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

Authors: Shujuan Wang; Miaoqing Tian; Rijian Su
Journal: Journal of Function Spaces
 (2022)

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## 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 ofuC0,T;L3ℝ3. 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.


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## Body

## 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)ρt+divρu=0,(2)ρut+divρu⊗u+∇P=∇×H×H+μ∆u,(3)Ht−H·∇u+u·∇H=νΔH,(4)Ht−H·∇u+u·∇H=ν∆H,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 takingρ≡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∈Hmℝ3 (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)ρ,u,Hx,0=ρ0,u0,H0xandρ0≤ρ¯forallx∈ℝ3,and far field conditions:
(6)ρ,u,H⟶0,0,0,asx⟶∞,where ρ¯ 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. Fork∈ℤ+ and r>1, the standard homogeneous and inhomogeneous Sobolev spaces for scalar/vector functions are denoted by:
(7)Lr=Lrℝ3,Dk,r=u∈Lloc1ℝ3∇kuLr<∞,Wk,r=Lr∩Dk,r,Hk=Wk,2,Dk=Dk,2,D1=u∈L6∇uL2<∞,uDk,r=∇kuLr..The strong solutions of the problem (1)–(4) are defined as follows.Definition 1.
A pair of functionsρ,u,H is called a strong solution to the problem (1)–(4) in ℝ3×0,T, if for some q0∈3,6,
(8)ρ∈C0,T;L1∩H1∩W1,q0,ρt∈C0,T;L2∩Lq0,ρ≥0,u∈C0,T;D1∩D2∩L20,T;D2,q0,ρut∈L∞0,T;L2,ut,Ht∈L20,T;D1,H∈C0,T;H2,Ht∈L∞0,T;L2,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 someq∈3,6 and the initial data ρ0,u0,H0 satisfying
(9)ρ0≥0,ρ0∈L1∩W1,q,divu0=0,u0∈D1∩D2,divH0=0,H0∈H2,(10)−△u0+∇P0−∇×H0×H0=ρ01/2g,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)ρ≥0,ρ∈C0,T∗;W1,q,ρt∈C0,T∗;Lq,u∈C0,T∗;D1∩D2∩L20,T∗;D2,q,ρut∈L∞0,T∗;L2,ut,Ht∈L20,T∗;D1,H∈C0,T∗;H2∩L20,T∗;W2,q,P∈C0,T∗;H1∩L20,T∗;W1,q.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 Proposition2 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∈C0,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 Proposition2 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)limT⟶T∗uC0,T;L3ℝ3=∞.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 Theorem3, the main two steps are to estimate the Lp-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.
ForU∈C0,T;L3, there exist U1∈C0,T;L3 and U2∈L∞0,T;L∞ such that for any δ>0,
(13)U=U1+U2,U1C0,T;L3≤δ,U2L∞0,T;L∞≤Cδ,M,where M≜uC0,T;L3 and Cδ,M is a positive constant depending only on δ,M.Thus, by choosingδ>0 suitably small, we then succeed in obtaining the estimates on ∇uL2 by utilizing the preliminary estimates of the vorticity ω=∇×u (see Lemma 7) to control the Lp-norm of ∇u in the proof of Lemma 10. With the estimate of ∇uL2 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 thatf∈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)fLp≤CfL26−p/2p∇fL23p−6/2p,gL∞≤CgL21/4∇2gL23/4.To complete some estimates in Section3, we need the following Lp-estimate for vorticity ω=∇×u. In fact, we deduce ω satisfy the following elliptic system by the momentum equation (2)
(15)∆ω=∇×ρut+ρu·∇u−H·∇H,due to ∇×H×H=H·∇H−∇H2/2. By virtue of the standard Lp-estimate of the elliptic system, we have the following.Lemma 7.
Letρ,u,H be a smooth solution of (1) and (3); if 0≤ρ≤ρ~, then there exists a generic positive constant depending only on ρ¯ such that
(16)∇ωL2≤CρutL2+ρu∇uL2+H∇HL2,∇ωL6≤CρutL6+ρu∇uL6+H∇HL6,where ρ~ is a given constant.Proof.
Using Lemma6, one deduces from (15) and the standard Lp-estimate of the elliptic system that
(17)∇ωL2≤CρutL2+ρu·∇uL2+H·∇HL2,≤Cρ1/2utL2+u∇uL2+H∇HL2,∇ωL6≤Cρu.L6+H·∇HL6≤C∇u.L2+H∇HL6,which immediately finish the proof of Lemma 7.


## 3. A Priori Estimates

Letρ,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)uC0,T;L3≤M<∞.Then, we will deduce a contradiction to the maximality ofT∗.Throughout this paper, we will denote byC 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)0≤ρx,t≤supx∈ℝ2ρ0x≤ρ¯forallx∈ℝ3,t∈0,T.Next, we give the standard energy estimate as follows.Lemma 8.
Letρ,u,H be a smooth solution of (1)–(4) on ℝ3×0,T. Then, there exist a constant C such that
(20)sup0≤t≤T∫12ρu2+12H2dx+∫0T∇uL22+∇HL22dt≤C.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)HL∞0,T;Lq≤C,∀q∈2,∞.Proof.
Multiplying (3) by qHq−2H and integrating the resulting equations over ℝ3 lead to
(22)ddt∫Hqdx=q∫ΔH+H·∇u−u·∇H·Hq−2Hdx=∑i=13Ii.
For the terms on the right-hand side of the equation above, we get by integrations by parts(23)I1=q∫ΔH·Hq−2Hdx=−q∫Hq−2∇H2dx−qq−22∫Hq−4∇H22dx,I2=q∫H·∇u·Hq−2Hdx=−∫qHq−2H·∇H·u+qq−22Hq−4H·∇H2u·Hdx,I3=−q∫u·∇H·Hq−2Hdx=−12∫u·∇Hqdx=0.
SubstitutingI1–I3 into (21) and using Young inequality lead to
(24)dd∫Hqdx+∫qHq−2∇H2+qq−22Hq−4∇H22dx=−∫qHq−2H·∇H·u+qq−22Hq−4H·∇H2u·Hdx≤12∫qHq−2∇H2+qq−22Hq−4∇H22dx+C∫u2Hqdx,which immediately implies that
(25)ddt∫Hqdx+∫∇Hq/22dx≤C∫u2Hq/22dx.
Due to the fact thatu∈C0,T;L3, we can decompose u into the following two parts:
(26)u≜U1+U2,with
(27)U1C0,T;L3≤δ,U2L∞0,T;L∞≤Cδ,M0,for M0≜uC0,T;L3 and any δ∈0,1.
From (25)–(27) and using Hölder inequality and imbedding inequality, we have
(28)ddt∫Hqdx+∫∇Hq/22dx≤CU1L32Hq/2L62+U2L∞0,T×ℝ32∫Hqdx≤Cδ∇Hq/2L22+C∫Hqdx.
This, together with takingδ 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)sup0≤t≤T∇uL22+HH12+∫0Tρ1/2utL22+HtL22+∇uH12+HH22dt≤C.Proof.
Multiplying (2) by ut in L2 and integrating the resulting equations by parts, we obtain after summing them up that
(30)12ddt∫∇u2dx+∫ρut2dx=−∫ρu·∇u·utdx−∫H·∇ut·Hdx,where we use the fact divut=0.
In addition, it follows from (3) that
(31)ddt∇HL22+HtL22+ΔHL2=∫Ht−ΔH2dx=∫H·∇u−u·∇H2dx.
Putting (30) and (31) together leads to
(32)ddt12∇u2+∇HL22+ρ1/2utL22+HtL22+ΔHL2=−∫ρu·∇u·utdx−∫H·∇ut·Hdx+∫H·∇u−u·∇H2dx=∑i=13Ji.
We estimate the three terms on the right-hand side of (32) term by term. Following from Young inequality, one has that
(33)J1≤14ρ1/2utL22+Cu∇uL22,J2=∫H·∇ut·Hdx=ddt∫H·∇u·Hdx−∫Ht·∇u·Hdx−∫H·∇u·Htdx≤ddt∫H·∇u·Hdx+14HtL22+CH∇uL22,J3=∫H·∇u−u·∇H2dx≤Cu∇HL22+H∇uL22.
PuttingJ1−J3 into (31), we obtain
(34)ddt12∇u2+∇HL22+12ρ1/2utL22+HtL22+ΔHL2≤−ddt∫H·∇u·Hdx+C1u∇HL22+u∇uL22+H∇uL22,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)u∇HL22≤∫U12∇H2dx+∫U22∇H2dx≤U1L32∇HL62+U2L∞2∇HL22≤δ2∇HL62+Cδ,M∇HL22,u∇uL22≤∫U12∇u2dx+∫U22∇u2dx≤U1L32∇uL62+U2L∞2∇uL22≤δ2∇uL62+Cδ,M∇uL22,H∇uL22≤CHL62∇uL32≤C∇uL2∇uL6≤δ2∇uL62+Cδ∇uL22.
Thus,(36)u∇HL22+u∇uL22+H∇uL22≤δ2∇uL62+∇HL62+Cδ∇uL22+∇HL22.
Next, we turn to estimate∇uL6 and ∇HL6. Note that, from (3), we obtain an elliptic system as follows:
(37)−ΔH=−Ht−u·∇H+H·∇u.
Applying standardLp-estimate to elliptic systems (15) and (37), we obtain that
(38)∇uL6≤CdivuL6+∇×uL6≤CωL6≤C∇ωL2≤Cρ1/2utL2+u∇uL2+H∇HL2≤Cρ1/2utL2+u∇uL2+HL3∇HL6≤Cρ1/2utL2+u∇uL2+∇HL6,∇HL6≤∇2HL2≤CHtL2+u∇HL2+H∇uL2,which imply that
(39)∇uL6+∇HL6≤C2ρ1/2utL2+HtL2+u∇uL2+u∇HL2+H∇uL2
Putting (39) into (36), we have by choosing δ>0 sufficiently small that
(40)u∇HL22+u∇uL22+H∇uL22≤14C1ρ1/2utL2+HtL2+Cδ∇uL22+∇HL22.where the C1 is given in (34).
Substituting (40) into (34) leads to
(41)ddt12∇u2+∇HL22+14ρ1/2utL22+HtL22+ΔHL2≤−ddt∫u·H·∇Hdx+C∇uL22+∇HL22.
By (21) and the Young inequality, we easily see that
(42)∫H·∇u·Hdx≤14∇uL22+C.
Taking this into account, we then conclude from (18)–(20), (41), and Gronwall’s inequality for any 0≤T<T∗ that holds
(43)sup0≤t≤T∇uL22+HH12+∫0Tρ1/2utL22+HtL22dt≤C.
Applying the standard ellipticL2-estimates to (37) leads to
(44)HH2≤HtL2+u·∇HL2+H·∇uL2≤HtL2+CuL6∇HL3+CHL6∇uL3≤HtL2+C∇uL2∇HL21/2∇2HL21/2+C∇uL21/2∇2uL21/2≤HtL2+14∇2uL2+∇2HL2+C,where we use (21), (43), and Gagliardo-Nirenberg inequality.
On the other hand, sinceu,P is a solution of the stationary Stokes equations
(45)−Δu+∇P=Fanddivu=0inℝ3,where F=−ρut−ρu·∇u−1/2∇H2+H·∇H. It follows from the classical regularity theory that
(46)∇uH1≤CρutL2+ρu·∇uL2+H·∇HL2≤Cρ1/2utL2+CuL6∇uL3+CHL6∇HL3≤Cρ1/2utL2+14∇uH1+14HH2+C.
Adding (44) to (46), we obtain
(47)∫0T∇uH12+HH22ds≤C∫0Tρ1/2utL22+HtL22+1ds≤C.
This, together with (43), immediately implies (27). This lemma is completed.The following lemma is concerned with theL2-estimate of ρ1/2ut and Ht.Lemma 11.
Under the assumption (22), it holds for any t∈0,T such that
(48)sup0≤t≤Tρ1/2utL22+HtL22+∫0T∇utL22+∇HtL22dt≤C.Proof.
Differentiating the momentum equations (2) with respect to t yields
(49)ρutt+ρu·∇ut+ρut·∇u+ρtut+u·∇u+∇Pt=Δut+H·∇H−12∇H2t.
Multiplying the equation above withut and integrating by parts, one gets
(50)12ddtρ1/2utL22+∇utL22=−∫ρtut2dx−∫ρut·∇u·utdx−∫ρtu·∇u·utdx+∫Ht·∇H+H·∇Ht·utdx,due to divut=0.
Differentiating (3) with respect to t and multiplying the resulting equation by Ht, we obtain after integrating by parts that
(51)12ddtHtL22+∇HtL22=−∫ut·∇H·Htdx+∫Ht·∇u·Ht+H·∇ut·Htdx,where we have used divHt=0 and divu=0.
Putting (50) and (51) together leads to
(52)12ddtHtL22+ρ1/2utL22+∇utL22+∇HtL22=−∫ρtut2dx−∫ρut·∇u·utdx−∫ρtu·∇u·utdx+∫Ht·∇H·ut−ut·∇H·Htdx+∫Ht·∇u·Htdx+∫H·∇ut·Ht+H·∇Ht·utdx=∑i=16Ri.
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)R1=∫ρu·∇ut2dx≤C∇utL2uL6ρutL3≤C∇utL2ρutL21/2ρutL61/2≤C∇utL23/2ρutL21/2≤ε∇utL22+Cερ1/2utL22.
Similarly, the estimate ofR2 is given as follows
(54)R2≤C∫ρut2∇udx≤CρutL2utL6∇uL3≤Cρ1/2utL2∇utL2∇uL21/2∇2uL21/2≤ε∇utL22+Cερ1/2utL24+Cε∇2uL22.
ForR3, we have that
(55)R3=∫ρu·∇u·∇u·utdx≤C∫u∇u2ut+u2∇2uut+u2∇u∇utdx≜∑i=13Ji.
From Lemma8 to 9, we can reduce that
(56)J1≤C∫u∇u2utdx≤C∇uL32uL6utL6≤C∇uL22∇2uL2∇utL2≤ε∇utL22+Cε∇2uL22,J2≤C∇2uL2uL62utL6≤C∇2uL2∇uL22∇utL2≤ε∇utL22+Cε∇2uL22,J3≤C∇uL6uL62∇utL2≤C∇2uL2∇uL22∇utL2≤ε∇utL22+Cε∇2uL22.
From theJ1–J3, we get estimate of R3(57)R3≤C∇2uL22+3ε∇utL22.
Similarly, we have(58)R4≤C∫Ht∇uutdx≤C∇uL3HtL2utL6≤C∇uL21/2∇2uL21/2HtL2∇utL2≤ε∇utL22+Cε∇2uL2HtL22≤ε∇utL22+CεHtL24+Cε∇2uL22,R5≤C∫Ht2∇udx≤CHtL42∇uL2≤CHtL21/2∇HtL23/2≤ε∇HtL22+CεHtL22.
It is easy to prove thatR6=0. Thus, taking ε suitable small and substituting the estimates of R1–R6 into (52) lead to
(59)ddtHtL22+ρ1/2utL22+∇utL22+∇HtL22≤C∇2uL22+C1+HtL22HtL22+C1+ρ1/2utL22ρ1/2utL22,which, together with Gronwall’s inequality, immediately leads to the desired estimate (50) since (27) implies ∇2u∈L20,T, 1+HtL22∈L10,T, and 1+ρ1/2utL22∈L10,T.Lemmas12 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)sup0≤t≤T∇uH1+HH2+∫0TuW2,62+HW2,62dx≤CT.Proof.
Applying classical regularity theory to (45) again, we have
(61)sup0≤t≤T∇uW1,6≤CρutL6+uL∞∇uL6+HL∞∇HL6+∇uL6+1≤C∇utL2+∇uH12+HH22.
Integrating the inequality above over0,t and by (27) lead to
(62)∫0tuW2,62ds≤C∫0t∇utL2+∇uH12+HH22ds≤C.
Similar proof leads to the same conclusion forH(63)∫0tHW2,62ds≤C.
By virtue of (44), (46), and (48), it is easily to prove that
(64)sup0≤t≤T∇uH1+HH2≤C.
Thus, Lemma11 is proved.Finally, the following lemma gives bounds of the first spatial derivatives of the densityρ.Lemma 13.
Under the assumption (18), it holds for any t∈0,T such that
(65)sup0≤t≤TρW1,q≤CT,forq∈2,6.Proof.
Differentiating (1) with respect to xii=1,2,3, multiplying it by ∂iρp−2∂iρ with p≥2, and integrating the resulting equation by parts, we obtain after summing over i from 1 to 3 that
(66)ddt∇ρLpp≤C∫∇u∇ρpdx≤C∇uW1,6∇ρLpp,which, together with Gronwall’s inequality, leads to
(67)∇ρLpp≤C∇ρ0LppexpC∫0TuW2,6dt≤C.
This finishes the proof of Lemma13.With all the a priori estimates in Section3 at hand, we are ready to prove the main result of this paper.Basing on Lemmas8–13 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.



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