1. Introduction
The generalized MHD system is
(1)
where
,
, u denotes the velocity field and b denotes the magnetic field. The magnetohydrodynamic (MHD) systems [1] control the dynamics of velocity and magnetic fields in conductive fluids such as plasma and reflect the basic laws of physical conservation.
In recent years, the MHD equations with partial dissipation regularity problem have attracted considerable interests. For example, the n-dimensional MHD Equation (1), when the coefficient satisfies
it has been proved that the solution has global regularity [2] . Wu [3] has been proved the 2D GMHD admits a global regularity for a three-case:
And it is also proved that the condition satisfying
has a global smooth solution with the direction of the magnetic field that remains sufficiently smooth. Cao, Regmi and Wu [4] have been proved that the 2D MHD with horizontal dissipation and horizontal magnetic diffusion in horizontal component of any solutions has a global regularity. The global regularity of the class solution of the MHD equation with magnetic diffusion and mixed partial dissipation is established by Wu [5] . In [6], the global existence and uniqueness of the smooth solution of 2D micropolar fluid flow with zero angular viscosity have been proved. Other related articles can be seen in [7] [8] [9], etc.
In this paper, we study the 2D MHD systems with vertical dissipation and vertical dissipation magnetic diffusion, namely
(2)
In this case, we only get the global
-bound of the solution in the y-direction, and the global regularity problem for the complete directional solution has not been achieved.
In the following article, let
, this will provide us with convenience. We have a symmetric equation by (2), namely
(3)
The new Equation (3) consists of two vectors, which is more complicated in the calculation process, therefore, we use fractionally derivative triple product estimation [4] to solve this difficulty. This paper takes Cao and Wu recent study of two-dimensional partially dissipated Boussinesq equation [8] as an example to discuss the influence of known vertical component
Lebesgue norm on global regularity. And in Section 4, we obtain the main Theorem 3, which proves that
for
. In fact, in Section 2 we get Theorem 1, which is about the solution of Equation (2) bounded by Lebesgue in the y-direction. The sameness of Theorem 1 and Theorem 3 is that boundedness is related to the r, but in Theorem 1, we get the case of
, and Theorem 3 has a slower bounded change with the increase of r.
The rest of this article is divided into four parts. In Section 2, we prove the global bounded for
, and the boundedness depends on the index of r. In Section 3, we show the global bounded for
and
with
. In Section 4, we prove that the solution of (2) in y-direction has a global Lebesgue bound. In Section 5, we prove the bounded condition of
under the
norm.
2. A Global Bound in the Lebesgue Spaces
In this section, we prove the classical solution of (2) at the y-direction exists globally bounded in
norm. The boundedness obtained here depends on the index of r. We have the following theorem.
Theorem 1. Assume that
and
,
be the corresponding solution of (2). For any
,
obeys global bound
(4)
where
and
are constants depending on
only.
To prove the Theorem 1, we need to estimate the global bounded under
norm.
Lemma 1. Let
and let
be the corresponding solution of (2). Then, for ant
,
obeys the
Here we omit the proof of Lemma 1 and now begin to prove Theorem 1.
Proof. Taking the product of the second component of the first equation of (3) with
, and integrating with respect to space variable, we obtain
(5)
note that
By Hölder’s and Sobolev’s inequalities, and using Young’s inequality, we got
where C is a constant independent of r. In order to bound the pressure, we take the divergence of (3), we get
(6)
Since, the Riesz transform [10] has bounds for any
on
, we have
(7)
Consequently,
Based on the above estimates, we get
Similarly,
Combine these two inequalities to get
Following the Gronwall's inequality, we obtain
According to Lemma 1, get (4). □
3. Global Bounds for the Pressure
In this section, we show the solution of the first components
has a
-bound with
or
, and establish the pressure has a global bound. The results can be stated as follows.
Theorem 2. Assume that
and let
be the corresponding solution of (2)
(8)
for any
, and
,
(9)
where
and
, and C is a constant related to T and initial value.
Here we use two calculus inequalities of the following lemma.
Lemma 2. [4] Assume that
,
and
, then
(10)
(11)
Proof. We use the symmetric Equation (3) to prove the case of
in Theorem 2. Take the inner product of the first Equation (3) with
, we obtain
(12)
Using
and integrate by parts, we get
by Hölder’s and Sobolev’s inequalities,
(13)
According to (7),
Therefore, by Young’s inequality,
To bound
, we first apply Hölder inequality,
(14)
According to Lemma 2 and
,
(15)
According to (7), we get
(16)
Therefore
(17)
Therefore, recalling Theorem 1 and Sobolev embedding theorem, we get a global bound for
.
Similarly, we can be established bound for
. To prove the
-bound in (8), we get from (3) that
Note that
Using Hölder’s inequality, (6) and Lemma 2, we obtain
The same can be proved that by Hölder’s inequality and (6), we get
Therefore, by Young’s and Gronwall’s inequalities,
(18)
We now proved the inequality (9), taking the divergence of the first two equations in (3), we get
Following the finiteness of Riesz transforms on
, we have
For
, according to Theorem 1 and (8),
and
is bounded, thus
.
Recall that the operator
is defined through the Fourier transform [11], namely
Combining (6), Hardy-Littlewood-Sobolev inequality [12] and the boundedness of Riesz transforms in
, we obtain
(19)
with
and C is a constant independent of s. □
4. An Improved Global Lebesgue Bound
From the conclusions of Sections 2 and 3, we have the main theorem of this paper.
Theorem 3. Assume that
be the corresponding solution of (2). Let
, then
(20)
where
is a smooth function of t and
depends only on
.
Before proving the Theorem 3, we first describe the lemma that will be used.
Lemma 3. [4] Let
and
. Assume
,
and
. Then,
(21)
where
and
are given by
(22)
and
denotes a fractional with respect to vertical dissipation and is defined by
(23)
Lemma 4. [8] Let
and
denote the ball centered at zero with radius R and by
the characteristic function on
with
and
. Write
(24)
where
and
denote the Fourier transform and the inverse Fourier transform. We have the following estimates for
and
.
1) For a pure constant
(independent of s)
(25)
2) For any
satisfying
, there is a constant
independent of s, q, R and f such that
(26)
Details can be seen in [8], we have omitted here.
Lemma 5. Let
. Let
and let
be defined as in (24). Then, there exists a constant C depending on q only such that
Next we prove the Theorem 3.
Proof. According to Theorem 1, we have
(27)
with
. The right side of Equation (27) will be estimated using a different method. First, we fix
and write
where
and
as defined in (24). To estimate
and
, let
(28)
By Hölder’s and Young’s inequalities, we obtain
Applying Lemma 4, we have
(29)
where
is a constant independent of s. In the rest of the proof, we focus on whether a constant is bounded uniformly as
. Using the interpolation inequality, we have
(30)
In summary, we obtain
(31)
where
is independent of s. Now we estimate
, apply Lemma 3 to obtain
where s and q satisfy (28),
and
are given explicitly in terms of s and q
(32)
and C is bounded uniformly as
. According to (30), we get
(33)
By Hölder’s inequality,
By Young’s inequality
(34)
where C is again bounded uniformly as
, and we make
(35)
For further estimation, we spilt
into two parts and bound one of them by Lemma 4. Moreover, we get any
,
(36)
Owing to the condition of s and q in (28), this boundary allows us to generate
with
. Inserting (36) in (34) yields
where C is bounded uniformly as
. We choose
so that the sum of the powers of
and of
is equal to 2, namely
Recalling (32) and (35), we have
(37)
The condition in (28) ensures that
, then
For
given by (37), we have
(38)
Combining (27), (31) and (38) we have
(39)
with a constant
is independent in s and C is bounded uniformly as
. Let
that is,
(40)
Using (32), (35) and (37) to simplify this index and get
Let
(41)
and therefore
. Obviously,
as
, and
(42)
Furthermore,
(43)
For generality, we assume
. Following (39) and get
(44)
where C is bounded uniformly as
, and
Since (44) holds for any
, we set
we get
(45)
Choose the right
, and according to Theorem 3,
is integrable at any time interval. This completes the proof of Theorem 3. □
5. Conditional Global Regularity
This section estimates the global boundedness of the vertical component
and
of
under the
norm. We have the following theorem.
Theorem 4. Assume
and
be the corresponding solution of (2). If
for some
, then
is finite on
.
We divide the proof of the theorem into two parts.
5.1. H1 in Terms of
In this section, we estimate that the solution has a
-bound, and we have the following proposition.
Proposition 5. Assume
and let
be the corresponding solution of (2). Then, for any
and
,
(46)
where
depends on T and the initial data only and
is a pure constant.
Proof. Taking the inner product of the first equation of (3) with
and integrating by parts, we find
where
Using the anisotropic Sobolev inequalities [5] and
, we can be bounded as follows,
Similarly, we can estimate
. Combining them yields
(47)
According to Gronwall’s inequality, get
has a
-bounded. Combining with the Lemma 1 to got (46). □
5.2. Proof of Theorem 4
In this section, we use the global bounds of Proposition 5 to prove the completion of the Theorem 4.
Proof. Taking the inner product of the first equation in (3) with
and integrating by parts, we find
(48)
We decompose the nonlinear term into different parts and estimate it using anisotropic dissipation. We write
with
We further divide
into four parts,
, where
Applying Hölder’s inequality and
, after integration by parts we get
Similarly, we obtain
To bound
and
, we use anisotropic Sobolev inequality and Proposition 5, we obtain
Combining with the estimates, we obtain
and
can be estimated in a similar way and here we will omit the details. Combining all of these estimates and applying Gronwall’s inequality, we have
(49)
where
Similarly,
(50)
and
combines with (50) and (49), we get
Applying Gronwall’s inequality and (4), (8), (47), we can prove that the solution
in (2) has a global
-bound. This completes the proof of Theorem 4. □
6. Conclusion
According to Wu [4], in this paper, we prove that the solution of the system (2) has regularity in the vertical direction. In order to get this result, we need to make a corresponding estimate of the pressure to prove that it’s bounded. Especially in the case of
, the solution has regularity in
.