Abstract
Regularity criteria of the weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamic (MHD) equations are discussed. Our results imply that the scalar pressure field plays an important role in the regularity problem of MHD equations. We derive that the weak solution is regular on , which is provided for the scalar pressure field in the Besov spaces.
1. Introduction
In this article, we consider the global regularity problem concerning the 3D incompressible MHD equations that satisfy the initial condition
Here, , and represent the velocity field, the magnetic field, and the pressure, respectively; are the corresponding initial data which satisfied in the sense of distribution.
MHD equations govern the dynamics of the velocity and magnetic fields of electrically conducting fluids such as plasmas, liquid metals, and salt water. Besides their important physical applications, the MHD equations also have important mathematical significance. It is well known [1] that problem (1) is locally well-posed for any given initial datum . However, whether a local strong solution can exist globally, or equivalently, whether global weak solutions are smooth is still a challenging open problem. Nevertheless, there exist plenty of results in the literature showing that the answer to this problem is positive if some additional conditions are imposed on the weak solutions [2–12]. Some of them are motivated by the works on the Navier-Stokes equations ()
Among these results, Zhou [7] and He and Xin [13] obtained some Ladyzhenskaya-Prodi-Serrin-type regularity criteria for the 3D MHD equations in terms of velocity and the gradient of velocity, independently. They proved that the velocity satisfies then the weak solution is regular on .
However, we are interested in regularity criteria involving only in terms of the pressure (for details, refer to [14–17]). For regularity criteria results for the 3D MHD equations, Jia and Zhou [14] obtained the following:
In [15], Liu established the new regularity criteria in terms of the pressure as follows:
Later, Gala and Ragusa [16] extended the regularity criteria to the BMO space and Besov space. If the pressure or the pressure gradient satisfies or then the weak solution is regular on
Very recently, Tong and Wang [17] showed the following regularity criterion for the 3D MHD: or
It is the aim of the paper to give the complete description on the regularity criteria of the weak solutions of the 3D MHD equations on the pressure field. Let us introduce the assignment of this paper; we first recall some preliminaries on functional settings and state the main results in Section 2 and prove the main results in Section 3.
2. Preliminaries and Main Result
Throughout this text, stands for a generic positive constant which may differ in value from one line to another. We use to denote the norm of the Lebesgue space , and the norm as follows:
In order to define the Besov space, let us first recall the Littlewood-Paley theory (see Ref. [18]). For a given function , its Fourier transformation of is defined by where is the Schwartz class of rapidly decreasing functions defined on . Choose a nonnegative radial function supported in such that for and . Setting the radial function for the integer set , we have
Let and define the dyadic blocks as follows:
We thus have the following Littlewood-Paley decomposition:
For we can now define the homogeneous Besov space as where
is the set of all scalar polynomials defined on . is the space of all tempered distributions on .
Definition 1. Assuming and , the measurable function defined on is called the weak solution of Equation (1), if
(1).(2) and haveEquation (1) holds in the sense of distributions.
For , we have
where .
(3)The strong energy inequality, that is,
In this paper, we establish the following theorem.
Theorem 2. Let with in the sense of distribution. Let be a weak solution of the MHD Equations (1) on which satisfies the strong energy inequality (21). If the corresponding pressure satisfies then the weak solution is regular on
When the time critical index , we can derive the following Theorem 3. And if , we prove that the weak solution is regular on in Theorem 4.
Theorem 3. Under the same assumption in Theorem 2, if the corresponding pressure satisfies then the weak solution is regular on
Theorem 4. Under all the assumptions in Theorem 2, the weak solution is regular on If the corresponding pressure satisfies the following condition:
Remark 5. In article [17], the regularity condition of is better than the regularity condition of Theorems 2–4. However, in article [17], the initial condition is required to satisfy , while Theorems 2–4 only need to satisfy .
3. Pressure Regularity Criteria
In order to avoid the difficulties caused by the magnetic field when we do energy estimates, by adding and subtracting (1)1 with (1)2, we convert the 3D MHD Equations (1) into a mathematically symmetric form as follows: with .
Let us introduce some auxiliary results. To establish some new regularity criteria in terms of pressure, an effective method is to find a “bridge” between the desired results and the known criteria. The following lemma plays such a role in the proof of Theorems 2–4.
Lemma 6 (Bernstein inequality [18]). For and an integer , the following estimate is true:
Lemma 7. Let and . Let be the corresponding solution to the 3D MHD Equations (1). If and satisfy Then, the weak solution is regular on .
Proof. By the definition of and , one can deduce that
By (4), we can derive that the weak solution is regular on .
Now, we are ready to prove the theorem. Taking the inner product of the first equation of (21) with and the second equation of (21) with in , adding them together, and noticing that we have
Here, we have used the following identities due to the fact that the divergence free condition :
By the Littlewood-Paley decomposition, can be written as follows:
Hence, by (29), we obtain
☐
3.1. The Proof of Theorem 2
We first consider that the velocity satisfies the growth condition (22):
For , by the Hölder inequality, we estimate of the right-hand side of (32) one by one.
Similarly, for , integration by parts gives
Hence, choosing the integer such that (34) is equal to (35), then
Plugging the estimates (34) and (35) into (32), and using Young inequality, we have
By letting , such that , and by using the Young inequality, it follows that
Hence, we obtain
Using the similar way with the estimate , we see that
Combination of (40), (41), and (29) implies that
Using the Gronwall inequality, we have
By assuming Theorem 2 and the definition of , we have
By using Lemma 6 and (22), we have
This completes the proof of Theorem 2.
3.2. The Proof of Theorem 3
Next, we are ready to prove Theorem 3. Let us begin with the case of growth condition (23):
For , by the Hölder inequality, we estimate one by one.
Similarly, for of (32), by (26) and integration by parts, gives
Hence, choosing the integer such that (47) is equal to (48) then
Plugging the estimates (47) and (48) into (32), we have
Using the similar way with the estimate , we see that
Combination of (51), (52), and (29) implies that
By letting , such that , and by using the Young inequality, it follows that
Using the triangle inequality we obtain
Hence,
Then, the bounds for the -norms of and follow from the standard Gronwall inequality. Thanks to (4), this completes the proof of Theorem 3.
3.3. The Proof of Theorem 4
For the final assumptions (24), that is,
After applying the Hölder inequality, we estimate one by one.
Similarly, for of (32), by (27), the Hölder inequality, and integration by parts, we have
Hence, choosing the integer such that (59) is equal to (60), then
Plugging the estimates (59) and (60) into (32), we have
Using the similar way with the estimate , we see that
Combination of (63), (64), and (29) implies that
Using the same triangle inequality we obtain
Hence,
By Gronwall’s inequality and (4), Theorem 4 is proven.
Data Availability
No data.
Conflicts of Interest
The authors declare that they have no conflict of interest.
Acknowledgments
This work is partially supported by the MOOC of Anhui Province (2018mooc604), the NSF of Anhui Province (KJ2018B0002), and the University Outstanding Talents Support Program Project of Anhui Province (gxyqZD2021144).