The Asymptotic Behavior of Solutions for 3D Globally Modified Bénard Problem with Delay ()
1. Introduction
Let
be an open and bounded set with regular boundary
. We consider the following problem for the Bénard system with homogeneous Dirichlet boundary conditions on
:
(1)
Here
are respectively the velocity, temperature of the fluid,
is the pressure of the fluid,
is the viscous coefficient of the fluid,
is a constant vector,
are the initial value. The external force terms
,
are a given function.
As we all know, Bénard system consists of the Navier-Stokes equations coupled with a parabolic equation. This system is a well-known model in hydrodynamics. It describes the behavior of the velocity, the pressure and the temperature for an incompressible flow. A detailed description of the physical background can be found in [1] and [2] . However, the well-posedness of the Bénard system is still an unsolved problem in the three-dimensional case. Therefore, it is particularly important to modify equations for Bénard system. We consider the following globally modified Bénard system:
(2)
Here the function
is defined by
In the modified term of Equation (2),
depends on the norm
, which in turn depends on
over the whole domain
and not just at or near the point
under consideration. Essentially, it prevents large gradients dominating the dynamics and leading to explosions.
The globally modified Bénard system can be better described when we add the appropriate delay term to the system. In this way, it can take into account not only the present state of the system but the history of the solution. Namely, we consider the following equations:
(3)
Here
is the initial time,
are the external force terms which depends on
, respectively.
is a delay function,
are respectively the velocity, temperature defined on
. In addition,
. When
, the Equation (3) is the Navier-Stokes equations with delays (see [3] [4] ).
This paper is inspired by the literature [3] . The [3] studies mainly the existence and uniqueness of strong solutions and the asymptotic behavior of solutions for three-dimensional globally modified Navier-Stokes equations with delays under local Lipschitz conditions. The purpose of this paper is to study the existence, uniqueness and asymptotic behavior of solutions of three-dimensional globally modified Bénard system with delays under local Lipschitz conditions. This paper is organized as follows. In Section 2, we recall some definitions of function spaces and some related properties. In Section 3, we give the proof of the existence and uniqueness of weak solutions. In Section 4, we consider the asymptotic behavior of solutions.
2. Preliminary
Let
We denote by
the scalar product,
the norm in H. For
,
. We denote by
the scalar
product,
the norm in V. For
,
.
The dual space of V is
. It follows that
, and the injections are dense and compact. We shall use
to denote the norm of
,
represent the dual product between V and
. P is a Leray-Helmotz projection from
to H,
is a Leray-Helmotz projection from
to
, suppose
,
, where
are linear continuous operators. Let
be the domain of the operator
in H,
be the domain of the operator
in
. In addition,
where
.
Because of the regularity of the boundary
, it is easy to prove that:
,
, furthermore, we obtain
The first eigenvalue of Stokes operator is defined by
, let
be the minimum of the first eigenvalues of the operators
. For all
, we put
. We know from [1] that
is a trilinear continuous form on
, and
for
,
. For any
, it follows that (see [5] )
(4)
Define
where
is linear in
, but it is nonlinear in v. Obviously,
. According to the property of b (see [6] ) and the definition of
, we can get
(5)
The following is a lemma, which is specifically proved in [5] .
Lemma 2.1. For all
and
, the following two conclusions hold:
1)
; 2)
.
For all
, let us define
. It has been given in [7] , and
for
.
Let
The following describes the properties of the external force terms
:
(c1) for any
,
are measurable;
(c2) there exists nonnegative functions
, and a nondecreasing function
, such that for all
, if
, then:
(c3) there exists nonnegative functions
, such that for any
(or
),
Supposing
,
, where
. We consider a delay function
for any
such that
, and there exists a constant
for any
satisfying:
.
We define
, where
,
is the differentiable and strictly increasing function, we obtain
(6)
Similarly, when
, there is also an analogy estimate for the
term.
Moreover, we call
as a set of solution for Equation (3) in
, when
and for any
, the following is true:
3. Existence and Uniqueness of Weak Solutions
Theorem 3.1. Under the conditions (c1)-(c3), assume that
,
and
,
. Then there exists a weak solution
of equation (3).
Proof. For simplicity, and without loss of generality, we assume
. We consider the proof of the existence of weak solutions of Equation (3). We use Galerkin method to prove the existence of solutions, which is standard (see [8] [9] ).
Consider the following equations:
(7)
Next, we need to make some priori estimates for the approximate solution
. So taking the inner product of the second equation of (7) with
, and using
, we obtain
that is
Integrating over
, and using (6), we obtain for
(8)
where
. It follows from Gronwall lemma that
(9)
Taking the inner product of the first equation of (7) with
, from (9) and
, we obtain
that is
Integrating over
, and using (6), we obtain for
(10)
where
. It follows from Gronwall lemma that
(11)
From (10) and (11), a subsequence of
is bounded in
and
(where
). Next from the first equation of (7) and
(5), we obtain that
is bounded in
. Therefore, there exists an element
and
, such that
Obviously,
(see [10] ). By the compactness Aubin-Lions (see [9] ), we can deduce
Similarly, from (8) and (9), there is also an analogy conclusion for the
term.
In summary, when
, there exists
for any
, such that
Since
weak converges to
in
, we cannot deduce that
or
and
at least almost everywhere. So we need make a stronger estimate.
Taking the inner product of the first equation of (7) with
, and using (9), we obtain
that is
(12)
Integrating over
, and using (6), we obtain for all
(13)
The following estimate
. Thanks to (10) and (11), we know that
is bounded in
, therefore, there exists
, such that for all integer
Integrating (13) for s over
, we obtain
Consequently,
for any
and any
.
On the other hand, using the uniformly Gronwall inequality for (12), we have
(14)
Taking the inner product of the second equation of (7) with
, we obtain
where
That is
Similarly, integrating over
, we obtain for all
(15)
where
Integrating (15) for s over
, we obtain
Consequently,
for all
and all
.
On the other hand, using the uniformly Gronwall inequality for (14), we have
(16)
From the above discussion, we can deduce that
is bounded in
and
. What’s more, when
, owing to
, we deduce that
is bounded in
; when
, it follows that
and injections are compact. Supposing there exists a positive sequence
, such that
Therefore
As long as there exists
, in
for
, we can get
Similarly, the same is true for the
term. In conclusion,
is a set of solution for Equation (3).
Theorem 3.2. Under the previous all assumptions, there exists a unique solution of equations (3).
Proof. Let
and
be two solutions of equations (3), and set
, according to the first equation of (3), we obtain
where
, using the property of trilinear form b, we easily get
From Lemma 2.1, (4) and the Young inequality, we obtain
Therefore
To estimate on the last term for above formula. For fixed
,
, there exists
, such that
, in addition, using (c2),
, the Hölder inequality, Young's inequality and (6), we obtain for any
Therefore
(17)
Similarly, according to the second equation of (3), we obtain
Since
where (16) is used, we can get
(18)
From (17) and (18), we obtain
Because the coefficients of
and
are bounded at the right of the equation, the conclusion follows from Gronwall lemma, since
.
4. Asymptotic Behavior of Solutions
In this section, we prove mainly the asymptotic behavior of the solution of Equations (3) when
.
Suppose that (c1)-(c2) hold with
, assume also that:
,
, where
, and let us denote respectively by
the unique solution of
(19)
(20)
Theorem 4.1. Under the above assumptions, for any
,
,
and any
, the solution
of equations (3) satisfies
(21)
(22)
Proof. Let
be the solution of equations (3) corresponding to the initial data
. From
we obtain
(23)
Integrating over
, we obtain
That is
Since
(24)
Using (19), we can get
That is
Similarly, from
we obtain
(25)
Integrating over
, we obtain
Using (20) and (24), we can get
That is