MASS CONCENTRATION PHENOMENON TO THE TWO-DIMENSIONAL CAUCHY PROBLEM OF THE COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS

. This concerns the global strong solutions to the Cauchy problem of the compressible Magnetohydrodynamic (MHD) equations in two spatial dimensions with vacuum as far ﬁeld density. We establish a blow-up criterion in terms of the integrability of the density for strong solutions to the compressible MHD equations. Furthermore, our results indicate that if the strong solutions of the two-dimensional (2D) viscous compressible MHD equations blowup, then the mass of the MHD equations will concentrate on some points in ﬁnite time, and it is independent of the velocity and magnetic ﬁeld. In particular, this extends the corresponding Du’s et al. results (Nonlinearity, 28, 2959-2976, 2015, [4]) to bounded domain in R 2 when the initial density and the initial magnetic ﬁeld are decay not too show at inﬁnity, and Ji’s et al. results (Discrete Contin. Dyn. Syst., 39, 1117-1133, 2019, [10]) to the 2D Cauchy problem of the compressible Navier-Stokes equations without magnetic ﬁeld.


YONGFU WANG
System (1.1) will be investigated with initial conditions ρ(x, 0) = ρ 0 (x), ρu(x, 0) = ρ 0 u 0 (x), H(x, 0) = H 0 (x), x ∈ R 2 , (1.4) for given the initial data ρ 0 , u 0 and H 0 , and the far-field conditions (ρ, u, H)(x, t) → (0, 0, 0) as |x| → ∞. (1.5) There is a considerable body of literature on the global regularity and large time behavior of the multi-dimensional compressible MHD flow. If there is no electromagnetic effect, namely, H ≡ 0, the MHD systems becomes the Navier-Stokes equations, there are huge literatures on the global regularity results, please see [1,3,8,9,11,12,13,18,22,23,24,25,26,28,29] and references therein. There are some important progresses on mathematical analysis on MHD flows in recent decades and we would like to recall some results on compressible MHD flows in multi-dimension briefly. Kawashima [14] obtained the global existence of smooth solutions to the 2D compressible MHD system, provided that the initial density was strictly positive. Hu and Wang [6] showed the global existence and the large time behavior of the renormalized solutions to the 3D compressible MHD system for general large initial data. The local strong solutions to the compressible MHD with large initial data were obtained by Vol'pert-Hudjaev [26] as the initial density is strictly positive and by Fan-Yu [5] as the initial density may contain vacuum in R 3 , respectively. Recently, Li, Xu and Zhang [15] established the global existence of the classical solution to three-dimensional compressible MHD flows with small total energy. Starting with the pioneering works by Beale-Kato-Majda [1] and Serrin [23], many articles have been study the blowup phenomenon to MHD equations in R 3 or bounded domain of R 2 (see [4,5,7,27,30] and the references therein). However, many fundamental and interesting problems are still open even for one-dimension case due to the lack of smoothing mechanism and the strong nonlinearity.
On anther side, Lü and Huang [19] have established the local strong solutions to Cauchy problem (1.1)-(1.5). Lü, Shi and Xu [20] established the global wellposedness to 2D MHD equations, as long as the initial total energy is small. Later, Lü, Xu and Zhong [21] obtained the global existence and uniqueness of strong solutions to the 2D Cauchy problem of nonhomogeneous incompressible magnetohydrodynamic equations provided that the initial density and the initial magnetic field decay not too slow at infinity. Furthermore, there are some progress to the Cauchy problem of 2D compressible Navier-Stokes equations, i.e., Li, Liang and Xin [16,17]. In particular, the regularity and uniqueness of the weak solution to the compressible MHD flows for general initial data are still open and challenge problems even in two dimensions case. Therefore, it is important to study the blowup mechanism and structure of possible singularities of strong (or classical) solutions to the compressible MHD equations in R 2 .
There are a series of blowup criteria for the 2D compressible MHD equations, especially, Du-Wang [4] gave a blowup criterion for strong solutions in a bounded domain of R 2 . More precisely, they proved that if T * is the life span of the strong solution to system (1.1), then (1.6) The goal of this paper is to extend the blowup criterion (1.6) in bounded domain of the 2D compressible MHD equations to Cauchy problem.
Before stating the main result, first, we introduce the following simplified notations Without loss of generality, we assume that the initial density ρ 0 satisfies The local strong solution of Cauchy problem to the 2D MHD equations with vacuum was obtained in [19]. We state those results as follows.
Proposition 1. For given positive constants 0 < η 0 ≤ 1, q > 2 and a > 1, we definē 8) and the initial data satisfy Then there exists a positive constant T 0 such that the Cauchy problem (1.1)-(1.5) has a unique strong solution (ρ, u, H) in R 2 × (0, T 0 ] satisfying that Our main result of this paper can be stated as follows. Theorem 1.1. Suppose that the initial data (ρ 0 , u 0 , H 0 ) satisfy (1.9), and (ρ, u, H) be a strong solution to the Cauchy problem (1.1)-(1.5) satisfying (1.10)-(1.11) in R 2 × (0, T * ). If T * < ∞ is the maximal existence time of the strong solution, then there exists some positive constant s 0 such that which implies that if the velocity field is regular, the characteristic line and the density can be defined as follows: dy ds (s; x, t) = u(y(s; x, t), s), which implies that if the singularity of the solution to the compressible MHD equations (1.1) formulates in finite time T * , we distinguish three possible cases in the following.
(1) The density may concentrate, that is to say, (1.13) (2) Vacuum states may appear in the non-vacuum region: there exist somex ∈ R 2 andx(t) satisfying ρ 0 (x) > 0 and y(0;x(t), t) =x, such that lim t→T * ρ(x(t), t) = 0. (1.14) (3) Vacuum states may vanish: there exist some x 0 ∈ R 2 and x 0 (t) satisfying A natural and interesting question may be asked: Which one or some of (1.13)-(1.15) will happen when the singularity formulates? Theorem 1.1 gives an answer to this question and show that the mass of the fluid will concentrate on some points before other cases (2) and (3) happen.
Remark 2. The approach can also be applied to deal with the bounded domain in R 2 . Roughly speaking, we generalize the results in [4] to the Cauchy problem of 2D compressible MHD equations. This criterion is analogous to the 2D barotropic compressible Navier-Stokes equations without magnetic field, in particular, it is independent of the magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations [10]. On the other hand, it would be interesting to study wether (1.12) is a necessary condition.
Remark 4. Indeed, the positive constant s 0 can be chosen in Section 3, which depends on λ, µ and γ.
We now comment on the analysis of this paper. Compared with [4,10] for 2D compressible barotropic MHD equations in bounded domain and 2D Cauchy problem of compressible barotropic Navier-Stokes equations, some new difficulties arise in the Cauchy problem of compressible MHD system. The first difficult lies in the fact that the Brezis-Waigner's inequality [2] fails for the 2D Cauchy problem, and it seems difficult to estimate u L q (R 2 ) for any q > 1 just in terms of √ ρu L 2 (R 2 ) and ∇u L 2 (R 2 ) . One way to overcome this difficulty is to estimate the momentum ρu instead of the velocity u, since ρ decays for large x, the momentum ρu decays faster than u itself. Moreover, we use the variant of Gagliardo-Nirenberg inequality and more finely estimate for the nonlinear coupling term ρu · ∇u. In particular, the high order estimates on ρ, u and H will not be improved as in the bounded domain case in [4]. Therefore, the Hardy-type inequality is introduced to control the L q -norm of ρu. On the other hand, compared with the 2D compressible Navier-Stokes system [10], there are some new nonlinear coupling terms, such as u · ∇H, H · ∇H and H · ∇u. Indeed, in order to control the |u||∇H| L 2 , our new observation is to obtain some weighted estimates on both H and ∇H, namely, Hx a 2 L 2 (R 2 ) and ∇Hx a 2 L 2 (R 2 ) . Furthermore, the initial density vacuum is allowed in this paper.
The rest of this paper is organized as follows. Some important inequalities and auxiliary lemmas will be given in Section 2. We will prove Theorem 1.1 in Section 3.

2.
Preliminaries. In this section, some elementary lemmas will be used later. One is the variant of Gagliardo-Nirenberg inequality. For its proof, refer to [22].
Next, the material derivativeḟ , the effective viscous flux G, and the vorticity ω are defined as follows.
We obtain two key elliptic system of G and ω.
where we have used (1.1) 2 and (2.2). From the standard L p -estimate of elliptic system (2.3), we have the following estimate.
Lemma 2.2. Let (G, ω) be a strong solution of (2.3). Then there exists a generic positive constant C depending only on µ, λ and p, such that We introduce the Hardy-type inequality as follows, which plays a crucial role in the estimate. The proof can be found in [16]. Lemma 2.3. Letx and η 0 be as in (1.8) and The main ingredient of the proof is that we decompose the velocity field into two parts, namely u = v + w, where v is the solution to Lamé system and w satisfies the following elliptic boundary value problem , we obtain the following estimates.
Lemma 2.4. Let v and w be a solution of (2.7) and (2.8), respectively. Then there exists a generic positive constant C depending only on µ, λ, p such that
Proof. We can prove (2.9) in a similar way as in the proof of Lemma 2.3 in [7]. The standard L p -estimate of elliptic system (2.8) yield to (2.10).
Remark 5. Indeed, taking p 0 ∈ (1, 2), using Hölder's inequality yield to Finally, we introduce the following Beale-Kato-Majda type inequality to estimate the term ∇u L ∞ , which was first proved in [1] when divu = 0. The proof for a general situation can be found in [13].
Lemma 2.5. Suppose that ∇u ∈ L 2 (R 2 )∩W 1,q (R 2 ) for any q ∈ (2, ∞), there exists a constant C depending only on q, such that 3. Proof of The main results. Let (ρ, u, H) be a strong solution of (1.1)-(1.3) on R 2 × [0, T * ). We will prove our main Theorem 1.1 by contradiction arguments. Suppose not, for some sufficiently large 1 < s 0 < ∞, there exists a positive constant M such that lim The combination of (1.7) and the mass conservation equation (1.1) 1 yields First, we obtain the following standard energy estimate for (ρ, u, H).
There exists a positive constant N 1 which holds that for 0 ≤ T < T * , Here and after, C and C i (i = 0, 1, 2, ...) denote the generic positive constants depending on M , µ, λ, ν, T * , N 1 and the initial data.
Proof. Indeed, the proof of (3.3) is standard. Multiplying the momentum equation by u, the magnetic field equation by H and using the (1.1) 1 , we can prove the estimate of (3.3).
Multiplying (1.1) 1 by η N1 , using Hölder's inequality, (3.3) and integrating by parts give that for N 1 suitably large. Therefore, we complete the proof of Lemma 3.1.
Next, there is an important spatial weighted estimates about ρ and H.
Next, multiplying (1.1) 3 by Hx a and integrating by parts gives 1 2 (3.10) Furthermore, due to (2.1), (2.6) and Hölder inequality, we estimate I 1 -I 4 as follows. and (3.14) Finally, multiplying (1.1) 3 by |H| 2 H and integrating by parts lead to d dt where we have used the following fact Furthermore, a direct computation gives that Substituting this into (3.15) yields that d dt Next, there is an important estimate about ρu, the similar arguments of the following estimate come from [25]. for any 0 ≤ T < T * .
With the help of Lemma 2.2 and Lemma 2.4, we can prove the following key estimate of ∇u and ∇H.
Consider the integral term where u is separated into v and w, which satisfy (2.7) and (2.8), respectively. First, for the first term, thanks to the equation in (2.7), we have

YONGFU WANG
For the second term, we obtain where have used the following fact P t + div(P u) + (γ − 1)P divu = 0.
Next, we can improve the regularity estimates on u and H.
Proof. Apply the operatoru j [∂ t + div(u·)] to (1.1) 2 ( j = 1, 2 ) and using the first equation of (1.1), integration by parts over R 2 , one has 1 2 First, integrating by parts, we have that due to (3.1), (3.13) and Hölder's inequality. Furthermore, for the second term N 2 , integrating by parts leads to due to Young's inequality. Similarly, we obtain Moreover, integration by parts and together with (2.1), (2.6), (3.7) and (3.13) yield that Similarly, we have Next, differentiating the third equation in (1.1) with respect to t, and multiplying the resulting equation by H t in L 2 , after integration by parts, using (2.1), (2.6), (3.7) and (3.19) yield that ≤C ∇w where p 1 ∈ ( 4 3 , 2) and s 4 = p1 2−p1 . Multiply (3.40) by 2C0 µ and adding it to (3.41), choosing η suitable small, and using (3.42), we obtain d dt which together with (3.19) give By Gronwall's inequality and multiplying (3.43) by t and integrating the resulting inequality over (0, T ) lead to Using the above estimate and (2.5), we obtain Therefore, the proof of Lemma 3.5 is finished.
Let p → ∞, we finish the proof of Lemma 3.6.
Suppose not, T * < ∞, in view of Lemma 3.3-Lemma 3.8, (ρ, u, H)(x, T * ) = lim t→T * (ρ, u, H)(x, t) satisfy the conditions imposed on the initial data at the time t = T * . Thus, with the help of Lemma 3.1-Lemma 3.8 and local existence in Proposition 1, we can extend the local strong solution of (ρ, u, H) beyond t > T * , which contradicts the maximality of T * . The proof of Theorem 1.1 is completed.