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]):

(1)

(2)

(3)

(4)

where ρ, u = (u¹, u², u³) ∈ ℝ³, P, and H = (H¹, H², H³) ∈ ℝ³ represent, respectively, the density, velocity, pressure, and magnetic field. The constants μ > 0 and ν > 0 denote the viscosity of fluid and the relative strengths of advection and diffusion of H. 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 ρ ≡ const, 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 (u₀, H₀) ∈ H^m(ℝ³) (m ≥ 2). 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 ν went to zero, the solution of (1)–(4) converged to the solution of ideal MHD system (i.e., μ = ν = 0). 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:

(5)

and far field conditions:

(6)

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 k ∈ ℤ⁺ and r > 1, the standard homogeneous and inhomogeneous Sobolev spaces for scalar/vector functions are denoted by:

(7)

The strong solutions of the problem (1)–(4) are defined as follows.

Definition 1. A pair of functions (ρ, u, H) is called a strong solution to the problem (1)–(4) in ℝ³ × (0, T), if for some q₀ ∈ (3, 6],

(8)

and (ρ, u, H) satisfies (1)–(4) a.e. in ℝ³ × (0, T).

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 q ∈ (3, 6] and the initial data (ρ₀, u₀, H₀) satisfying

(9)

(10)

for some g ∈ L². Then, there exist a time T_∗ > 0 and a unique strong solution (ρ, u, H) to (1)–(4) together with (5)–(6) in ℝ³ × (0, T_∗), such that

(11)

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 u ∈ C([0, T]; L³) 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 (ρ, u, H) be a strong solution to (1)–(4) with regularity (10). If T^∗ < ∞ is the maximal time of existence, then

(12)

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 L^p-norm of the magnetic field H and L²-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 U ∈ C([0, T]; L³), there exist U¹ ∈ C([0, T]; L³) and U² ∈ L^∞(0, T; L^∞) such that for any δ > 0,

(13)

where

and C(δ, M) is a positive constant depending only on δ, M.

Thus, by choosing δ > 0 suitably small, we then succeed in obtaining the estimates on by utilizing the preliminary estimates of the vorticity ω = ∇×u (see Lemma 7) to control the L^p-norm of ∇u in the proof of Lemma 10. With the estimate of at hand, we can give the higher-order estimate of (ρ, u, H) 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 f ∈ H¹ and g ∈ H² with q > 1 and r > 3. Then, for any p∈ [2,6], there exists a positive constant C, depending only on p, q, and r, such that

(14)

To complete some estimates in Section 3, we need the following L^p-estimate for vorticity ω = ∇×u. In fact, we deduce ω satisfy the following elliptic system by the momentum equation (2)

(15)

due to (∇×H) × H = H · ∇H − ∇|H|²/2. By virtue of the standard L^p-estimate of the elliptic system, we have the following.

Lemma 7. Let (ρ, u, H) be a smooth solution of (1) and (3); if , then there exists a generic positive constant depending only on ρ¯ such that

(16)

where

is a given constant.

Proof. Using Lemma 6, one deduces from (15) and the standard L^p-estimate of the elliptic system that

(17)

which immediately finish the proof of Lemma 7.

3. A Priori Estimates

Let (ρ, u, H) 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 T < T^∗ < ∞

(18)

Then, we will deduce a contradiction to the maximality of T^∗.

Throughout this paper, we will denote by C the various generic positive constants, which may depend on the initial data, M and T. 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

(19)

Next, we give the standard energy estimate as follows.

Lemma 8. Let (ρ, u, H) be a smooth solution of (1)–(4) on ℝ³ × (0, T]. Then, there exist a constant C such that

(20)

Proof. Multiplying (2) and (3) by u and H, respectively, integrating by parts, and adding them together, one immediately gets (20).

Under the assumption (18), we can improve the integrability of magnetic field H which will be frequently used in the sequel.

Lemma 9. Under the assumption (18), for any T < T^∗, it holds that

(21)

Proof. Multiplying (3) by q|H|^q−2H and integrating the resulting equations over ℝ³ lead to

(22)

For the terms on the right-hand side of the equation above, we get by integrations by parts

(23)

Substituting I₁–I₃ into (21) and using Young inequality lead to

(24)

which immediately implies that

(25)

Due to the fact that u ∈ C([0, T]; L³), we can decompose u into the following two parts:

(26)

with

(27)

for

and any δ ∈ (0, 1).

From (25)–(27) and using Hölder inequality and imbedding inequality, we have

(28)

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 u and H.

Lemma 10. Under the assumption (18), for any T < T^∗, it holds that

(29)

Proof. Multiplying (2) by u_t in L² and integrating the resulting equations by parts, we obtain after summing them up that

(30)

where we use the fact divu_t = 0.

In addition, it follows from (3) that

(31)

Putting (30) and (31) together leads to

(32)

We estimate the three terms on the right-hand side of (32) term by term. Following from Young inequality, one has that

(33)

Putting J₁ − J₃ into (31), we obtain

(34)

where C₁ 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:

(35)

Thus,

(36)

Next, we turn to estimate and . Note that, from (3), we obtain an elliptic system as follows:

(37)

Applying standard L^p-estimate to elliptic systems (15) and (37), we obtain that

(38)

which imply that

(39)

Putting (39) into (36), we have by choosing δ > 0 sufficiently small that

(40)

where the C₁ is given in (34).

Substituting (40) into (34) leads to

(41)

By (21) and the Young inequality, we easily see that

(42)

Taking this into account, we then conclude from (18)–(20), (41), and Gronwall’s inequality for any 0 ≤ T < T^∗ that holds

(43)

Applying the standard elliptic L²-estimates to (37) leads to

(44)

where we use (21), (43), and Gagliardo-Nirenberg inequality.

On the other hand, since (u, P) is a solution of the stationary Stokes equations

(45)

where F = −ρu_t − ρu · ∇u − (1/2)∇|H|² + H · ∇H. It follows from the classical regularity theory that

(46)

Adding (44) to (46), we obtain

(47)

This, together with (43), immediately implies (27). This lemma is completed.

The following lemma is concerned with the L²-estimate of ρ^1/2u_t and H_t.

Lemma 11. Under the assumption (22), it holds for any t ∈ (0, T] such that

(48)

Proof. Differentiating the momentum equations (2) with respect to t yields

(49)

Multiplying the equation above with u_t and integrating by parts, one gets

(50)

due to divu_t = 0.

Differentiating (3) with respect to t and multiplying the resulting equation by H_t, we obtain after integrating by parts that

(51)

where we have used divH_t = 0 and divu = 0.

Putting (50) and (51) together leads to

(52)

We now estimate each term on the right-hand side of (52) by using the previous estimates.

First, by virtue of (1), we obtain

(53)

Similarly, the estimate of R₂ is given as follows

(54)

For R₃, we have that

(55)

From Lemma 8 to 9, we can reduce that

(56)

From the J₁–J₃, we get estimate of R₃

(57)

Similarly, we have

(58)

It is easy to prove that R₆ = 0. Thus, taking ε suitable small and substituting the estimates of R₁–R₆ into (52) lead to

(59)

which, together with Gronwall’s inequality, immediately leads to the desired estimate (50) since (27) implies ∇²u ∈ L²(0, T),

, 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 t ∈ (0, T] such that

(60)

Proof. Applying classical regularity theory to (45) again, we have

(61)

Integrating the inequality above over (0, t) and by (27) lead to

(62)

Similar proof leads to the same conclusion for H

(63)

By virtue of (44), (46), and (48), it is easily to prove that

(64)

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 t ∈ (0, T] such that

(65)

Proof. Differentiating (1) with respect to x_i (i = 1, 2, 3), multiplying it by with p ≥ 2, and integrating the resulting equation by parts, we obtain after summing over i from 1 to 3 that

(66)

which, together with Gronwall’s inequality, leads to

(67)

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 (ρ, u, H) beyond t > T^∗ by the standard method. This leads to a contradiction of the assumption on T^∗. The proof of Theorem 3 is therefore complete.

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).

Open Research

Data Availability

All data, models, and code generated or used during the study appear in the submitted article.

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A Blow-Up Criterion for 3D Nonhomogeneous Incompressible Magnetohydrodynamic Equations with Vacuum

Abstract

1. Introduction

2. Auxiliary Lemmas

3. A Priori Estimates

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

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A Blow-Up Criterion for 3D Nonhomogeneous Incompressible Magnetohydrodynamic Equations with Vacuum

Abstract

1. Introduction

2. Auxiliary Lemmas

3. A Priori Estimates

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

Related

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