming that (4.12) holds true. We get from (4.11) and the induction assumtion (4.12) θ n (t) 2 C [A 1 (1 + t) 5/2 + θ 21 u 2 (1 + t) 2 + M 2 θ 21(1 ] + t) 7/4. This imlies (4.13) θ n (t) C [ A1 + θ 1 ( u + M) ] (1 + t) 7/8. This last inequality will be used to estimate u n as follows. First recall that d (4.14) dt un (t) θ n (t). After an integration in time we get, using (4.13), [ A1 ( u n (t) u + C + θ 1 u + M) ] t 1/8. For the induction argument we need to rove u n u + Mt 1/8. Hence we need that [ A1 ( C + θ 1 u + M) ] M. Choosing M large enough, for examle, M = max{ u, 2C A 1 }, our condition then boils down to the inequality θ 1 1/(4C). The validity this last inequality is insured by assumtion (4.2). This concludes the induction argument and establishes the validity of the estimate (4.12) for all n. Ste 3: Uniform bound for the L 2 -norm of the velocities u n. The result of Ste 2 imlies the existence of a constant A 2 > such that, for all n 1, (4.15) u n 1 (t) 2 A 2 (1 + t) 1/4. In the roof of the revious Ste we also deduced that, for some A 3 >, θ n (t) 2 A 3 (1 + t) 7/4. Combining such two estimates with inequality (4.9) we easily get θ n (t) 2 A 4 (1 + t) 2. Now using this imroved estimate for θ n 2 with (4.15) in (4.9) arrive at θ n (t) 2 A 5 (1 + t) 9/4.

17 DECAY AND GROWTH OF SOLUTIONS OF BOUSSINESQ (Setember 17, 211) 17 Going back to the differential inequality (4.14) we finally get, for some constant A 6 > indeendent on n, and t R +, u n (t) 2 A 6. Relacing in (4.9) we can further imrove the decay of θ n u to θ n (t) 2 A 7 (1 + t) 5/2. We now would like to imrove the result of the revious Proosition by establishing decay roerties for u n (t) 2. Secifically, if we assume in addition that the linear art of the velocity satisfies e t u 2 C(1 + t) 1/2 (this haens e.g. when u L 3/2 L 2 σ), then the same decay holds for the aroximate velocities u n. Proosition 4.2. Let (θ, u ) (L 1 1 L2 ) L 2 σ). Assume also that θ = and θ 1 < ε, where ε is the constant obtained in the revious roosition. Then the aroximate solutions of (3.1) satisfy, u n (t), as t, uniformly with resect to n. Moreover, if u L 3/2 L 2 σ, then (4.16) u n (t) 2 A(1 + t) 1/2, for some constant A > indeendent on n, and t. Proof. We denote by A > a constant deending only on the data that might change from line to line. The roof follows by Fourier Slitting. Since the estimates are indeendent of n, we simly denote the solutions by (θ, u). Multily the second equation in (3.1) by u, integrate in sace. By Proosition 4.1 we get d (4.17) dt u(t) u 2 C(t + 1) 5 4 Arguing as for the roof of inequality (4.6) we obtain (4.18) d dt [(t + 1) k u(t) 2 ] Ck(t + 1) k 1 S û(ξ, t) 2 dξ + C(t + 1) k 5 4 Where S was defined in (4.5). From now on, k = 7/2. We need to estimate û(ξ, t) for ξ S. Comuting the Fourier transform in the equation for u in (3.1), next alying the estimate u(t) 2 A obtained in Proosition 4.1 we get û(ξ, t) e t ξ 2 û + ξ t e t ξ 2 û + At ξ + u(s) 2 ds + t t θ(ξ, s) ds. θ(ξ, s) ds But comuting the Fourier transform in the equation for θ in (3.1) and alying once more the estimates of Proosition 4.1 we have θ(ξ, s) e s ξ 2 θ + A ξ θ (ξ) + A ξ Hence, [ (4.19) û(ξ, t) 2 ( A e 2t ξ 2 û 2 + θ 2) + t 2 ξ 2].

18 18 LORENZO BRANDOLESE AND MARIA E. SCHONBEK Integrating on S and alying inequality (4.1) we deduce ] û(ξ, t) 2 dξ A [ e t u 2 + (1 + t) 1/2. S Putting this inside (4.18) and integrating on an interval of the form [t ɛ, t], with ɛ > arbitrary and t ɛ chosen in a such way e t u 2 < ɛ for t t ɛ, we obtain u(t), as t. On the other hand, in the case u L 3/2 L 2 σ, we have û(ξ, t) 2 dξ A(1 + t) 1/2. S Now going back to (4.18) and integrating in time we finally get u(t) 2 A(1 + t) 1/2. We are now in the osition of deducing our result on weak solutions to the Boussinesq system (1.1). Proof of Theorem 2.2. Now this is immediate: assing to a subsequence, the aroximate solutions θ n and u n converge in L 2 loc (R+, R 3 ) to a weak solution (θ, u) of the Boussinesq system (1.1). Moreover, the revious Lemmata imly that θ n and u n satisfy estimates of the form v n (t) f(t), for all t >, where f(t) is a continuous function indeendent on n. Then the same estimate must hold for the limit θ and u, excet ossibly oints in a set of measure zero. But since weak solutions are necessarily continuous from [, ) to L 2 under the weak toology, θ(t) and u(t) are lower semi-continuous and hence they satisfy the above estimate for all t >. This observation on the weak semi-continuity is borrowed from [25]. 5. Strong solutions: reliminary lemmata The integral formulation for the Boussinesq system, formally equivalent to (1.1) reads t θ(t) = e t θ e (t s) (θu)(s) ds (5.1) t t u(t) = e t u e (t s) P (u u)(s) ds + e (t s) Pθ(s)e 3 ds. u = The above system will be solved alying the following abstract lemma, which slightly generalizes that of G. Karch et N. Prioux (see [26, Lemma 2.1]).

19 DECAY AND GROWTH OF SOLUTIONS OF BOUSSINESQ (Setember 17, 211) 19 Lemma 5.1. Let X and Y be two Banach saces, let B : X X X and B : Y X Y be two bilinear mas and L: Y X a linear ma satisfying the estimates B(u, v) X α 1 u X v X, B(θ, v) Y α 2 θ Y u X and L(θ) X α 3 θ Y, for some ositive constants α 1, α 2 and α 3. Let < η < 1 be arbitrary. For every (U, Θ) X Y such that the system η U X + α 3 Θ Y α 1η(1 η) 2 (2α 1 + α 2 ) 2, (5.2) θ = Θ + B(θ, u), u = U + B(u, u) + L(θ) has a solution (u, θ) X Y. This is the unique solution satisfying the condition η u X + α 3 θ Y η(1 η)(2α 1 + α 2 ). Proof. In the case < α 3 < 1, one can take η = α 3. In such articular case, this lemma is already known, see [26, Lemma 2.1]. Therefore, we only have to rove that we can get rid of the restriction < α 3 < 1. This is straightforward. We introduce on the sace Y an equivalent norm, defined by θ Y = α 3 η θ Y. By Lemma 2.1 of Karch and Prioux, alied in the sace (X, X ) and (Y, Y ), we have that if U X + Θ Y α 1(1 η) 2 (2α 1 + α 2 ) 2, then the system (5.2) has a unique solution such that u X + θ Y (1 η)(2α 1 + α 2 ). The conclusion of Lemma 5.1 is now immediate. Remark 5.2. Using our imroved version of Lemma 2.1 of [26], it is be ossible to get rid of the smallness assumtion β < 1 in the main results of Karch and Prioux [26]. Remark 5.3. The roof of Lemma 2.1 in [26] relies on the contraction maing theorem. In articular, the solution can be obtained assing to the limit with resect to the X Y-norm in the iteration scheme (k = 1, 2,...) (5.3) (u, θ ) = (U, Θ), (u k+1, θ k+1 ) = ( u + B(u k, u k ) + L(θ k ), θ + B(u k, θ k ) ). See [26] for more details. See also [33, Lemma 4.3] for similar abstract lemmata. Let a 1. We define X a as the Banach sace of divergence-free vector fields u = u(x, t), defined and measurable on R 3 R +, such that, for some C >, (5.4) u(x, t) C inf η a x η (1 + t) (η 1)/2. In the same way, for b 3 we define the sace Y b of functons θ L t (L 1 ) satisfying the estimates (5.5) θ(x, t) C inf η b x η (1 + t) (η 3)/2.

20 2 LORENZO BRANDOLESE AND MARIA E. SCHONBEK Such saces are equied with their natural norms. They are obviously decreasing with resect to inclusion as a and b grow. Recalling the definition of X and Y in Section 2.1, we see that X 1 = X L x,t with equivalence of the norms and that Y 3 = Y L x,t. The estimates (2.13)-(2.14) in Proosition 2.5 are thus equivalent to the conditions u X a and θ Y b. We start with some elementary embeddings. Lemma 5.4. Let L,q be the Lorentz sace, with 1 < < and 1 q. Then the following four inequalities hold: u(t) L,q C u X t 1 2 ( 3 1), 3 <, (5.6) u(t) L,q C u Xa (1 + t) 1 2 ( 3 1), 3 a <, θ(t) L,q C θ Y t 1 2 ( 3 3), 1 < θ(t) L,q C θ Y3 (1 + t) 1 2 ( 3 3), 1 < for some constants C deending only on and q. In articular, choosing = q one gets the correonding estimates for the classical L -saces. Proof. The above estimates for the weak-lebesgue saces L, are simle. Indeed, if u X, then u(x, t) C x 3/ t 1 2 ( 3 1) and one has only to recall that any function bounded by x 3/ belongs to L,. The other L, -estimates for u and θ contained in (5.6) In the case 1 q <, we use that L,q is a real interolation sace between L ε, and L +ε,, for all 1 < ε < < + ε <. Therefore estimates (5.6) for all 1 q follow from the corresonding estimates in the articular case q = via the interolation inequality. The first useful estimate in view of the alication of Lemma 5.1 is the following. Lemma 5.5. Let 1 a < 3 and θ Y 3. We have, for some constant C > deending only on a, (5.7) L(θ) Xa C θ Y3. Moreover, (5.8) L(θ) X C θ Y. Proof. We rove only (5.7) since the roof of (5.8) is essentially the same. By a renormalization, we can and do assume that θ Y3 = 1. Let K(x, t) be the kernel of the oerator e t P. Then we can write t L(θ)(x, t) = K(x y, t s)θ(y, s)e 3 dy ds. We have the well known estimates for K (see, e.g., [7, Pro. 1]) (5.9) K(x, t) C x a t (3 a)/2, for all a 3

21 DECAY AND GROWTH OF SOLUTIONS OF BOUSSINESQ (Setember 17, 211) 21 where C > is come constant indeendent on x, t and a 3. We also recall the scaling relation (5.1) K(x, t) = t 3/2 K(x/ t, 1) and the fact that K(, t) C (R 3 ) for t >. The usual L estimates for K are (5.11) K(t) Ct 3/2+ 3 2, 1 <. Using the L 2 -L 2 convolution inequality, we get L(θ)(t) C t (t s) 3/4 (1 + s) 3/4 ds C(1 + t) 1/2. Owing to this estimate, the conclusion L(θ) X a will follow rovided we rove the ointwise inequality, This leads us to decomose L(θ) (x, t) C a x a t (a 1)/2, (x, t) s.t. x 2 t. L(θ) = I 1 + I 2 + I 3, where I 1 = t y x /2..., I 2 = t x y x /2... and I 3 = t y x /2, x y x /2.... Using θ L t (L 1 ) we get (5.12) I 1 (x, t) C x 3 t, which is even better in the region {(x, t): x 2 t} than what we need (recall that 1 a < 3). Using now θ(x, t) C x 3 and the scaling roerties of K we obtain by a change of variables t I 2 (x, t) C x 3 K(y, 1) dy y x /(2 s) C x 3 t log( x / t) C 3 a x a t (a 1)/2 for x 2 t, and 1 a < 3. Next, using again θ(x, t) C x 3 and K(x, t) C x 3, t I 3 (x, t) C y 6 dy ds C x 3 t. Therefore, L(θ) (x, t) Lemma 5.5 in now established. y x /2 C 3 a x a t (a 1)/2, x 2 t. We collect in the following Lemma all the estimates on B(u, v) that we shall need. (We will aly estimate (5.15) in the roof of Proosition 2.4, estimate (5.14) for Proosition (2.4) and estimate (5.14) in Theorem 2.6.

22 22 LORENZO BRANDOLESE AND MARIA E. SCHONBEK Lemma 5.6. Let 1 a < 3. For some constant C >, deending only on a we have (5.13) B(u, v) Xa C u X v Xa and (5.14) B(u, v) X(2a) C u Xa v Xa, where (2a) = min{2a, 4}. Moreover, (5.15) B(u, v) X C u X v X. Proof. We begin with the roof of the first estimate. As before, we can assume u X = v Xa = 1. We start writing t (5.16) B(u, v)(x, t) = F (x y, t s)(u v)(y, s) dy ds, where F (x, t) is the kernel of the oerator e t P. The well known counterart of relations (5.9)-(5.1) are (see, e.g., [29], [7, Pro. 1]) (5.17) F (x, t) C x η t (4 η)/2, for all η 4 and some constant C > indeendent on x, t and on a 4. Moreover, (5.18) F (x, t) = t 2 F (x/ t, 1). These bounds imly the useful estimates (5.19) F (t) Ct (1 ). Alying the first of (5.6) with = q = 6, we get u(t) 6 t 1/4. Similarily, v(s) (1 + t) 1/2. Hence, (5.2) B(u, v) C C t t F (t s) 6/5 u v(s) 6 ds (t s) 3/4 s 1/4 (1 + s) 1/2 ds C(1 + t) 1/2. It remains to establish a ointwise estimate in the in the region {(x, t): x 2 t}. Let us decomose B(u, v) = I 1 + I 2, by slitting the integrals as t y x /2... and t y x /2.... For the estimate of I 1 we use u s 1/2, v y a s (a 1)/2 and F (x, t) C x 3 t 1/2. For the estimate of I 2 we use again u s 1/2, v C y a s (a 1)/2, and the L 1 -estimate for F (see (5.19)). This leads to B(u, v) (x, t) Ct x a t (a 1)/2. The conclusion follows combining this with estimate (5.2). The roof of estimate (5.14) is similar. Notice the limitation (2a) 4, which is due to the restriction on η in inequality (5.17). The roof of (5.15) also follows along the same lines and is left to the reader.

23 DECAY AND GROWTH OF SOLUTIONS OF BOUSSINESQ (Setember 17, 211) 23 We finish with B(θ, u). Lemma 5.7. Let a 1, b 3. For some constant C > deending only on a, b, we have (5.21) B(θ, u) Yb C u X θ Yb and (5.22) Moreover, (5.23) B(θ, u) Ya+b C u Xa θ Yb. B(θ, u) Y C u X θ Y. Proof. As before, we give details only for the first estimate. Denoting F (x, t) the kernel of e t, we can write t (5.24) B(θ, u)(x, t) = F (x y, t s)(θ u)(y, s) dy ds, Notice that F rescales exactly as F. Moreover, (5.25) F (x, t) C η x η t (4 η)/2, for all η < These are the same estimates as for F, but there is now no limitation to the satial decay rate (i.e., the restriction η 4 aearing in (5.17) can be removed). Therefore, the sace-time ointwise decay estimates for B(θ, u)(x, t) can be roved essentially in the same way as in the revious Lemma. The L 1 -estimate (useful for estimating the Y-norm is straightforward: B(θ, u)(t) 1 C t (t s) 1/2 u(s) θ(s) 1 ds C u X θ Y. This allows us to conclude. Proof of Proosition 2.4. We need two elementary estimates on the linear heat equation. Namely, ( ) (5.26) e t θ Y C θ 1 + ess su x 3 θ (x) x and (5.27) e t u X Cess su x u (x). x Both estimates immediately follow from direct comutations on the heat kernel g t (x) = (4π t) 3/2 e x 2 /(4t). (See, e.g. [4, 29]) Here one only needs to use g t (x) C x 3 and the usual L 1 -L estimates for g t. Letting U = e t u and Θ = e t θ, by assumtion (2.1) we get, for some C >, U X + Θ Y Cɛ. The system (5.1) can be written in the abstract form (5.2). By inequalities (5.8), (5.15) and (5.23), all the assumtions of Lemma 5.1 are satisfied rovided ɛ > is small enough. The conclusion of Proosition 2.4 readily follows.

24 24 LORENZO BRANDOLESE AND MARIA E. SCHONBEK Proof of Part (a) of Proosition 2.5. By construction, the solution (u, θ) of Proosition 2.4 is obtained as the limit in X Y of the sequence (u k, θ k ) defined in (5.3). By the first of assumtions (2.12), and alying straightforward estimates on the heat kernel (see also [4, 29]), e t u (x) C(1 + x ) a and e t u C(1 + t) a/2. (Here we need a < 3). These two conditions imly in articular that e t u X a. Similarily one deduces from the second inequality in (2.12) that e t θ Y b (when b = 3 one here needs also θ L 1 ). By estimate (5.21) and assumtion (2.1), for all k = 1, 2... we get θ k+1 Yb e t θ Yb + Cɛ θ k Yb. If ɛ > is small enough then Cɛ < 1 (the size of the admissible ɛ thus deend on b and, as we will see later, also on a). Iterating this inequality shows that the sequence (θ k ) is bounded in Y b. Combining estimates (5.7) with (5.13) we get u k+1 Xa e t u Xa + Cɛ u k Xa + C θ k Y3. Assuming Cɛ < 1, we deduce from the boundness of (θ k ) in Y 3 that (u k ) is bounded in X a. Thus, the solution (u, θ) belongs to X a Y b and the first art of Proosition 2.5 follows. Proof of Part (b) of Proosition 2.5. The roof of the second art of Proosition (2.5) is quite similar but relies on the use of slightly different function saces. So, let a 2. We define X a as the Banach sace of divergence vector fields u = u(x, t) such that, for some C >, (5.28) u(x, t) C inf η a x η (1 + t) (η 2)/2. For b 4 we define the sace Ỹb of functons θ L t (L 1 1 ) satisfying the estimates (5.29) θ(x, t) C inf η b x η (1 + t) (η 4)/2. Such saces are equied with their natural norms. Notice that the saces X a and Ỹb differ from the their counterarts X a and Y b only by the fact that the time decay conditions are slightly more stringent in the former case. The counterart of estimates (5.6) are (5.3) u(t) L,q C u Xa (1 + t) 1 2 ( 3 2), max{1, 3 a } <, θ(t) L,q C θ Ỹ4 (1 + t) 1 2 ( 3 4), 1. The first of (5.3) can be comleted by (5.31) u(t) 1 C u Xa (1 + t) 1+ 3 a, if 3 < a < 4. This last estimate follows immediately by slitting the integral u into the regions x t 1/a and x t 1/a. We also notice the continuous embedding (5.32) Ỹ b L t (L 1 1), if b > 4

25 DECAY AND GROWTH OF SOLUTIONS OF BOUSSINESQ (Setember 17, 211) 25 that follows easily by slitting the integral (1 + x ) θ(x, t) dx into x (1+t) 1/2... and x (1+t) 1/2..., and using the bound θ(x, t) C(1+t) 2 for the first term and θ(x, t) C x b for the second one. Lemma 5.8. Let 2 a < 4. If θ(t) dx = for all t, then for some C >, (5.33) L(θ) Xa C θ Ỹ4. Assume, without restriction, θ Ỹ4 = 1. The time decay estimate L(θ) = t K(t s) θ(s) ds C(1 + t) 1 immediately follows by the L 2 -L 2 Young inequality. It only remains to rove that L(θ) can be bounded by C x a (1 + t) (a 2)/2 for all (x, t) belonging to the arabolic region x 2 t. Thus, we decomose (5.34) L(θ) = I 1 + I 2 + I 3 = (I 1,1 + I 1,2 + I 1,3 ) + I 2 + I 3. where I 1, I 2 and I 3 are as in Lemma 5.5 and and the terms contributing to I 1 are defined below. First, ( t I1,1 )j (x, t) K j,3 (x, t s) θ(y, s) dy ds = by the zero-mean assumtion on θ. Next ( t I1,2 )j (x, t) K j,3 (x, t s) and ( I1,3 ) j (x, t) t 1 y x /2 θ(y, s) dy ds ( ) K j,3 (x λy, t s) dy y θ(y, s) dλ ds, y x /2 where we have used the Taylor formula to write the difference K j,3 (x y, t s) K j,3 (x, t s). From the bounds K(x, t) C x 3 and K(x, t) C x 4 we get I 1,2 + I 1,3 C x 4 t, where we used the continuous embedding of L 1 1 into Ỹ4 that follows from the definition of such sace. The above ointwise estimate is even better, in our arabolic region x 2 t, than what we actually need. Next, t (I 2 ) j = K j,3 (x y, t s)θ(y, s) dy ds x y x /2 can be bounded as follows t I 2 C x 4 x y x /2 C x 4 t log( x / t) C x a t (a 2)/2 for all (x, t) such that x 2 t (recall that 2 a < 4). K j,3 (x y, t s) dy ds