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Proc. R. Soc. A (2008) 464, 1983–1999
doi:10.1098/rspa.2007.0366
Published online 8 April 2008
Existence of solutions to the convection
problem in a pseudomeasure-type space
B Y L UCAS C. F. F ERREIRA 1
AND
E LDER J ESÚS V ILLAMIZAR R OA 2, *
1
Departamento de Matemática, Universidade Federal de Pernambuco,
CEP 50740-540, Recife-PE, Brazil
2
Universidad Industrial de Santander, Escuela de Matemáticas,
A.A. 678 Bucaramanga, Colombia
We study the well-posedness of a convection problem in a pseudomeasure-type space
PMa, without assuming that the gravitational field is bounded. Considering the PMa
space with right homogeneity, the existence of self-similar solutions is proved. Finally, an
analysis about asymptotic stability is made.
Keywords: convection problem; pseudomeasure-type space; Navier–Stokes
1. Introduction
In this paper, we study the following system related to a generalized convection
problem on Rn:
ut C nðKDÞg u C ðu$VÞu C r K1 Vp Z kqf C f1 ;
div u Z 0;
x 2 Rn ;
x 2 Rn ;
tO 0;
tO 0;
ð1:1Þ
ð1:2Þ
qt C cðKDÞg q C ðu$VÞq Z h 1 ;
x 2 Rn ;
uðx; 0Þ Z u 0 ;
x 2 Rn
ð1:4Þ
qðx; 0Þ Z q0 ;
x 2 Rn ;
ð1:5Þ
tO 0;
ð1:3Þ
and
with gO0 and nR2. In (1.1)–(1.5), the unknowns u(x, t)2Rn, p(x, t)2R and
q(x, t)2R represent, respectively, the velocity field, the pressure and the
temperature of fluid. The initial velocity and the initial temperature are denoted
by u 0(x) and q0(x), respectively. Moreover, we assume that the initial data of the
velocity u 0 satisfy the condition div u 0Z0 in the distributional sense. In (1.1), f
is the gravitational field at a point x, f1 represents an external force and in (1.3),
h1 is the reference temperature. The parameters r, n, k and c are positive
physical constants that represent density, kinematic viscosity, the coefficient of
volume expansion and thermal conductance. The Riesz potential operator (KD)r
* Author for correspondence ([email protected]).
Received 12 December 2007
Accepted 13 March 2008
1983
This journal is q 2008 The Royal Society
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1984
L. C. F. Ferreira and E. J. Villamizar Roa
is defined as Ð usual through the Fourier transform as ððKDÞr f Þ^ðxÞZ jxj2rf^ðxÞ,
where f^ðxÞZ Rn e Kixx f ðxÞ dx. We use the standard notations V, D and div for
gradient, Laplacian and divergence operators, respectively. The ith components
of (u$V)u and (u$V)q, in Cartesian coordinates, are given, respectively, by
3
3
X
X
vu
vq
½ðu$VÞui Z
uj i and ½ðu$VÞq Z
uj
:
vxj
vxj
jZ1
jZ1
In the case gZ1, system (1.1)–(1.5) corresponds to the model that governs the
motion of fluid and the diffusion of heat. Indeed, it was first pointed out by
Boussinesq that there are many situations of practical occurrence in which system
(1.1)–(1.5) (gZ1) is obtained after some simplifications; therefore, to consider the
convection phenomena mathematically, the model equations derived from the
Boussinesq approximation (Landau & Lifshitz 1968; Chandrasekhar 1981) are
usually applied. This approximation says that the density variations are neglected,
with the exception of the gravitational term, where they are assumed to be
proportional to the temperature variation. Equation (1.2) comes from the wellknown continuity equation rtCdiv (ru)Z0, which, due to the homogeneity of the
fluid (r(x, t)Zconst.), is equivalent to the incompressibility condition div uZ0.
Several papers are devoted to the existence and uniqueness of solutions to the
non-stationary problem (1.1)–(1.5); see, for instance, Cannon & DiBenedetto
(1980), Hishida (1997), Ferreira & Villamizar-Roa (2006a) and papers cited
therein. In Cannon & DiBenedetto (1980), using singular integral operators, a
construction of solutions of class Lp ð0; T; Lq ðRn ÞÞ with suitable exponents p and q
was made. In Hishida (1997), the convection problem in an exterior domain
of R3, in the framework of L( p,N) spaces, was analysed. Later, in Ferreira &
Villamizar-Roa (2006a,b), the problem (1.1)–(1.5) was considered and the
well-posedness and asymptotic behaviour of solutions, in the framework of L( p,N)
spaces, including the existence of self-similar solutions, were analysed.
In this paper, we study the problem (1.1)–(1.5) considering initial data in a
pseudomeasure-type space PMa, introduced in Cannone & Karch (2004),
a
0
1
n
a
PM h v 2 S : v^ 2 L loc ðR Þ; kvka h ess sup jxj j^
v ðxÞj!N ;
x2Rn
where a2R is a given parameter. In Cannone & Karch (2004), the existence of
singular solutions for the three-dimensional Navier–Stokes equations with initial
data in the space PM2, and the asymptotic stability of small solutions, was
studied. An important characteristic of the PMa space is that it contains
homogeneous functions of degree aKn. In particular, the field Kx/j x j1C2g belongs
to the PMnK2g space. From a physical standpoint, if we take f in (1.1) as being
the gravitational field f ðxÞZKGðx=j x j3 Þ, where G is the gravitational constant,
we can interpret the convection problem (1.1)–(1.5) as a mathematical version of
the Bénard problem in Rn. Fractional powers of the Laplacian operator, which
correspond to a lesser dissipation, can model physical real phenomena, and in
this case, to maintain the right homogeneity of the equation and hence obtain the
existence of self-similar solutions, we consider the generalized gravitational field
f ðxÞZKGðx=jxj1C2g Þ. Like examples of fluid mechanics models with fractional
dissipation were analysed in the papers of Constantin & Wu (1999), Córdoba &
Córdoba (2004), Córdoba et al. (2005) and Wu (2006).
Proc. R. Soc. A (2008)
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Existence of solutions to convection problem
1985
Observing equation (1.1), the coupling term kqf carries out an important
role on the restrictions of the functional spaces where the solution is searched.
In general, when the convection problem is studied, it is assumed that the
gravitational field f belongs to the space f2LN(U) (see Cannon & DiBenedetto
1980; Hishida 1997). For instance, in Hishida (1997), the author takes UO0,
an exterior domain, and f ðxÞZKGðx=jx j3 Þ 2 LNðUÞ. This hypothesis is not
verified by the homogeneous field f in UZRn. For the Navier–Stokes equations,
the non-existence of a coupling term allows the existence of a solution in the
Banach space BCðð0;NÞ; PMa Þ3 , where aZ2 in such a way that the norm of the
subspace is invariant by the scaling property of the three-dimensional Navier–
Stokes equations. On the other hand, considering the convection problem
(1.1)–(1.5) with f being a homogeneous function of degree K2g, to obtain the
existence in BC ðð0;NÞ; PMnKð2gK1Þ ÞnC1 , the restriction nO4gK1 is necessary,
which in the case gZ1 is equivalent to the condition nO3. In order to
overcome this difficulty and include the case nZ3, besides using a fractional
dissipation (g), we study the existence of (1.1)–(1.5) in a space of vector
functions where the velocity field u and the temperature q are taken in
different functional spaces, without losing the existence of self-similar
solutions. These subjects are analysed in this paper; indeed, new aspects
around the convection problem (1.1)–(1.5) are considered in this work. Firstly,
we show the existence and uniqueness of small and large mild solutions in
pseudomeasure-type PMa space, discuss some aspects about the smoothness
and posteriorly analyse our results in the context of the homogeneous field
f ðxÞZKGðx=j x j 1C2g Þ. Finally, we study the asymptotic stability and the
existence of self-similar solutions of (1.1)–(1.5) in PMa spaces; consequently, if
we take small perturbations of initial data, we obtain a vanishing criterion.
From another point of view, we show that PMa spaces allow the existence of
initially singular solutions. These solutions are instantaneously smoothed out if
they are small enough initially. We show the existence of global solutions in PMa
space for which we do not know whether or not the singularity persists (see
remark 2.8).
Recently, we have known of the existence of the paper by Karch & Prioux
(2008), related to problems (1.1)–(1.5), x2R3, in PMa spaces. Karch & Prioux
assumed the field f as being a constant function that was small enough, and
they obtained the existence of a self-similar global solution (u, q) in class
suptO0 ðkuk2 C kqk0 C t r=2 kqkr Þ!N, 1!r!2. Note that when f is a constant,
that is, f is a homogeneous function of degree 0, the scaling of system (1.1)–(1.5)
is different to the case when the field f is a homogeneous function of degree 2
(or K2g); thus, from the point of view of scaling invariant techniques, our
problem is different to the one considered by Karch & Prioux. Indeed, it is
necessary to analyse the invariance of the scaling by choosing a new existence
class that is different to the functional setting of Karch & Prioux, and therefore
our results are entirely complementary.
The outline of this paper is given as follows. The basic properties of PMa
spaces will be reviewed in §2. In the same section, we show the results of
well-posedness. Finally, in §3, we analyse the existence of self-similar solutions
and give a result of the asymptotic stability of solutions.
Proc. R. Soc. A (2008)
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1986
L. C. F. Ferreira and E. J. Villamizar Roa
2. Preliminaries, definitions and some results
In this section, we introduce the functional spaces that will be used to construct
the solutions to system (1.1)–(1.5). We list some facts about convolution and
give the notion of solution at these spaces. We start recalling that the Leray
projector of a smooth vector field u is given by PuZ uKVD K1 ðdiv uÞ. Moreover,
we remark that P is a matrix n!n with elements Pk;j Z dkj C Rk Rj , where R j ( jZ
1, 2, ., n) are the Riesz transforms that are pseudodifferential operators defined
d
^
as R
j f ðxÞZ ðixj =jxjÞf ðxÞ. In this way, in order to prove the continuity of Leray
operator P on the PMa spaces, it is sufficient to prove the continuity of the Riesz
transform R j (jZ1, ., n).
Lemma 2.1. The Riesz transform R j (jZ1, 2, ., n) is continuous at PMa
spaces, a2R with kR j kZ1.
Proof. Estimating directly the norm k$ka of R j, we have
a d
a xj
v ðxÞj Z kvka ;
kRj vka Z ess sup jxj jRj vðxÞj Z ess sup jxj v^ðxÞ % ess sup jxja j^
n
n
n
jxj
x2R
x2R
x2R
which proves the lemma.
&
Now we recall a fact about convolution, which will be useful to carry out some
estimates in PMa spaces.
Proposition 2.2 (convolution of singular kernels, Lieb & Loss 2001). Let
0!a!n, 0!b!n and 0!aCb!n. Then we have
ð
aKn
bKn
jxj ÞðyÞ Z
jzjaKn jyKzjbKn dz Z C ða; b; nÞjyjaCbKn :
ð2:1Þ
ðjxj
Rn
Now, we describe our results of well-posedness of system (1.1)–(1.5) in PMa
spaces. We start with the definition of time-dependent functional spaces needed
to study the initial-value problem (1.1)–(1.5). Spaces of scalar-value and vectorvalue distributions will be denoted in the same way.
Definition 2.3. Let aZ nKð2g K1Þ, 0!T%N, a%q, r!N, aq Z ðqKaÞ=g and
ar Z ðr KaÞ=g. We define the following Banach spaces of time-dependent
distributions:
ar =2
ET
u 2 BC ðð0; TÞ; PMr Þn ; t aq =2 q 2BC ðð0; TÞ; PMq Þg
r;q Z fðu; qÞ : t
and
r n
r
ET
r Z fðu; qÞ : u 2 BC ðð0; TÞ; PM Þ ; q 2BC ðð0; TÞ; PM Þg;
with the norms defined, respectively, as
kðu; qÞkE Tr;q Z sup t ar =2 kuðtÞkr C sup t aq =2 kqðtÞkq
0!t!T
0!t!T
and
kðu; qÞkE Tr Z sup kuðtÞkr C sup kqðtÞkr ;
0!t!T
0!t!T
which are weakly continuous in the distributional sense at tZ0. When TZN, we
T
denote the spaces E T
r;q and E r simply by Er,q and Er , respectively.
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1987
Existence of solutions to convection problem
With no loss of generality, we will take the constants r, n and c in (1.1)–(1.5)
to be equal to one. Moreover, we will take h1 and f1 to be zero. For a general
case, our results remain valid with slight modifications. Therefore, applying the
Leray projector in equation (1.1), and using PðVpÞZ 0 and div (u)Z0, system
(1.1)–(1.5) is formally reduced to the following one:
ut C ðKDÞg u C Pðu$VÞu Z kPðqf Þ;
qt C ðKDÞg q C ðu$VÞq Z 0;
x 2 Rn ;
x 2 Rn ;
tO 0;
tO 0;
ð2:2Þ
ð2:3Þ
uðx; 0Þ Z u 0 ;
x 2 Rn
ð2:4Þ
qðx; 0Þ Z q0 ;
x 2 Rn ;
ð2:5Þ
and
where the velocity filed satisfies div (u)Z0. Let us recall that applying the
divergence operator in equation (1.1), we obtain the elliptical equation DpZKdiv
ððu$VuÞKkqf Þ, and therefore the pressure p may be recovered by
Vp ZKVD K1 div ððu$VuÞKkqf Þ Z ðPKI Þððu$VuÞKkqf Þ:
Now, with the help of Duhamel’s principle, we introduce the notion of a solution for
system (2.2)–(2.5) in Fourier variables.
Definition 2.4. Let 0!T%N and 1/2!g%1. A mild solution to system (2.2)–
(2.5), with initial data y0Z(u 0, q0) and div u 0Z0, in PMa spaces, is a timedependent distribution yZ ðu; qÞ 2 E T
r;q such that
u0
u^ðx; tÞ Z expðKtjxj2g Þ^
ðt
h i
dÞðx; sÞ ds
d ðx; sÞ C k ðqf
^
C expðKðtKsÞjxj2g Þ PðxÞ
ix$ u5u
ð2:6Þ
0
and
^ tÞ Z expðKtjxj2g Þq^0 C
qðx;
ðt
dÞðx; sÞ ds
expðKðtKsÞjxj2g Þix$ðqu
ð2:7Þ
0
for all 0!t!T, which satisfies div uZ0 and y(t).y0 when t/0C, in the
distributional sense.
^
^
Above, we denote by PðxÞ
the matrix n!n with components ð PðxÞÞ
i;j Z
dij Kðxi xj =jxj2 Þ, which is the symbol of the operator P. Let us write y0Z(u 0, q0),
y1Z(u 1, q1) and y2Z(u 2, q2). From now on, we will use the notation given below.
Gd
ðtÞðy0 Þ Z Gd
ðtÞðu 0 Þ; Gd
ðtÞð0 Þ Z ðexpðKtjxj2g Þu^0 ; expðKtjxj2g Þq^0 Þ;
ð2:8Þ
d FðqÞ
d Zk
FðyÞZ
ðt
0
dÞ ðx; sÞ ds
^ ðqf
expðKðtKsÞjxj2g Þ PðxÞ
ð2:9Þ
and
d
d
d
Bðy
;
y
Þ
Z
B
ðu
;
u
Þ;
B
ðu
;
Þ
1 2
1 1 2
2 1 2 ;
Proc. R. Soc. A (2008)
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1988
L. C. F. Ferreira and E. J. Villamizar Roa
where
B1d
ðu1 ; u2 Þ Z
and
B2d
ðu1 ; 2 Þ Z
ðt
0
ðt
0
^
expðKðtKsÞjxj2g Þ PðxÞix$
ud
1 5u 2 ðx; sÞ ds
ð2:11Þ
expðKðtKsÞjxj2g Þix$ qd
u
2 1 ðx; sÞ ds:
ð2:12Þ
Now we state our main results about the well-posedness of system (2.2)–(2.5).
Theorem 2.5. Let 1/2!g%1, aZnK(2gK1), rOn/2, qCrOn, a%q, r!n and
0!b!n. Let y0Z(u 0, q0), with div u 0Z0 as any vector function in the PMa space.
(i) (Small solutions) Let n!bCq!nCa and bCqKn%r%bCqKaC1.
Assume that t bKaC1=2g f ðtÞ 2 BC ðð0;NÞ; PMb Þn with the norm suptO0
t bKaC1=2g kf ðtÞkb sufficiently small. Then there exists a constant 3O0,
such that if ky0 ka Z max fku 0 ka ; kq0 ka g! 3, the initial-value problem
(2.2)–(2.5) has a global solution yðt; xÞZ ðuðt; xÞ; qðt; xÞÞ 2 Er ;q in the
sense of definition 2.4. Moreover, if kykEr ;q is sufficiently small, then the
solution is unique.
(ii) (Regularization) Moreover, if a%q and r!d!n, there exists 0!3d%3,
such that if ky0 ka ! 3d , then the previous solution yðtÞZ ðuðtÞ; qðtÞÞ verifies
t ad =2 ðu; qÞ 2 BC ðð0;NÞ; PMd ÞnC1 .
(iii) (Large solutions) Let a!r!n, bCrOn and hR0, such that 1KððnKbÞ=2gÞ
KhR 0. Assume that t h f 2 BC ðð0; TÞ; PMr Þn with sup 0!t!T t h kf ðtÞkb
(sufficiently small when 1KððnKbÞ=2gÞK hZ 0). Then there exists
0!T1%T, such that the initial-value problem (2.2)–(2.5) has a mild solution
1
ðu; qÞ 2 E T
r .
(iv) Furthermore, in the previous cases, if we assume y0 Z ðu 0 ; q0 Þ2
ðPMa h PMp ÞnC1 , with 2gK1!p!n, then there exists 0!3p%3, such that
if ky0 ka ! 3p , the previous solutions yðtÞZ ðuðtÞ; qðtÞÞ also verify
ðu; qÞ 2 BCðð0; TÞ; PMp ÞnC1 , with TZN in the first case.
Corollary 2.6 (Bénard problem). Let bZ a K1Z nK2g and fZKG(x/j x j1C2g)
be the generalized gravitation field (note that if gZ1, f is the Newtonian
gravitation field ).
(i) (Small solutions) Let nO4gK2 max fa; 2gg! q! n and assume that kG
and ky0ka are sufficiently small, where k is the constant of (1.1), which
represents the coefficient of volume expansion. Then the initial-value
problem (2.2)–(2.5) has a global solution ðuðt; xÞ; qðt; xÞÞ 2 Ea;q .
On the other hand, if nO4gK1, then the initial-value problem (2.2 )–(2.5 )
has a global solution in Ea Z BC ðð0;NÞ; PMa ÞnC1 .
(ii) (Large solutions) Let max f2g; ag! r ! n. Then there exists 0!T%N,
such that the initial-value problem (2.2)–(2.5) has a mild solution
r nC1
ðu; qÞ 2 E T
.
r;r Z BC ðð0; TÞ; PM Þ
Remark 2.7 (generalizations). Note that, in part iv of theorem 2.5, we did not
assume any smallness hypothesis on the initial data in the norm k$kp. Ever since
1/2!g%1, in the absence of the gravitational field, that is f Z0, we obtain a
generalization of the results of Cannone & Karch (2004) for the case of fractional
Proc. R. Soc. A (2008)
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Existence of solutions to convection problem
1989
dissipation. Furthermore, all the theorems given previously remain true if we
consider system (2.2)–(2.5) with an external field f1 and an external force h1 to be
non-null with suitable smallness conditions on the respective norms.
Remark 2.8 (smoothness).
— An interesting point related to the solutions of the convection problem (2.2)–
(2.5) is to know if they are solutions in the classical sense. Indeed, we can
adapt the arguments of Kato (1992) in order to prove that the regularized
solutions given by theorem 2.5 are C N-smooth instantly and they are the
solutions of system (2.2)–(2.5) for tO0, in the classical sense. We do not know
whether the respective small solutions (first part) have the same property. We
observe that for tO0, the regularized solutions lie in ðPMr h PMd Þn !ðPMq h
PMd Þ 3ðLl ðRn ÞÞn !Ll ðRn Þ with a%q, r!d and n! l ! ðn=ðnKdÞÞ, where Ll
ðRn Þ denotes the well-known Lebesgue space; this last fact does not hold for
small solutions.
— We can obtain an analogous version of the regularized solutions for the large
solutions and therefore, if we take 0!T!N as small enough, the solutions are
also C N-smooth instantly and they are the solutions of system (2.2)–(2.5) in
the classical sense.
(a ) Proofs of theorem 2.5 and corollary 2.6
In this section, we will develop the proofs of theorem 2.5 and corollary 2.6. For
this, we recall the following lemma in a generic Banach space, which can be found
in Cannone & Planchon (1999) and Ferreira & Villamizar-Roa (2006a). For a
generalization of that lemma, in the case of a p-nonlinearity, the reader is
referred to Ferreira & Villamizar-Roa (2006b). The proof is also based on the
standard Picard iteration technique completed by the Banach fixed point
theorem.
Lemma 2.9. Let X be a Banach space with norm k$kX, F : X/X a linear
continuous map with norm t!1 and B : X!X/X a continuous bilinear map,
that is, there exists a constant KO0, such that for all x 1 and x 2 in X, we have
kBðx 1 ; x 2 ÞkX % Kkx 1 kX kx 2 kX . Then, if 0! 3! ðð1KtÞ2 =4KÞ, for any vector
y2X, ys0, such that kykX ! 3, there exists a solution x2X for the equation
x Z yC Bðx; xÞC FðxÞ, such that kxkX % ð23=ð1KtÞÞ. The solution x is unique
ð23=ð1K tÞÞÞ. Moreover, the solution depends continuously on y in
in the ball Bð0;
the following sense: if k~
y kX % 3, x~Z y~C Bð~
x ; x~ÞC Fð~
x Þ and k~
x kX % 23=ð1KtÞ,
then
1K t
kx K x~kX %
kyK y~kX :
ð1K tÞ2 K4K3
We will also need the following preliminary lemmas.
Lemma 2.10. Let 0!T%N, 0!a%q and r!n. If y0 Z ðu 0 ; q0 Þ 2 ðPMa ÞnC1 ,
then
kGg ðtÞy0 kE Tr;q % C ky0 ka ;
where C is a positive constant that is equal to 1 when qZrZa, and GðtÞy0 . y0
when t/0C, in a distributional sense.
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1990
L. C. F. Ferreira and E. J. Villamizar Roa
Proof. Estimating directly the norm k$kq of Gg(t), we have
kGg ðtÞu 0 kq % t KððqKaÞ=2gÞ sup ðjt 1=2g xjqKa expðKtjxj2g ÞÞku 0 ka Z C t KððqKaÞ=2gÞ ku 0 ka :
x2Rn
Analogously, we have that kGg ðtÞq0 kr % C t KððrKaÞ=2gÞ kq0 ka .
To complete the proof, we need to prove the weak continuity in tZ0. Note
that for 4 2 SðRn Þ, we have
ð
^ jhGg ðtÞy0 K y0 ; 4ij Z ðexpðKtjxj2g Þ K 1Þ^
y 0 4dx
4
jexpðKtjxj2g Þ K1j
^ % t ess sup
ky
k
/ 0; as t/ 0C;
0 a
2g
aK2g n
jxj
tjxj
x2R
L1 ðRn Þ
and the proof is finished.
&
Now, we show the continuity of the bilinear form B($ , $) defined by (2.10).
Lemma 2.11. Let B($ , $) be the bilinear form defined by (2.10), 1/2!g%1,
aZ nKð2g K1Þ% q, r!n, rOn/2, rCqOn, y1 Z ðu 1 ; q1 Þ and y2 Z ðu 2 ; q2 Þ. Then,
(i) There exists a positive constant KEr ;q , such that
kBðy1 ; y2 ÞkEr ;q % KEr ;q sup t ððrKaÞ=2gÞ ku 1 kr supðt ððrKaÞ=2gÞ ku 2 kr
Ct
tO0
ððqKaÞ=2gÞ
tO0
kq2 kq Þ;
c y1 ; y2 2 Er;q :
ð2:13Þ
(ii) If 2gK1!p!n, then there exists a positive constant KEp , such that
kBðy1 ; y2 ÞkEp % KEp sup ku 1 kp supðt ðrKaÞ=2g ku 2 kr C t ðqKaÞ=2g kq2 kq Þ;
tO0
c y1 2 Ep ;
tO0
c y 2 2 Er ; q :
ð2:14Þ
(iii) If 0!T!N and a!r!n, then there exists a positive constant KE Tr , such
that
n C 1Kr
O 0;
kBðy1 ; y2 ÞkE Tr % KE Tr T d ky1 kE Tr ky2 kE Tr ; d Z 1K
2g
c y1 ; y2 2 E T
r :
ð2:15Þ
Proof. We will omit the proof of estimate (2.14) because we can get it in a way
analogous to the inequality (2.13); therefore, we just need to prove the
inequalities (2.13) and (2.15). For this, let 0!b!n, 0!l!n and bClKnO0.
Using elementary properties of the Fourier transform and proposition 2.2, we
obtain
ð
1
1
d
dz ku 1 ðsÞkb kq2 ðsÞkl
q2 u 1 ðx; sÞ% C
b
l
Rn jxKzj jzj
1
% C bClKn ku 1 ðsÞkb kq2 ðsÞkl :
ð2:16Þ
jxj
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Existence of solutions to convection problem
1991
Consequently, we have
ðt
d
jxj B2 ðu1 ; 2 Þ% C expðKðtKsÞjxj2g ÞjxjlC1
l
0
1
bClKn
jxj
ku 1 ðsÞkb kq2 ðsÞkl ds:
Therefore for a% j % min fb; lg, we obtain
sup t ðlKjÞ=2g kB2 ðu 1 ; q2 ÞðtÞkl % K1
tO0
sup ðI ðx; tÞt ðbKjÞ=2g ku 1 kt ðlKjÞ=2g kq2 kl Þ; ð2:17Þ
tO0;x2Rn
where
I ðx; tÞ Z t
ðlKjÞ=2g
ðt
expðKðtKsÞjxj2g ÞjxjnC1Kb s KððlKjÞ=2gÞKððbKjÞ=2gÞ ds:
0
Then we have three cases.
First case. If lZq, bZ nKð2g K1Þ, jZa and noting that ab Z ðbKaÞ=gZ 0
when bZa, we have
ð 1=2
expðKð1KsÞtjxj2g Þtjxj2g s Kðaq =2Þ ds
0
ð1
exp ðKtð1KsÞjxj2g Þtjxj2g ds% C:
C 2aq =2
I ðx; tÞ%
1=2
Second case. If lZq, bZ r O nKð2g K1Þ and jZa, we obtain
ð1
ðKnC1KrÞ=2g Ks2g
I ðx; tÞ% sup s
e
ð1KsÞðKnC1KrÞ=2g s KðaqCar Þ=2 ds% C :
sO0
0
Third case. If we take a! j Z l Z bZ r ! n in the expression I(x,t) of (2.17),
we get
ðt
I ðx; tÞ% expðKðtKsÞjxj2g ÞjxðtKsÞ1=2g jnC1Kr ðtKsÞ KðððnC1ÞKrÞ=2gÞ ds
0
% CT 1KðððnC1ÞKrÞ=2gÞ :
Following analogous arguments, we can prove that
sup t ðrKaÞ=2g kB1 ðu 1 ; u 2 ÞðtÞkr % C sup t ðrKaÞ=2g ku 1 kr sup t ððrKaÞ=2gÞ ku 2 kr
tO0
tO0
tO0
and
sup kB1 ðu 1 ; u 2 ÞðtÞkr % CT 1KðððnC1ÞKrÞ=2gÞ sup ku 1 kr sup ku 2 kr ;
tO0
and hence we conclude the proof of the lemma.
Proc. R. Soc. A (2008)
tO0
tO0
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1992
L. C. F. Ferreira and E. J. Villamizar Roa
Lemma 2.12. Let aZ nKð2g K1Þ, 1/2!g%1, 0!b, q!n, n! bC q! nC a,
bC qKn% r % bC qKaC 1 and F($) be as defined in (2.9). Then
(i) there exists a positive constant Kr, such that
sup t ððrKaÞ=2gÞ kFðqÞðtÞkr % Kr sup t ððqKaÞ=2gÞ kqðtÞkq sup t ððbKaC1Þ=2gÞ kf ðtÞkb ;
tO0
tO0
tO0
ð2:18Þ
(ii) if 2gK1!p!n, then there exists a positive constant Kp, such that
sup kFðqÞðtÞkp % Kp sup kqðtÞkp sup t ððbKaC1Þ=2gÞ kf ðtÞkb ;
tO0
tO0
ð2:19Þ
tO0
(iii) if 0!T!N and a!r!n, then there exist hR0, such that d h
1KððnKbÞ=2gÞK hR 0, and a positive constant Kr,T satisfying
sup kFðqÞðtÞkr % Kr ;T T d sup kqðtÞkr sup t h kf ðtÞkb :
tO0
0!t!T
ð2:20Þ
tO0
Proof. We will omit the proof of estimate (2.19), since we can obtain it in a
way analogous to the inequality (2.18). Working as in lemma 2.11, we find that
ð t
r
2g d
jxj k expðKðtKsÞjxj Þ ðqf Þðx; sÞ ds
0
% CI ðx; tÞ sup t ððqKaÞ=2gÞ kqðtÞkq sup t ððbKaC1Þ=2gÞ kf ðtÞkb ;
tO0
where
I ðx; tÞ Z
ðt
tO0
expðKðtKsÞjxj2g ÞjxjrCnKbKq s KððqCbK2aC1Þ=2gÞ ds:
0
We analyse the convergence of I(x, t) in two cases.
First case. If r C nKbKq! 2g,
ðt
I ðx; tÞ Z expðKðtKsÞjxj2g ÞjxjrCnKbKq s KððqCbK2aC1Þ=2gÞ ds
0
% sup sððrCnKbKqÞ=2gÞ e Ks
2g
ðt
ðtKsÞ KððrCnKbKqÞ=2gÞ s KððqCbK2aC1Þ=2gÞ ds
0
sO0
% C t KððrCnKbKqÞ=2gÞKððqCbK2aC1Þ=2gÞC1 Z Ct KððrKaÞ=2gÞ :
Second case. If r C nKbKqZ 2g, then denoting bZ ððqC bK2aC 1Þ=2gÞ,
we have
ðt
ð1
2g
2g Kb
Kb
I ðx; tÞ Z expðKðtKsÞjxj Þjxj s ds% t
expðKð1KsÞtjxj2g Þtjxj2g s Kb ds
0
% Ct
0
Kb
Z Ct
Proc. R. Soc. A (2008)
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:
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1993
Existence of solutions to convection problem
On the other hand, if hR0 and 1KððnKbÞ=2gÞK hR 0, we can estimate
kFðqÞðtÞkr % sup I ðx; tÞ sup kqðtÞkr sup t h kf ðtÞkb ;
0!t!T
where
I ðx; tÞ Z
ðt
0!t!T
ð2:21Þ
0!t!T
exp ðKðtKsÞjxj2g ÞjxjnKb sKh ds% CT 1KððnKbÞ=2gÞKh ;
0
which proves (2.20).
&
Proof of theorem 2.5.
(i) Small solutions. Using lemma 2.10, we can take the initial data (u 0,q0) as
sufficiently small such that the hypothesis of lemma 2.9 is verified. In
lemma 2.11, the inequality (2.13) implies the continuity of the bilinear
form in the Er,q space. Now, using inequality (2.18) of lemma 2.12, we
obtain the continuity of the linear term F($) whose norm depends on the
size of the norm suptO0 t ððbKaC1Þ=2gÞ kf ðtÞkb . Consequently, a direct
application of lemma 2.9 in the Banach space Er,q, implies the wellposedness of the integral equations (2.6) and (2.7) in Er,q. On the other
hand, we need to show that Bðy; yÞðtÞ. 0 and FðyÞ. 0, as t/0C in the
distributional sense, but we omit the proof because this follows as in the
second part of the proof of lemma 2.10.
(ii) Large solutions. The proof of the third part of theorem 2.5 is also a direct
application of lemmas 2.9 and 2.10, inequality (2.15) of lemma 2.11 and
inequality (2.20) of lemma 2.12. Indeed, we can take TO0 as sufficiently
small in such a way that the conditions of lemma 2.9 are verified without a
smallness assumption on the data.
(iii) Regularization. Using the hypotheses a%q and r!d!n, we have the
continuity of the bilinear form B($,$) and the linear term F($) in the space
Ed,d. Consequently, a direct application of lemma 2.9 completes the wellposedness of the integral equations (2.6) and (2.7). Initial data are taken
in the same space as in the small solutions and therefore the proof of the
regularization of solutions is finished.
The final part of theorem 2.5, that is, ðu; qÞ 2 BC ðð0; TÞ; PMp ÞnC1 , 0!T%N,
for initial data y0 2 ðPMp h PMa ÞnC1 with 2gK1!p!n, can be proved as
follows. We consider the case TZN corresponding to the case of small solutions
(the case 0!T!N follows using analogous arguments). Since the solution given
by lemma 2.9 is obtained by the following sequence:
y1 ðt; xÞ Z GðtÞy0 and ykC1 ðt; xÞ Z y1 ðt; xÞKBðyk ; yk Þ C Fðyk Þ;
where ykZ(u k,qk) and k2N; we can use lemmas 2.10–2.12 in order to obtain the
existence of a positive constant K (f ) such that
sup ky1 ðtÞkp % ky0 kp ;
tO0
sup kukC1 ðtÞkp % ku 0 kp C KEp sup kuk ðtÞkp sup t ar =2 kuk ðtÞkr C Kf sup kqk ðtÞkp
tO0
and
tO0
tO0
tO0
sup kqkC1 ðtÞkp % kq0 kp C KEp sup kuk ðtÞkp sup t aq =2 kqk ðtÞkq :
tO0
Proc. R. Soc. A (2008)
tO0
tO0
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1994
L. C. F. Ferreira and E. J. Villamizar Roa
Now, let us choose 0!3p%3 and take f small enough in its respective norm so
that ð2KEp 3p =1KKf ÞC Kf ! 1 and assume that ky0 ka ! 3p . The proof of the first
part of theorem 2.5 (see lemma 2.9) shows that kykEr;q % 23p =ð1KKf Þ. Therefore,
!
2KEp 3p
sup kykC1 ðtÞkp % ky0 kp C
C Kf sup kyk ðtÞkp :
1KKf
tO0
tO0
Let us denote M0 Z ky0 ðtÞkp and Mk Z suptO0 kyk ðtÞkp , then the sequence fMk gN
kZ0
satisfies MkC1 % M0 Cðð2KEp 3p =ð1KKf ÞÞCKf ÞMk : Taking RZ ð2KEp 3p =ð1KKf ÞÞC
Kf ! 1, we can write
1
M ; for all k 2 N;
Mk % ð1 C R C R2 C/C Rk ÞM0 %
1KR 0
and thus,
wkC1 Z ykC1 K yk ZKBðyk ; yk Þ C BðykK1 ; ykK1 Þ C Fðyk ÞKFðykK1 Þ
ZKBðwk ; yk ÞKBðykK1 ; wk Þ C Fðwk Þ:
Finally, lemmas 2.11 and 2.12 imply that
sup kwkC1 kp %
tO0
2KEp M0
1KR
kwk kEr;q C K ðf Þ sup kwk ðtÞkp :
tO0
Now, we denote GZ lim supk/NðsuptO0 kwk kp Þ. Since K ðf Þ! 1 and limk/N
kwk kEr;q Z 0, then G% K ðf ÞG and consequently GZ0. Therefore, the sequence
{yk} is a Cauchy sequence in the space BCðð0;NÞ; PMp ÞnC1 and thus it converges
to some y~ðt; xÞ 2 BCðð0;NÞ; PMp ÞnC1 . The uniqueness of the limit in the distributional sense gets the desired conclusion.
&
Proof of corollary 2.6.
Small solutions. Note that under the assumptions on b, q, n and g, and taking rZa,
the hypothesis of the first part of theorem 2.5 is satisfied. As bZ a K 1Z nK2g,
then ðbKaC 1Þ=2gZ 0 and moreover f ZKGðx=jxj1C2g Þ 2 ðPMnK2g Þn . If kG is
small enough then suptO0 t ððbKaC1Þ=2gÞ kf ðtÞkb is small, and hence we can apply the
first part of theorem 2.5 in order to obtain the existence of small solutions for the
Bénard problem.
Large solutions. As f ZKGðx=jxj1C2g Þ 2 ðPMnK2g Þn , then hZ0. Moreover, as
a!r!n then 1KððnC 1KrÞ=2gÞK hO 0. Now, applying the third part of
theorem 2.5, we obtain the intended result. Note that we did not need smallness
assumptions on G.
&
3. Self-similar solutions and stability in PMa spaces
(a ) Self-similar solutions
Let us consider that f ðt; xÞZ l2g f ðl2g t; lxÞ is a smooth function and that the
pair ðuðt; xÞ; qðt; xÞÞ is a smooth solution of the convection problem (2.2)–(2.5).
We can see that the pair of functions ðu; qÞl ðt; xÞZ l2gK1 ðuðl2g t; lxÞ; qðl2g t; lxÞÞ
is also a solution of system (2.2)–(2.5). The particular solutions of system
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Existence of solutions to convection problem
1995
(2.2)–(2.5), satisfying
ðuðt; xÞ; qðt; xÞÞ Z ðuðt; xÞ; qðt; xÞÞl ðt; xÞ;
ð3:1Þ
for any tO0, x2Rn and lO0, are called self-similar solutions of the system. We
can check that taking t/0C formally in (3.1), ðuð0; xÞ; qð0; xÞÞ should be a
homogeneous function of degree K(2gK1).
This fact gives the hint that a suitable space to find self-similar solutions
should be the one containing homogeneous functions with that exponent, and
this gives another justification to study the convection problem in PMa spaces.
The aim of this subsection is to describe the principal results of the existence of
self-similarity solutions in PMa spaces.
Theorem 3.1. Let y0 Z ðu 0 ; q0 Þ 2 ðPMa ÞnC1 and aZ nKð2g K1Þ. Assume that
y0 is a homogeneous vector function of degree K(2gK1), that is, y0 ðlxÞZ
l Kð2gK1Þ y0 ðxÞ for all x2Rn, xs0 and all lO0. If f satisfies the assumptions of
theorem 2.5 (respectively corollary 2.6) and the scale relation
f ðt; xÞ Z l2g f ðl2g t; lxÞðrespectively f ZKGxjxj Kð1C2gÞ Þ;
then the solution given by theorem 2.5 (respectively corollary 2.6) is self-similar, that
is, yðt; xÞZ l2gK1 yðl2g t; lxÞ, for all x2Rn, xs0 and all lO0.
Proof. The proof of theorem 2.5 is based on lemma 2.9, in which the solution is
obtained by successive approximations. Hence, we consider the following Picard
iteration:
y1 ðt; xÞ Z Gg ðtÞy0 ðxÞ and ykC1 ðt; xÞ Z y1 ðt; xÞKBðyk ; yk Þ C Fðyk Þ;
where kZ1, 2,. . We can verify that y1(t,x) satisfies y1 ðt; xÞZ l2gK1 y1 ðl2g t; lxÞ.
By an induction process, we can prove that yk satisfies yk ðt; xÞZ l2gK1 yk ðl2g t; lxÞ,
for all k. Consequently, as the mild solution y(t,x) is obtained as the limit of
2gK1
sequence fyk gN
yðl2g t; lxÞ, for all
kZ1 , we have that y(t,x) must verify yðt; xÞZ l
n
lO0, all tO0 and x2R .
&
(b ) Asymptotic stability in PMa spaces
In the present section, we show some stability properties of mild solutions
when the initial velocity and the temperature are perturbed. Our results now
read as below.
Theorem 3.2. Let yZ(u, q) and wZ(v, f) be two small global solutions of
(2.2 )–(2.5 ) as in theorem 2.5, corresponding to the initial conditions y0Z(u 0, q0)
and w0Z(v0, f0)2(PMa)nC1, respectively. If limt/Nt ar =2 kGg ðtÞðu 0 K v0 Þkr Z 0
and limt/Nt aq =2 kGg ðtÞðq0 K f0 Þkq Z 0, then
lim t ar =2 kðuðtÞKvðtÞÞkr Z 0
t/N
and
lim t aq =2 kqðtÞKfðtÞkq Z 0:
t/N
ð3:2Þ
Moreover, if y0 Z ðu 0 ; q0 Þ 2 ðPMa h PMl ÞnC1 k$ka with l!a, then (3.2 ) holds.
n nC1
Remark 3.3. Note that as a consequence of theorem 3.2, if y0 2 ðC N
,
0 ðR ÞÞ
then the respective solution yZ(u, v) decays to zero, that is,
limðt ar =2 kuðtÞkr C t aq =2 kqðtÞkq Þ Z 0:
t/N
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1996
L. C. F. Ferreira and E. J. Villamizar Roa
Proof. Let us define Nr;q ðtÞZ t ar =2 kuKvkr C t aq =2 kqKfkq . Subtracting the
integral equations in definition 2.4 for w from the analogous expression for y,
taking the norm t ar =2 k$kr in the first coordinate and t aq =2 k$kq in the second
coordinate of the resulting system, we obtain
t ar =2 kðuKvÞkr % t ar =2 kGg ðtÞðu 0 K v0 Þkr C t ar =2 kFðqKfÞkr
ð dt
ar =2
C K sup t
expðKðtKsÞjxj2g Þjxj1CnKr ðkuðsÞkr
x2Rn
0
C kvðsÞkr ÞkuðsÞKvðsÞkr ds C K sup t
ar =2
ðd
x2Rn
expðKðtKsÞjxj2g Þ
dt
!jxj1CnKr ðkuðsÞkr C kvðsÞkr ÞkuðsÞKvðsÞkr ds
ð3:3Þ
and
t aq =2 kðqKfÞkq % t aq =2 kGg ðtÞðq0 K f0 Þkq
ð dt
C K sup t aq =2
expðKðtKsÞjxj2g Þjxj1CnKr ðkuðsÞkr
x2Rn
0
C kvðsÞkr ÞkqðsÞKfðsÞkq ds C K sup t aq =2
x2Rn
1CnKr
!xj
ðd
expðKðtKsÞjxj2g Þj
dt
ðkuðsÞkr C kvðsÞkr ÞkqðsÞKfðsÞkq ds;
ð3:4Þ
where the small constant d will be chosen later.
Assume that r Z aZ nKð2g K 1Þ! q! n and arZ0 (the case rOa follows in
an analogous way). In the first integral on the r.h.s. of (3.3), we change the
variables sZtz and use the identity
sup jxj2g expðKð1KsÞtjxj2g Þ Z
x2Rn
1
C
sup jw j2g e Kjw j 2g Z
ð1KsÞt w2Rn
ð1KsÞt
in order to estimate it by
ðd
K sup
x2Rn
0
ðd
%C
0
tjxj2g expðKtð1KzÞjxj2g ÞkuðtzÞKvðtzÞka dz supðkvka C kuka Þ
tO0
ð1KzÞ K1 kuðtzÞKvðtzÞka dz:
We deal with the second integral of the inequality (3.3), estimating it directly by
ðd
2g
2g
K sup
jxj expðKðtKsÞjxj Þ ds
sup kuðsÞKvðsÞka 43
x2Rn
Z
dt
43K
sup kuðsÞKvðsÞka ;
1K t dt%s%t
Proc. R. Soc. A (2008)
dt%s%t
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1997
Existence of solutions to convection problem
where t is the constant given by the application of lemma 2.9. On the other hand,
we estimate the term kFðqKfÞðtÞka as
ðt
ððbKaC1Þ=2gÞ
kFðqKfÞðtÞka % C sup t
kf kb ðtKsÞKððaCnKbKqÞ=2gÞ s KððbKaC1Þ=2gÞ
0
tO0
!kðqKfÞðtsÞkq ds
Z C sup t ððbKaC1Þ=2gÞ kf kb
tO0
ð1
ð1KzÞ KððaCnKbKqÞ=2gÞ z KððbKaC1Þ=2gÞKðaq =2Þ
0
!ðtzÞaq =2 kðqKfÞðtzÞkq dz:
Therefore, we have
ððbKaC1Þ=2gÞ
kuðtÞKvðtÞka % kGg ðtÞðu 0 K v0 Þka C C sup t
kf kb
tO0
ð1
! ð1KzÞ KððaCnKbKqÞ=2gÞ z KððqCbK2aC1Þ=2gÞ Na;q ðtzÞ dz
0
ðd
43K
C C ð1KzÞ K1 Na;q ðtzÞ dz C
sup kuðsÞKvðsÞka ;
1K t dt%s%t
0
for all tO0. We also have
t
aq =2
kqðtÞKfðtÞkq % t
aq =2
kGg ðtÞðq0 K f0 Þkq C C
ðd
0
ð1KzÞ K1 z Kðaq =2Þ Na;q ðtzÞ dz
43K
sup saq =2 kqðsÞKfðsÞkq :
C
1Kt dt%s%t
Now, we define
A Z lim supðNa;q ðtÞÞ Z
t/N
lim
supðNa;q ðtÞÞ:
k2N; k/N tRk
We will show that AZ0. Using the dominated convergence theorem, we obtain
!
ðd
ðd
1
K1
K1
;
lim sup ð1KzÞ Na;q ðtzÞ dz % A ð1KzÞ dz Z A log
1Kd
t/N 0
0
ð1
lim sup ð1KzÞ KððaCnKbKqÞ=2gÞ z KððqCbK2aC1Þ=2gÞ Na;q ðtzÞ dz
t/N 0
Ð
%A 01 ð1KzÞKððaCnKbKqÞ=2gÞ z KððqCbK2aC1Þ=2gÞ dz Z CA
and
ðd
lim sup
ð1KzÞ
K1 Kðaq =2Þ
z
t/N 0
%CA log
Proc. R. Soc. A (2008)
Na;q ðtzÞ dz % A
!
!
1
1Kd
ðd
C d1Kðaq =2Þ :
0
ð1KzÞ K1 z Kðaq =2Þ dz
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1998
L. C. F. Ferreira and E. J. Villamizar Roa
Since
sup sup Na;q ðsÞ% sup Na;q ðsÞ;
tRk dt%s%t
dk%s!N
we have
lim sup sup Na;q ðsÞ% A:
t/N dt%s%t
From the proof of the first part of theorem 2.5, we know that
ð1
ððbKaC1Þ=2gÞ
C sup t
kf kb ð1KzÞ KððaCnKbKqÞ=2gÞ z KððbKaC1Þ=2gÞKðaq =2Þ dz % t;
0
tO0
and therefore, summing (3.3) and (3.4), computing lim supt/N in the resulting
inequality and using the last inequalities, we obtain
1
43K
A% C log
C d1Kðaq =2Þ C
C t A:
1Kd
1Kt
If we take dO0 as sufficiently small, since ð43K=ð1KtÞÞC t! 1 and A is nonnegative, then AZ0. This completes the proof of the first part of theorem 3.2. In
order to prove the second part, we use lemma 2.10 and a density argument to
obtain
lim t ar =2 kGg ðtÞðu 0 K v0 Þkr Z 0
t/N
and
lim t aq =2 kGg ðtÞðq0 K f0 Þkq Z 0;
t/N
and therefore (3.2) holds.
&
The second author E.J.V.R. was partially supported by COLCIENCIAS, Colombia, Proyecto
COLCIENCIAS-BID III etapa.
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