Global Attractors and Dimension Estimation of the 2D Generalized MHD System with Extra Force ()
1. Introduction
In this paper, we study the following magnetohydrodynamic system:
(1.1)
here
is bounded set,
is the bound of
, where u is the velocity vector field, v is the magnetic
vector field,
are the kinematic viscosity and diffusivity constants respectively.
. Let
.
When
, problem (1.1) reduces to the MHD equations. In particular, if
, problem (1.1) becomes the ideal MHD equations. It is therefore reasonable to call (1.1) a system of generalized MHD equations, or simply GMHD. Moreover, it has similar scaling properties and energy estimate as the Navier-Stokes and MHD equations.
The solvability of the MHD system was investigated in the beginning of 1960s. In particular in [1] -[4] the global existence of weak solutions and local in time well-posedness was proved for various initial boundary value problems. However, similar to the situation with the Navier-Stokes equations, the problem of the global smooth solvability for the MHD equations is still open.
Analogously to the case of the Navier-Stokes system (see [5] -[8] ) we introduce the concept of suitable weak solutions. We prove the existence of the global attractor (see [9] ) and getting the upper bound estimation of the Hausdorff and fractal dimension of attractor for the MHD system.
2. The Priori Estimate of Solution of Problem (1.1)
Lemma 1. Assume
so the smooth solution
of problem (1.1) satisfies
![](//html.scirp.org/file/11-7402686x17.png)
Proof. We multiply u with both sides of the first equation of problem (1.1) and obtain
(2.1)
We multiply v with both sides of the second equation of problem (1.1) and obtain
(2.2)
According to
we obtain
(2.3)
According to (2.1) + (2.2), so we obtain
(2.4)
According to Poincare and Young inequality, we obtain
(2.5)
(2.6)
(2.7)
From (2.5)-(2.7), we obtain
![]()
![]()
Let
, according that we obtain
![]()
Using the Gronwall’s inequality, the Lemma 1 is proved. ![]()
Lemma 2. Under the condition of Lemma 1, and ![]()
,
,
, so the solution
of problem (1.1) satisfies
![]()
Proof. For the problem (1.1) multiply the first equation by
with both sides, for the problem (1.1) multiply the second equation by
with both sides and obtain
(2.8)
![]()
According to the Sobolev’s interpolation inequalities,
(2.9)
(2.10)
According to (2.9)-(2.10), we have
(2.11)
Here
![]()
In a similar way, we can obtain
(2.12)
Here
![]()
(2.13)
Here
![]()
![]()
(2.14)
Here
![]()
![]()
According to the Poincare’s inequalities
(2.15)
(2.16)
(2.17)
From (2.12)-(2.17), we have
![]()
Here
![]()
So
![]()
We obtain
![]()
Using the Gronwall’s inequality, the Lemma 2 is proved. ![]()
3. Global Attractor and Dimension Estimation
Theorem 1. Assume that
and
so problem (1.1)
exist a unique solution ![]()
Proof. By the method of Galerkin and Lemma 1-Lemma 2,we can easily obtain the existence of solutions. Next, we prove the uniqueness of solutions in detail.
Assume
are two solutions of problem (1.1), let
, Here
so the difference of the two solution satisfies
(3.1)
(3.2)
The two above formulae subtract and obtain
(3.3)
For the problem (3.3) multiply the first equation by u with both sides and obtain
(3.4)
For the problem (3.3) multiply the second equation by v with both sides and obtain
(3.5)
According to
(3.6)
According to (3.1) + (3.2), we have
(3.7)
According to Sobolev inequality, when n < 4
(3.8)
(3.9)
According to (3.8)-(3.9),we can get
(3.10)
(3.11)
(3.12)
(3.13)
From (3.10)-(3.13),
![]()
Here ![]()
So, we have
![]()
![]()
Let
, so we obtain
![]()
According to the consistent Gronwall inequality,
![]()
So we can get
the uniqueness is proved. ![]()
Theorem 2. [9] Let E be a Banach space, and
are the semigroup operators on E.
here I is a unit operator. Set
satisfy the follow conditions
1)
is bounded. Namely
,
, it exists a constant
, so that
;
2) It exists a bounded absorbing set
namely
it exists a constant t0, so that
;
3) When
,
is a completely continuous operator A.
Therefore, the semigroup operators
exist a compact global attractor.
Theorem 3. Assume
,
. Problem (1.1) have global attractor ![]()
Proof.
1) When
From Lemma 1,
![]()
So
in E is uniformly bounded.
2)
has E in a bounded absorbing set
![]()
From Lemma 2, when
there is
![]()
Since
is tightly embedded, so
is
in the tight absorbing set in E.
3) So the semigroup operator
is completely continuous. ![]()
In order to estimate the Hausdorff and fractal dimension of the global attractor A of problem (1.1), let problem (1.1) linearize and obtain
(3.14)
Assume
is the solutions of the problem (3.14). We know
. It is easy to prove the problem (3.14) has the uniqueness of solutions
.
To prove
in
has differential, let
so there has
![]()
Theorem 4. Assume
and T are constants, so it exists a constant
and
has
so there is
(3.15)
Proof. Meet the initial value problem (3.14) of respectively for
,
solutions for
,
, let
,
. So
,
satisfies
(3.16)
Here
(3.17)
(3.18)
For the problem (3.16) multiply the first equation by
with both sides and for the problem (3.16) multiply the second equation by
with both sides and obtain
(3.19)
Then
(3.20)
Here
.
For the problem (3.16) multiply the first equation by
with both sides and for the problem (3.16) multiply the second equation by
with both sides and obtain
(3.21)
According to the Sobolev’s interpolation inequalities
(3.22)
(3.23)
According to (3.22)-(3.23), we have
(3.24)
In a similar way, we can obtain
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)
So, we can get
![]()
Here
, we obtain
![]()
According to the Poincare’s inequalities
(3.32)
Let
,
![]()
According to Gronwall’s inequalities, we obtain
(3.33)
Let
be the solutions of the linear Equation (3.14), and satisfies
, Assume
(3.34)
So, we can get
(3.35)
Here
(3.36)
(3.37)
For the problem (3.33) multiply the first equation by w1 with both sides and for the problem (3.33) multiply the second equation by w2 with both sides and obtain
(3.38)
According to (3.8)-(3.9), then
(3.39)
(3.40)
(3.41)
(3.42)
(3.43)
(3.44)
(3.45)
(3.46)
(3.47)
(3.48)
According to, we obtain
![]()
Here ![]()
![]()
We obtain
![]()
So
(3.49)
Let
be the solutions of the linear Equation (3.33) correspond- ing to the initial value
so there is
(3.50)
is linear mapping that is defined in the problem (3.34),
represents the outer product, tr represents the trace, QN is the orthogonal projection from
to the span ![]()
Theorem 5. Under the assume of Theorem 3, the global attractor A of problem (1.1) has finite Hausdorff and fractal dimension, and
![]()
Here J0 is a minimal positive integer of the following inequality
![]()
Proof. By theorem [8] , we need to estimate the lower bound of
Let
be the orthogonal basis of subspace of ![]()
(3.51)
According to (3.8)-(3.9), we can get
(3.52)
(3.53)
(3.54)
(3.55)
(3.56)
(3.57)
Under the bounded condition, select
is the standard eigenfunction of
,
and the corresponding eigenvalues are
and
![]()
![]()
Let
. Therefore, we can get
![]()
Let
.
![]()
By
and ![]()
![]()
![]()
So, we can obtain
![]()
We have
![]()
Therefore
Funding
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11161057.
NOTES
*Corresponding author.