The Global Attractors of the Solution for 2D Maxwell-Navier-Stokes with Extra Force Equations ()
1. Introduction
In recent years, the Maxwell-Navier-Stokes equations have been studied extensively, and the studies have obtained many achievements [1] [2] . The Maxwell-Navier-Stokes equations are a coupled system of equations consisting of the Navier-Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. The coupling comes from the
Lorentz
force in the fluid equation and the electric current in the
Maxwell
equations. In [1] , the authors studied the non-resistive limit of the 2D Maxwell-Navier-Stokes equations and established the convergence rate of the non-resistive limit for vanishing resistance by using the Fourier localization technique. In [2] , the author has proved the existence and uniqueness of global strong solutions to the non-resistive of the
2D Maxwell-Navier-Stokes equations for initial data
with
. The
long time behaviors of the solutions of nonlinear partial differential equations also are seen in [3] -[10] .
In this paper,we will study the 2D Maxwell-Navier-Stokes equations with extra force and dissipation in a bounded area under homogeneous Dirichlet boundary condition problems:
(1.1)
here
is bounded set,
is the bound of
,
is the velocity of the fluid,
is the viscosity,
and
are resistive constants,
is the electric current which is given by Ohm’s law,
is the electric field,
is the magnetic field and
is the Lorentz force.
Let
and
.
2. The priori estimate of solution of questions (1.1)
Lemma 1. Assume
so the solution
of the Dirichlet
bound questions (1.1) satisfies
![]()
here
.
Proof. For the system (1.1) multiply the first equation by
with both sides and obtain
(2.1)
For the system (1.1) multiply the second equation by
with both sides and obtain
(2.2)
For the system (1.1) multiply the third equation by
with both sides and obtain
(2.3)
Because
, so
is
(2.4)
According to Poincare’s inequality, we obtain
(2.5)
According to
, we obtain
(2.6)
According to Young’s inequality, we obtain
(2.7)
(2.8)
(2.9)
From (2.4) (2.5) (2.6) (2.7) (2.8) (2.9), we obtain
![]()
so
![]()
Let
, according that we obtain
![]()
so
![]()
Using the Gronwall’s inequality, the Lemma 1 is proved.
Lemma 2. Under the condition of Lemma 1, and
,
so the solution
of the Dirichlet bound questions (1.1) satisfies
![]()
here ![]()
Proof. For the system (1.1) multiply the first equation by
with both sides and obtain
(2.10)
For the system (1.1) multiply the second equation by
with both sides and obtain
(2.11)
For the system (1.1) multiply the third equation by
with both sides and obtain
(2.12)
According
and (2.10) (2.11) (2.12) we obtain
(2.13)
here
![]()
so
![]()
According to the Sobolev’s interpolation inequalities
![]()
so
(2.14)
(2.15)
According to the Sobolev’s interpolation inequalities and Young’s inequalities
(2.16)
According to the Holder’s inequalities and inequalities
(2.17)
and
(2.18)
According to the (2.13) (2.14) (2.15) (2.16) (2.17) (2.18), we obtain
(2.19)
here
![]()
According to the Poincare’s inequalities
(2.20)
According to the Young’s inequalities
(2.21)
In a similar way,we can obtain
(2.22)
(2.23)
From (2.19)-(2.23), we have
![]()
Let
, because
, so existing
satisfied
![]()
So
![]()
According to the Gronwall’s inequality,we can get the Lemma 2.
3. Solution’s existence and uniqueness and attractor of questions (1.1)
Theorem 1. Assume that
and
so questions (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 questions (1.1), let
. Here
so the diffe-
rence of the two solution satisfies
![]()
![]()
The two above formulae subtract and obtain
(3.1)
For the system (3.1) multiply the first equation by
with both sides and obtain
(3.2)
For the system (3.1) multiply the second equation by
with both sides and obtain
(3.3)
For the system (3.1) multiply the third equation by
with both sides and obtain
(3.4)
According to (3.2) + (3.3) + (3.4), we obtain
(3.5)
here
, and
so
From that, we have
(3.6)
(3.7)
Notice that
(3.8)
From the (3.5), (3.6), (3.7) and (3.8), we can obtain
![]()
Let
![]()
so, we have
![]()
According to the consistent Gronwall inequality, the uniqueness is proved.
Theorem 2. [8] Let
be a Banach space, and
are the semigroup operators on
.
here
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
so that
;
3) When
is a completely continuous operator
.
Therefor, the semigroup operators
exist a compact global attractor.
Theorem 3. Under the assume of Theorem 1, questions (1.1) have global attractor
![]()
is the bounded absorbing set of
and satisfies
1) ![]()
2)
here
and it is a bounded set, ![]()
Proof. Under the conditions of Theorem 1 and Theorem 2, it exists the solution semigroup
of ques-
tions (1.1), ![]()
From Lemma 1 - Lemma 2, to
is a bounded set that includes in the ball
![]()
![]()
This shows
is uniformly bounded in ![]()
Furthermore, when
there is
![]()
therefore,
![]()
is the bounded absorbing set of semigroup ![]()
Since
is tightly embedded, which is that the bounded set in
is the tight set in
, so the semigroup operator
to
is completely continuous.
4. Discussion
If we want to estimate the Hausdorff and fractal dimension of the attractor A of question (1.1), we need proof of the solution of question (1.1) that is differentiable. We are studying the solution’s differentiability hardly and positively. Over a time, we will get some results.
Acknowlegements
This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 11161057.
NOTES
*Corresponding author.