Abstract
For the strong solutions of the nonhomogeneous incompressible magnetohydrodynamics (MHD) system with vacuum, we establish a blow-up criterion for this system in terms of . Moreover, the result generalizes previous ones in Giga (1986) and He and Xin (2005) where homogeneous incompressible Navier-Stokes equations and homogeneous incompressible MHD system are considered, respectively, and demonstrates that the velocity field plays a more dominant role in the MHD system.
1. Introduction
Magnetohydrodynamics (MHD) is concerned with the interaction between fluid flow and magnetic field, and the motion of the nonhomogeneous incompressible MHD can be stated as follows (see, e.g., [1–4]): where , , , and represent, respectively, the density, velocity, pressure, and magnetic field. The constants and denote the viscosity of fluid and the relative strengths of advection and diffusion of . Since the presence of all the physical constants does not create essential mathematical difficulties, for notational simplicity, we will normalize all constants in the system to be one in the sequel.
In recent years, the MHD system has drawn the attention of engineers and applied mathematicians due to its important physical background and mathematical feature. If taking , the system (1)-(4) is reduced to the homogeneous incompressible MHD. For this case, Duvaut and Lions [5] constructed a class of global weak solutions, similar to the Leray-Hopf weak solutions to the three-dimensional Navier-Stokes equations. Sermange and Temam [6] first gave a local existence of the strong solution with any given initial data (). It should be pointed out that whether this unique local solution can exist globally with general initial data is an outstanding challenging problem in three dimensions. Thus, there are many works to study the regularity criteria for weak or classical solutions, see [7–12]. We also notice that if partial viscosity and resistivity are zero, the global regularity issues have been established in [13].
For the nonhomogeneous case (1)–(4), there are a lot of literature which includes the existence, uniqueness, and regularity of solutions [1, 14–17]. Zhang [18] established local classical solutions of (1)–(4) and showed that as the viscosity μ and resistivity went to zero, the solution of (1)–(4) converged to the solution of ideal MHD system (i.e., ). Gerbeau [3] and Desjardins and Le Bris [19] considered the global existence of weak solutions of finite energy in the whole space or in the torus. Abidi and Paicu [20] proved the global existence of strong solutions with small initial data in some Besov spaces. Recently, Huang and Wang [21] demonstrated the unique global strong solution with general initial data to (1)–(4) in two dimensions.
In this paper, we are interested in the Cauchy problem of (1)–(4) subject to the following initial conditions: and far field conditions: where is a given constant.
To state the main results in a precise way, we first introduce some notations and conventions which will be used throughout the paper. For and , the standard homogeneous and inhomogeneous Sobolev spaces for scalar/vector functions are denoted by:
.
The strong solutions of the problem (1)–(4) are defined as follows.
Definition 1. A pair of functions is called a strong solution to the problem (1)–(4) in , if for some , and satisfies (1)–(4) a.e. in .
Before stating the main result of this paper, we first state a local existence of strong solutions to (1)–(4). The following local well-posedness theorem of strong solutions was given in [16].
Proposition 2. Assume that for some and the initial data satisfying for some . Then, there exist a time and a unique strong solution to (1)–(4) together with (5)–(6) in , such that
Although significant progress has been made in the study of multidimensional nonhomogeneous incompressible MHD system, many physically important and mathematically fundamental problems are still open due to the lack of smoothing mechanism and the strong nonlinearity. Similar to that for the three-dimensional incompressible Navier-Stokes equations, whether the unique local strong solution obtained in Proposition 2 can exist globally is an outstanding challenging open problem. If the answer is negative, then it simultaneously raises the interesting questions of the mechanism of blowup and the structure of possible singularities.
In the recent paper [7], He and Xin proved a blow-up criterion to nonhomogeneous incompressible magnetohydrodynamic equations; that is, if is bounded above, then the local strong solution, in fact, is a global one. This criterion is analogous to the criterion on the weak solutions to the 3D incompressible Navier-Stokes equations (see [22]). Motivated by these works on the blow-up criterion of local strong solutions to the Navier-Stokes equation and homogeneous incompressible MHD system, we will generalize this result in [7, 22] to the 3D nonhomogeneous incompressible MHD system (1)–(4). Our main result of this paper is stated as follows.
Theorem 3. Suppose that the assumptions in Proposition 2 are satisfied. Let be a strong solution to (1)–(4) with regularity (10). If is the maximal time of existence, then
Remark 4. The proof of this theorem together with the result in [23], we can extend this result from incompressible magnetohydrodynamic equations to the compressible case, which is our work in the future.
To prove Theorem 3, the main two steps are to estimate the -norm of the magnetic field and -norm of the gradient of the velocity. To do this, the key observation of the present paper lies in the following simple fact:
Proposition 5. For , there exist and such that for any , where and is a positive constant depending only on .
Thus, by choosing suitably small, we then succeed in obtaining the estimates on by utilizing the preliminary estimates of the vorticity (see Lemma 7) to control the -norm of in the proof of Lemma 10. With the estimate of at hand, we can give the higher-order estimate of and thus finish the proof of Theorem 3.
2. Auxiliary Lemmas
We state the well-known Gagliardo-Nirenberg inequality (see, for instance, [24]).
Lemma 6. Assume that and with and . Then, for any [2,6], there exists a positive constant , depending only on , , and , such that
To complete some estimates in Section 3, we need the following -estimate for vorticity . In fact, we deduce ω satisfy the following elliptic system by the momentum equation (2) due to . By virtue of the standard -estimate of the elliptic system, we have the following.
Lemma 7. Let be a smooth solution of (1) and (3); if , then there exists a generic positive constant depending only on ρ¯ such that where is a given constant.
Proof. Using Lemma 6, one deduces from (15) and the standard -estimate of the elliptic system that which immediately finish the proof of Lemma 7.
3. A Priori Estimates
Let be strong solutions to the problem (1)–(4) as described in Proposition 2. We will prove Theorem 3 by a contradiction argument. To this end, we suppose that for any
Then, we will deduce a contradiction to the maximality of .
Throughout this paper, we will denote by the various generic positive constants, which may depend on the initial data, and . Special dependence will be pointed out explicitly in this paper if necessary.
First of all, by the method of characteristics, it is easy to see that
Next, we give the standard energy estimate as follows.
Lemma 8. Let be a smooth solution of (1)–(4) on . Then, there exist a constant such that
Proof. Multiplying (2) and (3) by and , respectively, integrating by parts, and adding them together, one immediately gets (20).
Under the assumption (18), we can improve the integrability of magnetic field which will be frequently used in the sequel.
Lemma 9. Under the assumption (18), for any , it holds that
Proof. Multiplying (3) by and integrating the resulting equations over lead to
For the terms on the right-hand side of the equation above, we get by integrations by parts
Substituting into (21) and using Young inequality lead to
which immediately implies that
Due to the fact that , we can decompose into the following two parts:
with
for and any .
From (25)–(27) and using Hölder inequality and imbedding inequality, we have
This, together with taking δ suitably small and applying Gronwall’s inequality, immediately leads to the desired estimate (21).
Under the assumption (18) and Lemmas 8 and 9, we prove the following crucial estimate concerning the estimates of the gradients of and .
Lemma 10. Under the assumption (18), for any , it holds that
Proof. Multiplying (2) by in and integrating the resulting equations by parts, we obtain after summing them up that
where we use the fact .
In addition, it follows from (3) that
Putting (30) and (31) together leads to
We estimate the three terms on the right-hand side of (32) term by term. Following from Young inequality, one has that
Putting into (31), we obtain
where is a positive constant depending only on the initial data.
By virtue of (21) and (27), we can deal with the second term on the right-hand side of (34) as follows:
Thus,
Next, we turn to estimate and . Note that, from (3), we obtain an elliptic system as follows:
Applying standard -estimate to elliptic systems (15) and (37), we obtain that
which imply that
Putting (39) into (36), we have by choosing sufficiently small that
where the is given in (34).
Substituting (40) into (34) leads to
By (21) and the Young inequality, we easily see that
Taking this into account, we then conclude from (18)–(20), (41), and Gronwall’s inequality for any that holds
Applying the standard elliptic -estimates to (37) leads to
where we use (21), (43), and Gagliardo-Nirenberg inequality.
On the other hand, since is a solution of the stationary Stokes equations
where . It follows from the classical regularity theory that
Adding (44) to (46), we obtain
This, together with (43), immediately implies (27). This lemma is completed.
The following lemma is concerned with the -estimate of and .
Lemma 11. Under the assumption (22), it holds for any such that
Proof. Differentiating the momentum equations (2) with respect to yields
Multiplying the equation above with and integrating by parts, one gets
due to .
Differentiating (3) with respect to and multiplying the resulting equation by , we obtain after integrating by parts that
where we have used and .
Putting (50) and (51) together leads to
We now estimate each term on the right-hand side of (52) by using the previous estimates.
First, by virtue of (1), we obtain
Similarly, the estimate of is given as follows
For , we have that
From Lemma 8 to 9, we can reduce that
From the , we get estimate of Similarly, we have
It is easy to prove that . Thus, taking ε suitable small and substituting the estimates of into (52) lead to
which, together with Gronwall’s inequality, immediately leads to the desired estimate (50) since (27) implies , , and .
Lemmas 12 and 13 deal with the higher-order estimates of the solutions which are needed to guarantee the extension of a local strong solution to a global one.
Lemma 12. Under the assumption (22), it holds for any such that
Proof. Applying classical regularity theory to (45) again, we have Integrating the inequality above over and by (27) lead to Similar proof leads to the same conclusion for By virtue of (44), (46), and (48), it is easily to prove that Thus, Lemma 11 is proved.
Finally, the following lemma gives bounds of the first spatial derivatives of the density ρ.
Lemma 13. Under the assumption (18), it holds for any such that
Proof. Differentiating (1) with respect to , multiplying it by with , and integrating the resulting equation by parts, we obtain after summing over from 1 to 3 that which, together with Gronwall’s inequality, leads to This finishes the proof of Lemma 13.
With all the a priori estimates in Section 3 at hand, we are ready to prove the main result of this paper.
Basing on Lemmas 8–13 and using the local existence theorem (cf. Proposition 2), one can easily extend the strong solutions of beyond by the standard method. This leads to a contradiction of the assumption on . The proof of Theorem 3 is therefore complete.
Data Availability
All data, models, and code generated or used during the study appear in the submitted article.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors have contributed their parts equally and have also read and approved the final manuscript.
Acknowledgments
This research was partly supported by the National Natural Science Foundation of China (Grant Nos. 61773018, 11971446, 11701525, and 12101569), the Natural Science Foundation of Henan (212300410301), and the China Scholarship Council (Grant No. 201709440003).