A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density

We study an initial boundary value problem for the 3D magnetohydrodynamics (MHD) equations of compressible fluids in $\R^3$. We establish a blow-up criterion for the local strong solutions in terms of the density and magnetic field. Namely, if the density is away from vacuum ($\rho= 0$) and the concentration of mass ($\rho=\infty$) and if the magnetic field is bounded above in terms of $L^\infty$-norm, then a local strong solution can be continued globally in time.

On the other hand, the global existence of smooth solution to the MHD system (1.1)-(1.4) with arbitrary smooth data is still unknown. For the corresponding Navier-Stokes system, Z. Xin [16] proved that smooth solution will blow up in finite time in the whole space when the initial density has compact support, while Rozanova [11] showed similar results for rapidly decreasing initial density. Recently Fan-Jiang-Ou [2] established some blow-up criteria for the classical solutions to 3D compressible flows, which were further extended by Lu-Du-Yao [7] for MHD system.
The main goal of the present paper is to generalize the corresponding results of Sun-Wang-Zhang [15] to the MHD system (1.1)-(1.4). When the initial vacuum is allowed, Y. Sun, C. Wang and Z. Zhang obtained a blow-up criterion in terms of the upper bound of the density for the strong solution to the 3-D compressible Navier-Stokes equations. With the presence of magnetic field, we are able to obtain parallel results as in [15] except that we do not allow vacuum in the initial density.
We now give a precise formulation of our results. First concerning the assumptions on the parameters, we have (1.6) There exists K > 0 such that P (ρ) = Kρ for all ρ > 0; ε, λ, ν > 0 and λ < ε. (1.7) For the initial data, we assume that and we also write We make use of the following standard facts (see Ziemer [17] . (1.10) We denote the material derivative of a given function v byv = v t + ∇v · u, and if X is a Banach space we will abbreviate X 3 by X. Finally if I ⊂ [0, ∞) is an interval, C 1 (I; X) will be the elements v ∈ C(I; X) such that the distribution derivative v t ∈ D ′ (R 3 × int I) is realized as an element of C(I; X).
We recall a local existence theorem for (1.1)-(1.4) by Kawashima [10], pg. 34-35 and pg. 52-53: (Kawashima) Assume that ε, λ, ν are strictly positive and that the pressure P satisfies (1.6). Then givenρ > 0 and C 3 > 0, there is a positive time T depending onρ, C 3 and the parameters ε, λ, ν, P such that if the initial data (ρ 0 −ρ, u 0 , B 0 ) is given satisfying (1.8) and The following is the main result of this paper: Theorem 1.2 Assume that the system parameters satisfy (1.6)-(1.7). Givenρ > 0, suppose (ρ 0 −ρ, u 0 , B 0 ) satisfies (1.8). Assume that (ρ −ρ, u, B) is the smooth solution as constructed in Theorem 1.1, and let T * ≥ T be maximal existence time of the solution. If T * < ∞, then we have The rest of the paper is organized as follows. We begin the proofs of Theorem 1.2 in section 2 with a number of a priori bounds for local-in-time smooth solutions. We make an important use of estimates on the Lamé operator L which are mainly inspired by [3] and [15]. Finally in section 3 we prove Theorem 1.2 via a contradiction argument by deriving higher order H 3 -bounds for smooth solutions.

A prior estimates
In this section we derive a prior estimates for the local solution (ρ −ρ, u, B) on [0, T ] with T ≤ T * as described by Theorem 1.1. Here T * is the maximal time of existence which is defined in the following sense: We will prove Theorem 1.2 using a contradiction argument. Therefore, for the sake of contradiction, we assume that (2.1) To facilitate our exposition, we first define some auxiliary functionals for 0 ≤ t ≤ T ≤ T * : The following is the main theorem of this section: Theorem 2.1 Assume that the hypotheses and notations in Theorem 1.1 are in force. Given C > 0 andρ > 0, assume further that (ρ −ρ, u, B) satisfies (2.1).
Then there exists a positive number M which depends on C 0 , C, T * and the system parameters P, ε, λ, ν such that, for 0 ≤ t ≤ T ≤ T * , We prove Theorem 2.1 in a sequence of lemmas. We first derive the following lemma which gives estimates on the solutions of the Lamé operator L = ε∆ + (ε + λ)∇div. More detailed discussions can also be found in Sun-Wang-Zhang [15].

Lemma 2.2 Consider the following equation:
ε∆v HereC is a positive constant which depends only on ε, λ, p Proof. A proof can be found in [15] pg. 39 and we omit the details here.
We proceed to the following a prior estimates which is the energy-balanced law: where M (C) is a constant which depends on C.
Proof. Let G = G(ρ) be a functional defined by Multiplying the momentum equation (1.2) by u j , summing over j, integrating and making use of the continuity equation (1.1), we get: Similarly, we multiply the magnetic field equation (1.3) by B and integrate to get We then obtain (2.4) by adding (2.5) to (2.6) and using the fact that We obtain the following L 4 bounds for u and B: Lemma 2.4 Assume that the hypotheses and notations of Theorem 2.1 are in force. Then for any Proof. Multiply (1.2) by 2|u| 2 u and integrate to obtain The third term on the left side of (2.8) can be estimated from below by By assumption (1.7) we have ε < λ, hence it implies On the other hand, we multiply (1.3) by 4|B| 2 B and integrate to get Adding (2.10) to (2.9) and integrate with respect to t, we get (2.11) Using the assumption (2.1), the right side of (2.11) can be bounded by (2.12) Using (2.12) on (2.11) and applying Cauchy Inequality, we get and (2.7) now follows by Gronwall's inequality.
We obtain estimates on the functional A 1 in terms of H: Lemma 2.5 Assume that the hypotheses and notations of Theorem 2.1 are in force. Then for any 0 ≤ t ≤ T ≤ T * , Proof. We multiply (1.2) byu j , sum over j and integrate to get Niext we multiply (1.3) by B t and integrate, Adding (2.14) and (2.15), we obtain (2.16) The second term on the right side of (2.16) is bounded by where the last inequality follows by Lemma 2.3. For the last integral on the right side of (2.16), using assumption (2.1), it can be bounded by Recall from Lemma 2.4 that, for 0 ≤ t ≤ T ≤ T * , Therefore, using (1.10), , and by Cauchy inequality, we obtain t 0 R 3 |∇B| 2 |u| 2 dxds ≤ M 1 + A 1 (t) 3 4 . (2.17) Applying (2.17) to (2.16) and absorbing terms, (2.13) follows.
We derive the following estimates on the effective viscous flux which were first described by Hoff [3] and later modified by Sun-Wang-Zhang [15].
Proof. Using the momentum eqaution (1.2), The first term on the right side of (2.20) can be estimated as follows: where the last inequality follows by Lemma 2.2 and assumption (2.1). Therefore (2.21) becomes The third and the fourth term on the right side of (2.20) are bounded by It remains to estimate the term t 0 R 3 −ρv t · w t dxds on the right side of (2.20). By the definition of v and P (ρ), we have Hence we can apply Lemma 2.2 and Lemma 2.3 to get and so by Lemma 2.2, and we apply the above to (2.26) to conclude

which (2.18) follows by Gronwall's inequality.
We finally obtain an estimate on the functional A 2 which is sufficient to prove Theorem 2.1: Lemma 2.7 Assume that the hypotheses and notations of Theorem 2.1 are in force. Then for any 0 ≤ t ≤ T ≤ T * , Proof. Taking the convective derivative in the momentum equation (1.2), multiplying it byu j , summing over j and integrating, sup 0≤s≤t R 3 Next we differentiate the magnetic field equation (1.3) with respect to t, multiply by B t and integrate, Adding the above to (2.28) and absorbing terms, The third term on the right side of (2.29) is bounded by where the last inequality follows by (2.17) and (2.22). The last term on the right side of (2.29) is bounded by and So it remains to estimate H. Let w and v be as defined in Lemma 2.6. Then The second term on the right side of (2.32) is bounded by And for t 0 R 3 |∇w| 4 dxds, using (2.18), Notice that, by rearranging the terms in (2.19), and so by Lemma 2.2, Therefore we conclude that and (2.2) follows.

Higher Order Estimates and proof of Theorem 1.2
In this section we continue to obtain higher order estimates on the smooth local solution (ρ−ρ, u, B) as described in section 2. Together with Theorem 2.1, we show that, under the assumption (2.1), the smooth local solution to (1.1)-(1.4) can be extended beyond the maximal time of existence T * as defined in section 2, thereby contradicting the maximality of T * . The following is the main theorem of this section: Theorem 3.1 Assume that the hypotheses and notations in Theorem 2.1 are in force. Given C > 0 andρ > 0, assume further that (ρ −ρ, u, B) satisfies (2.1).
Then there exists a positive number M ′ which depends on C 0 , C, T * and the system parameters P, ε, λ, ν such that, for 0 ≤ t ≤ T ≤ T * , Proof. We give the proof in a sequence of steps. Most of the details are reminiscent of Suen and Hoff [14] and we omit those which are identical to or nearly identical to arguments given in [14]. We first begin with the following estimates on the effective viscous flux F and the vorticity matrix ω: Step 1: Define where M (q) is a positive constant depending on q and proof of Step 1. We give the proof of (3.2) as an example. Using the definition of F and ω, Differentiating and taking the Fourier transform we then obtain (2ε + λ)û j x l (y, t) = y j y l |y| 2F (y, t) + (2ε + λ) y k y l |y| 2 ω j,k (y, t) + y k y l |y| 2 ( P −P )(y, t) and (3.2) then follows immediately from the Marcinkiewicz multiplier theorem (Stein [13], pg. 96). Similarly, (3.3) can be proved by the same method. Also, by the definition of F , we have ∆F = div(g), (3.6) where and similarly, sup 0≤s≤t R 3 |∇ω| 2 dx ≤ M ′ , which proves (3.4).
Step 2: The velocity gradient satisfies the following bound proof of Step 2. The proof is identical to Suen and Hoff [14] pg. 51-53, and we omit the details here.
Step 3: We further obtain proof of Step 3. These follow immediately from the momentum equation (1.2) and the ellipticity of the Lamé operator ε∆ + (ε + λ)∇div.
Step 4: The following H 2 -bound for density holds proof of Step 4. We take the spatial gradient of the mass equation (1.1), multiply by ∇ρ and integrate by parts to obtain Applying the above to (3.10) and using the result of Step 2, sup 0≤s≤t ||∇ρ(·, s)|| L 2 ≤ M ′ .
Step 5: The velocity and magnetic field satisfy proof of Step 5. Define the forward difference of quotient D h t by D h t (f )(t) = (f (t + h) − f (t))h −1 and let E j = D h t (u j ) + u · ∇u j . By differentiating the momentum equation, we obtain where O(h) → 0 as h → 0. Therefore by taking h → 0, sup 0≤s≤t ||∇u(·, s)|| L 2 + t 0 R 3 The bound for ∇B t can be derived in an exactly same way.
Step 6: Finally we have the following bounds twice with respect to space, expressing the fourth derivatives of u and B in the terms second derivatives ofu, B t , ∇ρ and lower order terms, and applying the bounds in (2.1) and (3.11) For (3.13), it can be obtained by applying two space derivatives and one spatial difference operator D h xj defined by Taking h → 0 and applying Gronwall's inequality, we obtain the required bound for the term ||D 3 x ρ(·, s)|| L 2 . proof of Theorem 1.2. Using Theorem 3.1, we can apply an open-closed argument on the time interval which is identical to the one given in Suen and Hoff [14] pp. 31 to extend the local solution (ρ −ρ, u, B) beyond T * , which contradicts the maximality of T * . Therefore the assumption (2.1) does not hold and this completes the proof of Theorem 1.2.