Well-posedness and asymptotic behaviour for the Bousinessq system in Rn

XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matem´ atica Aplicada Sevilla, 24-28 septiembre 2007 (pp. 1–8) Well-posedness and asymptotic behaviour for the Boussinesq system in Rn. E. J. Villamizar-Roa1, L. C. Ferreira 2 1Escuela de Matem´aticas, Universidad industrial de Santander, A.A. 678, Bucaramanga-Colombia. E-mails: [email protected]. 2Dpto. de Matem´atica, Universidade Federal de Pernambuco, Recife-Brazil. E-mail: [email protected]. Palabras clave: Well-posedness, asymptotic behaviour, Boussinesq system Resumen We analyze the well-posedness of the initial value problem for a Convection Problem. Mild solutions are obtained in the weak-Lp(Rn) spaces and the existence of self-similar solutions is showed, while the only small self-similar solution in the Lebesgue space Lp(Rn) is the null solution. The asymptotic stability of solutions is analyzed and, as a consequence, a criterium of self-similarity persistence at large times is obtained. 1. Introduction We consider a viscous incompressible fluid filling the whole space Rn, n ≥2.Due to the Boussinesq approximation (Chandrasekhar [3]), density variations are neglected except in the gravitational term (buoyancy term) and they are assumed to be proportional to temperature variations. The relationship among the velocity field u(x, t)∈Rn,the pressure p(x, t)∈Rand the temperature θ(x, t)∈R,can be described by the following initial value problem ∂u ∂t +u∇u−ν∆u+1 ρ∇p=βθf +f1, x ∈Rn, t > 0,(1) ∇ · u= 0, x ∈Rn, t > 0,(2) ∂θ ∂t +u∇θ−χ∆θ=h, x ∈Rn, t > 0,(3) θ(x, 0) = θ0(x), u(x, 0) = u0(x), x ∈Rn,(4) where frepresents a gravitational vector field at x,hthe reference temperature and f1 an external force. ρ, ν, β, χ are positive physical constants which represent, respectively, 1 E.J. Villamizar-Roa, L. C. Ferreira the density, the kinematic viscosity, the coefficient of volume expansion and the thermal conductance. Without loss of generality, we will assume the constants ρ, ν, β, χ to be one and the reference temperature hand the external force f1to be zero. The initial data u0 satisfies the condition ∇ · u0= 0 in the distributional sense. New aspects to studies on the Convection Problem (1)-(4) are considered in this work, in fact, we will study the system (1)-(4) in the whole space Rnin the framework of weak−Lpspaces. Firstly we present results of well-posedness in these spaces and make some considerations around the well-posedness in the Lebesgue Spaces Lp(see [2]). On the other hand, we show some results about the existence, uniqueness, the asymptotic stability and the self-similarity persistence of solutions for the Problem (1)-(4) in weak−Lp spaces. Moreover, as a consequence of results of asymptotic stability, we will show that the only self-similar solutions in Lebesgue spaces Lpis the null solution, reinforcing the need of more singular initial data to allow the existence of self-similar solutions. These self-similar solutions correspond, for instance, to homogeneous initial functions of degree −1. Finally, from a physical standpoint, our analysis can be applied for several classes of external forces f. In fact, we can take fas the gravitational field f=f(x) = −G∇xφ=Gx |x|3∈L(n 2,∞)(Rn), where Gis the gravitational constant, in order to show the existence of global solutions (u, θ) which are constructed in different functional spaces (see Theorem 3.5, Theorem 3.3 and Remark 4.5). This case can be regarded as an interesting physical case of the B´enard Problem. More details about the physical and mathematical analysis of system (1)-(4) see [2] and the references therein. 2. Function Spaces and Definitions In this section, we introduce the functional spaces relevant to our study of solutions regarding the Cauchy problem for system (1)-(4). For each Lebesgue mensurable function fdefined on Rn, the rearrangement f∗is defined by f∗(t) = ´ınf{s > 0 : m({x∈Rn:|f(x)|> s})≤t}, t > 0. The Lorentz space L(p,q)≡L(p,q)(Rn) is the set of all fsuch that kfk(p,q)=   ³p qR∞ 0[t1 pf∗∗(t)]qdt/t´1 q, if 1 < p < ∞,1≤q < ∞, supt>0t1 pf∗∗(t) , if 1 < p ≤ ∞,q=∞. is finite, where f∗∗(t) = 1 tRt 0f∗(s)ds, for t > 0.We observe that Lp=L(p,p).L(p,∞)are called the Marcinkiewicz spaces or weak-Lpspaces. Moreover, L(p,q1)⊂Lp⊂L(p,q2)⊂ L(p,∞)for 0 < q1≤p≤q2≤ ∞. See [4]. Proposition 2.1 [4] (Generalized Holder’s inequality). Let 1< p1,p2,r < ∞. Let f∈ L(p1,q1)and g∈L(p2,q2)where 1 p1+1 p2<1, then the product h=fg belongs to L(r,s)where 1 r=1 p1+1 p2, and s≥1is any number such that 1 q1+1 q2≥1 s. Moreover, khk(r,s)≤r0kfk(p1,q1)kgk(p2,q2), 2 Well-posedness and asymptotic behaviour for the Boussinesq system being r0the conjugate index of r. Let us recall the Helmholtz decomposition Lr(Ω) = Lr σ(Ω)⊕Gr(Ω),1< r < ∞,where Gr(Ω) = {∇p∈Lr(Ω) : p∈Lr loc(Ω)}.Pr(or simply P) denotes the projection operator from Lronto Lr σ.The Stokes operator is denoted by Ar(or simply Aor −P∆) and the Laplace operator is denoted by Br(or simply Bor −∆). We know that −Ar,−Brgenerate uniformly bounded holomorphic semigroups {e−tAr}t≥0,{e−tBr}t≥0of class C0in Lr σand Lr,respectively. Borchers and Miyakawa [1] established the following Helmholtz decomposition of the Lorentz spaces. We can extend Prto a bounded operator on L(r,d)(Ω), which we denote by Pr,d.Set L(r,d) σ(Ω) = Range(Pr,d) and G(r,d)(Ω) = Kernel(Pr,d). Then, L(r,d)(Ω) = L(r,d) σ(Ω) ⊕G(r,d)(Ω).Based on [1], −A, −Bgenerate uniformly bounded analytic semigroups on L(r,d) σ(Ω) and L(r,d)(Ω),respectively. However, notice that these semigroups are not strongly continuous at t= 0 if d=∞,since in this case Dr,∞(A) and Dr,∞(B) are not dense in L(r,∞) σand L(r,∞),respectively. We recall that in our case Ω = Rn, {e−tB}t≥0is the heat semigroup given as the convolution with the Gauss-Weierstrass kernel: G(x, t) = (4πt)−n/2exp(−|x|2/(4t)). Finally, let 1 < p < ∞,1< q < ∞and 1 ≤d≤ ∞.The following notation is adopted for the norm of product in Lorentz spaces L(p,d) σ(Rn)×L(q,d)(Rn) : ° ° °hu θi° ° °(p,q),d =kuk(p,d)+kθk(q,d). If p=qand d=∞,we simply denote this norm as ° ° °hu θi° ° °(p,∞)=kuk(p,∞)+kθk(p,∞). 3. Results of well-posedness We define the operator M:L(p,∞) σ×L(q,∞)→L(p,∞) σ×L(q,∞)by Mhu θi=h−P∆u −∆θi. With the use of the semigroup {e−tM }t≥0,the Cauchy problem (1)-(4) is converted to the integral equation hu(t) θ(t)i=e−tM hu0 θ0i−Zt 0 e−(t−s)M³h(u· ∇u) (u· ∇θ)i−h(θf) 0i´ds, t > 0,(5) in L(p,∞) σ(Rn)×L(q,∞)(Rn).The term in (5) −Zt 0 e−(t−s)Mh(u· ∇u)(s) (u· ∇θ)(s)ids will be called the bilinear vector, and the term in (5) Zt 0 e−(t−s)P∆(θf)ds, 3 E.J. Villamizar-Roa, L. C. Ferreira we will called of coupling term. Let us remember the following L(p,d)−L(r,d)estimates of the semigroup {e−tM }t≥0 and give the proof by completeness. Lemma 3.1 [2] Let 1≤d≤ ∞. For all (ϕ, φ)∈L(p,d) σ(Rn)×L(q,d)(Rn),and all t > 0, there exists a constant C(p, r, s, q)such that ° ° °∇je−tM hϕ φi° ° °(r,s),d ≤Ct−n 2(γ+j n)° ° °hϕ φi° ° °(p,q),d, where γ= 1/p −1/r = 1/q −1/s, with 1< p ≤r < ∞and 1< q ≤s < ∞. Next, let us introduce suitable time dependent functional spaces in which we will need to study the initial value problem (1)-(4). Definition 3.2 Let n < q < ∞and α= 1 −n/q. We define the spaces E≡ {(u, θ) : (u, θ)∈BC((0,∞), L(n,∞) σ×L(n,∞))}, Eq≡ {(u, θ)∈E:tα/2(u, θ)∈BC((0,∞), L(q,∞) σ×L(q,∞))}, Fq≡ {(u, θ) : u∈BC((0,∞); L(n,∞) σ), tα/2θ∈BC((0,∞); L(q,∞))}, which are Banach spaces with the norms in E, Eq, Fqdefined, respectively, as ° ° °hu θi° ° °E= sup t>0° ° °hu θi° ° °(n,∞),° ° °hu θi° ° °Eq =° ° °hu θi° ° °E+ sup t>0 tα/2° ° °hu θi° ° °(q,∞), ° ° °hu θi° ° °Fq = sup t>0 kuk(n,∞)+ sup t>0 tα/2kθk(q,∞). Theorem 3.3 (i) Let n > 2a positive integer number, (u0, θ0)be any pair in L(n,∞) σ× L(n,∞)and fsmall enough with respect the following norm kfkb= sup t>0 tβ 2kf(t)k(b,∞)<∞, β = 2 −n b, b > n 2. Then, there are constants 0< τ =τ(f)<1,δ > 0and ε=ε(δ)>0(ε→0when δ→0) such that if ° ° °hu0 θ0i° ° °(n,∞)< δ, the initial value problem (1)-(4) has a global solution (u(t, x), θ(t, x)) ∈Esatisfying (5), with l´ım t→0(u(t), φ) = (u0, φ),l´ım t→0(θ(t), ϕ) = (θ0, ϕ), for all φ∈L(n0,1) σ(Rn), ϕ ∈L(n0,1) σ(Rn).Moreover, if ° ° °hu θi° ° °E<2ε 1−τ,then the solution is unique. (ii) If we assume that (u0, θ0)∈(L(n,∞) σ×L(p,∞) σ)∩(L(n,∞)×L(p,∞))with 1< p0< n, there are 0< δp≤δand 0< τp=τp(f)≤τsuch that if ° ° °hu0 θ0i° ° °(p,∞)< δp,then previous solution (u, θ)verifies that (u, θ)∈BC((0,∞), L(p,∞) σ×L(p,∞)). 4 Well-posedness and asymptotic behaviour for the Boussinesq system Theorem 3.4 (Regularization) Under the assumptions of Theorem 3.3, let n < q < ∞, such that 1 b+1 q>1 n. If kf(t)kb= supt>0tβ 2kf(t)k(b,∞)is small enough, there are constants 0< τq(f)<1and 0< δq≤δsuch that if ° ° °hu0 θ0i° ° °(n,∞)< δq,then the solution (u, θ)of Theorem 3.3 belongs to Eq. In the case n > 2, the assumption kfkb<∞can be changed by the following one: supt>0kf(t)k(n 2,∞)<∞. Indeed we will prove the following theorem: Theorem 3.5 Let (u0, θ0)∈L(n,∞) σ×L(n,∞)where n > 2and assume that fbelongs to BC((0,∞), L(n 2,∞)). If n < q < ∞and supt>0kfk(n 2,∞)is sufficiently small, then there are constants 0< τ =τ(f)<1,δ > 0and ε=ε(δ)>0(ε→0when δ→0) such that if ° ° °hu0 θ0i° ° °(n,∞)< δ, then the initial value problem for (1)-(4) has a global solution (u(t, x), θ(x, t)) ∈Fqsatisfying (5) together with l´ım t→0(u(t), φ) = (u0, φ),l´ım t→0(θ(t), ϕ) = (θ0, ϕ), for all φ∈L(n0,1) σ(Rn), ϕ ∈L(n0,1) σ(Rn).Moreover, if ° ° °hu θi° ° °Fq ≤2ε 1−τ,then the solution is unique in the space Fq. Furthermore, if we assume that (u0, θ0)∈(L(n,∞) σ∩L(p,∞))×(L(n,∞) σ∩L(p,∞)), with q0< p0<n 2, there are 0< δp≤δand 0< τp=τp(f)≤τsuch that if ° ° °hu0 θ0i° ° °(n,∞)< δp, then previous solution (u, θ)satisfies (u, θ)∈BC((0,∞), L(p,∞) σ×L(p,∞)). 3.1. Sketch of Proofs of the well-posedness Theorems The proofs of the well-possedness Theorems follows basically from the next lemma in a generic Banach space and lemmas 3.7, 3.8, 3.9 ( see [2]). Lemma 3.6 Let Xbe a Banach space with norm k · kX,T:X→Xa linear continuous map with norm τ < 1and B:X×X→Xa continuous bilinear map, that is, there exists a constant K > 0such that for all x1and x2in XkB(x1, x2)kX≤Kkx1kXkx2kX. Then, for 0< ε < (1−τ)2 4Kand for any vector y∈X,y6= 0, such that kykX< ε, there exists a solution x∈Xfor the equation x=y+B(x, x) + T(x)such that kxkX≤2ε 1−τ. The solution xis unique in the closed ball B2ε 1−τ:= B(0,2ε 1−τ)⊂X. Moreover, the solution depends continuously on yin the following sense: If k˜ykX≤ε,˜x= ˜y+B(˜x, ˜x) + T(˜x), and k˜xkX≤2ε 1−τ, then kx−˜xkX≤1−τ (1−τ)2−4Kε ky−˜ykX. Lemma 3.7 If (u0, θ0)∈L(n,∞) σ×L(n,∞).Then e−tM hu0 θ0i∈E, with ° ° °e−tM hu0 θ0i° ° °E≤ C° ° °hu0 θ0i° ° °(n,∞)and e−tM hu0 θ0i*hu0 θ0iwhen t→0+,where the limit is taken in the 5 E.J. Villamizar-Roa, L. C. Ferreira weak-star topology of the L(n,∞) σ×L(n,∞).Moreover ° ° °e−tM hu0 θ0i° ° °Eq ≤C° ° °hu0 θ0i° ° °(n,∞),and if (u0, θ0)∈(L(p,∞) σ×L(p,∞))then ° ° °e−tM hu0 θ0i° ° °(p,∞)≤C° ° °hu0 θ0i° ° °(p,∞). Lemma 3.8 Let n, b be as in the Theorem 3.3 and T(θ) = Rt 0e(t−s)P∆(θf)(s)ds. Then kT(θ)k(n,∞)≤Ckfkbsup t>0 kθk(n,∞),kT(θ)k(p,∞)≤Ckfkbsup t>0 kθk(p,∞). Moreover, if n, b satisfy the assumptions of Theorem 3.4, then kT(θ)kEq≤Ckfkbsup t>0 tα 2kθk(q,∞). Lemma 3.9 If 1< p < q < ∞then for all φ∈L(p,1)(Rn)hold: s1 2(n p−n q+1)k∇e−sM φk(q,1) ≤Ckφk(p,1), s1 2(n p−n q)ke−sM φk(q,1) ≤Ckφk(p,1), Z∞ 0 s1 2(n p−n q)−1 2k∇e−sM φk(q,1)ds ≤Ckφk(p,1),Z∞ 0 s1 2(n p−n q)−1ke−sM φk(q,1)ds ≤Ckφk(p,1). 4. Self-Similarity Assuming that f(t, x) = λ2f(λ2t, λx) is smooth and that (u(t, x), θ(t, x)) is a smooth solution of the convection problem (1)-(4), it is straightforward to check that (u, θ)λ(t, x) = λ(u(λ2t, λx), θ(λ2t, λx)) is also a solution of the System (1)-(4). In fact, we can look for particular solutions of the System (1)-(4) satisfying (u(t, x), θ(t, x)) = (u(t, x), θ(t, x))λ(t, x),(6) for any t > 0, x∈Rnand λ > 0.These solutions are called self-similar solutions of the system and it is clear that taking t→0+,formally in (6), (u(0, x), θ(0, x)) should be a homogeneous function of degree −1. This remark gives the hint that a suitable space to find self-similar solutions should be one containing homogeneous functions with that exponent. The space L(n,∞)is the only weak-Lpspace such that |x|−1∈L(p,∞). Moreover, in case that such a self-similar solution exists, its norm is invariant by the scaling transformation, (u(t, x), θ(t, x)) →(u(t, x), θ(t, x))λ=λ(u(λ2t, λx), θ(λ2t, λx)). Moreover, homogenous functions of any order do not belong to any strong Lpspace. All of these facts reinforce the idea that weak-Lpspaces with the right homogeneity are the most relevant spaces for finding global non-trivial self-similar solutions to the Convection Problem. 4.1. Decay Estimates in weak −Lpand Lp Theorem 4.1 Let (u0, θ0)as in the Theorem 3.4 and r≥pis finite and satisfies 1 p+1 q− 1 r<1 nand 1 b+1 p−1 r<2 n. Then the solution of the Theorem 3.4 satisfies t(n 2p−n 2r)u∈BC((0,∞); L(r,∞) σ)n, t(n 2p−n 2r)θ∈BC((0,∞); L(r,∞)). Moreover, this theorem is true by relaxing the assumptions to n≥2, and even substituting weak-Lpspaces by their stronger counterparts. 6 Well-posedness and asymptotic behaviour for the Boussinesq system 4.1.1. Proof of Theorem 4.1. Let (u0, θ0)∈L(p,∞) σ×L(p,∞). As a direct consequence of Lemma 3.1 we have that supt>0t r−p 2γ° ° °e−tM hu0 θ0i° ° °(r,∞)≤C° ° °hu0 θ0i° ° °(p,∞).Now, by the second part of Theorem 3.3, we already know that if the initial data (u0, θ0)∈(L(p,∞) σ∩L(n,∞) σ)×(L(p,∞)∩L(n,∞)), then the solution (u(t), θ(t)) satisfies supt>0(ku(t)k(p,∞)+kθ(t)k(p,∞))<∞.Thus, in order to conclude the proof of the Theorem 4.1, we need a lemma where we estimate the norm supt>0t r−p 2γk·kr,∞of bilinear vector term and the linear operator term T(θ), using the norm k · kEq+ supt>0k·kp,∞of the solution. For this, we prove the following lemmas. Lemma 4.2 Let pand bas in the Theorem 4.1 and r≥p, then sup t>0 t(n 2p−n 2r)kT(θ)(t)k(r,∞)≤Ckfkbsup t>0 kθk(p,∞). Lemma 4.3 Let pas in the Theorem 4.1 and r≥p, then sup t>0 tρ ° ° °Zt 0 e−(t−s)Mh(u1· ∇u2) (u2· ∇θ1)i° ° °(r,∞)≤Csup t>0° ° °hu1 θ1i° ° °(p,∞)sup t>0 tα 2ku2k(q,∞), where ρ= ( n 2p−n 2r). 4.2. Self-Similar Solution in the spaces L(n,∞). The aim of this subsection is to describe the principal results relative to the existence and the uniqueness of self-similarity solutions in the L(n,∞)-spaces. Theorem 4.4 Let (u0, θ0)∈L(n,∞) σ×L(n,∞). Assume that u0, θ0are homogeneous functions of degree −1, that is, u0(λx) = λ−1u0(x), θ0(λx) = λ−1θ0(x)for all x∈Rn,x6= 0 and all λ > 0and fas in Theorem 3.3 and Theorem 3.5, satisfies the scale relation f(t, x) = λ2f(λ2t, λx).Then, if ° ° °hu0 θ0i° ° °(n,∞)< ε the solution (u, θ)given by Theorem 3.3 and Theorem 3.5 is self-similar, i.e., u(t, x) = λu(λ2t, λx), θ(t, x) = λθ(λ2t, λx), for all x∈Rn,x6= 0 and all λ > 0. Moreover, in case of Theorem 3.3, if the initial data is smaller ° ° °hu0 θ0i° ° °(n,∞)< εq, the previous unique self-similar solution becomes regularized. Remark 4.5 [B´enard Problem] Note that, in the case of Theorem 3.5, we can take f as the gravitational field f=f(x) = −G∇x(1 |x|) = Gx |x|3∈L(n 2,∞)(Rn),where Gis the gravitational constant. This case can be regarded as the B´enard problem (see [3]) which corresponds to the interesting physical case. This consideration is also true for the modified Theorem 3.3, where we assume f∈BC([0,∞); L(n 2,∞)(Rn)),instead of supt>0tβ 2kf(t)k(b,∞)<∞, and we search solution in the space Eq. 7 E.J. Villamizar-Roa, L. C. Ferreira 5. Stability in L(n,∞). We analyze the large time behavior of solutions of Section 3. In short, we will show that perturbations of the initial data are negligible for large times. Theorem 5.1 Assume that (u, θ)and (v, φ)are solutions of (1)-(4) as in the Theorem 3.3 corresponding to the initial conditions (u0, θ0)and (v0, φ0)∈L(n,∞) σ×L(n,∞),respectively. Suppose that l´ım t→∞ ° °et∆(θ0−φ0)° °(n,∞)= l´ım t→∞ ° °etP∆(u0−v0)° °(n,∞)= 0, then l´ım t→∞ ku(t)−v(t)k(n,∞)= 0,l´ım t→∞ kθ(t)−φ(t)k(n,∞)= 0. Moreover, assume (u, θ)and (v, φ)are solutions of (1)-(4) given by Theorem 3.4 corresponding to initial conditions (u0, θ0)and (v0, φ0)∈L(n,∞) σ×L(n,∞)satisfying that l´ım t→∞ tα 2° °et∆(θ0−φ0)° °(q,∞)= l´ım t→∞ ° °etP∆(u0−v0)° °(q,∞)= 0, then l´ım t→∞ tα 2ku(t)−v(t)k(q,∞)= 0,l´ım t→∞ tα 2kθ(t)−φ(t)k(q,∞)= 0. Theorem 5.2 Assume that (u, θ)and (v, φ)are solutions of (1)-(4) as in the Theorem 3.5 corresponding to the initial conditions (u0, θ0)and (v0, φ0)∈L(n,∞) σ×L(n,∞),respectively. Suppose that l´ımt→∞ tα 2° °et∆(θ0−φ0)° °(q,∞)= 0 and that l´ımt→∞ ° °etP∆(u0−v0)° °(n,∞)= 0, then l´ım t→∞ ku(t)−v(t)k(n,∞)= 0,l´ım t→∞ tα 2kθ(t)−φ(t)k(q,∞)= 0. Corollary 5.3 Let (u0, θ0)∈Ln σ×Ln(Lebesgue space) be as in the Theorem 3.4. Then the corresponding solution satisfies l´ımt→∞ ku(t)kLn= 0,l´ımt→∞ kθ(t)kLn= 0.As a consequence, the unique self-similar solution in Ln(Rn)×Ln(Rn)is the null solution. Agradecimientos E J Villamizar-Roa was supported by COLCIENCIAS, Colombia, Proyecto COLCIENCIASBID III etapa and UIS and L C F Ferreira was supported by CAPES, Brazil. Referencias [1] W Borchers, and T Miyakawa, 1995 On stability of exterior Navier-Stokes flows. Acta Math. 147, 311-382. [2] L C F Ferreira, E J Villamizar-Roa, 2006 Well-posedness and asymptotic behaviour for the convection problem in Rn,Nonlinearity 19, 2169-2191. [3] S Chandrasekhar, 1981 Hidrodinamic and Hydromagnetic Stability. Dover, New York. [4] R O’Neil, 1963 Convolution operators and L(p, q)spaces. Duke Math. J. 30, 129-142. 8