A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity
aa r X i v : . [ m a t h . A P ] A ug A REGULARITY CRITERION FOR THE 3D FULLCOMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONSWITH ZERO HEAT CONDUCTIVITY
JISHAN FAN, FUCAI LI, AND GEN NAKAMURA
Abstract.
We establish a regularity criterion for the 3D full compressiblemagnetohydrodynamic equations with zero heat conductivity and vacuum ina bounded domain. Introduction
In this paper, we consider the 3D full compressible magnetohydrodynamic equa-tions in a bounded domain Ω ⊂ R : ∂ t ρ + div ( ρu ) = 0 (1.1) ∂ t ( ρu ) + div ( ρu ⊗ u ) + ∇ p − µ ∆ u − ( λ + µ ) ∇ div u = rot b × b, (1.2) C V [ ∂ t ( ρθ ) + div ( ρuθ )] − κ ∆ θ + p div u = µ |∇ u + ∇ u t | + λ (div u ) + ν | rot b | , (1.3) ∂ t b + rot ( b × u ) = ν ∆ b, div b = 0 , (1.4)with the initial and boundary conditions u = 0 , κ ∂θ∂n = 0 , b · n = 0 , rot b × n = 0 on ∂ Ω × (0 , ∞ ) , (1.5)( ρ, u, θ, b )( · ,
0) = ( ρ , u , θ , b ) in Ω ⊂ R . (1.6)Here the unknowns ρ, u, p, θ , and b stand for the density, velocity, pressure, tem-perature, and magnetic field, respectively. The physical constants µ and λ are theshear viscosity and bulk viscosity of the fluid and satisfy µ > λ + µ ≥ C V > κ > ν > ∇ u t denotes the transpose of the matrix ∇ u .We assume that Ω is a bounded and simply connected domain in R with smoothboundary ∂ Ω. We use n to denote the outward unit normal vector to ∂ Ω.The full compressible magnetohydrodynamic equations (1.1)-(1.4) can be rigor-ous derivation from the compressible Navier-Stokes-Maxwell system [14]. Due to
Date : September 1, 2018.2010
Mathematics Subject Classification.
Key words and phrases. compressible magnetohydrodynamic equations, zero heat conductivity,regularity criterion. the physical importance of the magnetohydrodynamics, there are a lot of litera-ture on the system (1.1)-(1.4), among others, we mention [8] on the local strongsolutions, [4, 9, 10] on the global weak solutions, [15, 16] on low Mach number limit,and [19] on the time decay of smooth small solutions.Assume that the pressure take the form p = Rρθ with R being the generic gasconstant.In [11], Huang and Li proved the following regularity criterion ρ ∈ L ∞ (0 , T ; L ∞ ) and u ∈ L s (0 , T ; L r ) with 2 s + 3 r = 1 and 3 < r ≤ ∞ , (1.7)with b satisfying the homogeneous Dirichlet boundary condition b = 0 on ∂ Ω × (0 , ∞ ). Later this result was generalized in [7] to the case of the boundary condition(1.5), i.e., b · n = 0 , rot b × n = 0 on ∂ Ω × (0 , ∞ ) . (1.8)When considering the system (1.1)-(1.4) in a two dimensional domain, Lu, Chenand Huang [18] showed the following regularity criteriondiv u ∈ L (0 , T ; L ∞ ) (1.9)with b satisfying the boundary condition b = 0 on ∂ Ω × (0 , ∞ ). Here we remark thatsame result can be proved for b satisfying the boundary condition: b · n = 0 , rot b = 0on ∂ Ω × (0 , ∞ ). An related weak result was obtained in [6].Very recently, Huang and Wang [12] establish the following regularity criterion ρ, θ, b ∈ L ∞ (0 , T ; L ∞ ) with 2 µ > λ. (1.10)for the system (1.1)-(1.4) in the whole space R with κ = ν = 0.The aim of this paper is to show that the regularity criterion (1.10) still holdfor the system (1.1)-(1.4) in a bounded domain with the boundary condition (1.5)when κ = 0 and ν = 1. We will prove Theorem 1.1.
Let κ = 0 and ν = 1 . For q ∈ (3 , , assume that the initial data ( ρ ≥ , u , p = Rρ θ ≥ , b ) satisfy ρ , p ∈ W ,q (Ω) , u ∈ H (Ω) ∩ H (Ω) , b ∈ H with div b = 0 in Ω ,b · n = 0 , rot b × n = 0 on ∂ Ω (1.11) and the compatibility condition − µ ∆ u − ( λ + µ ) ∇ div u + ∇ p − rot b × b = √ ρ g, (1.12) with g ∈ L (Ω) . Let ( ρ, u, p, b ) be a local strong solution to the problem (1.1) - (1.6) .If (1.10) holds true with < T < ∞ , then the solution ( ρ, u, p, b ) can be extendedbeyond T > . EGULARITY CRITERION FOR FULL MHD EQUATIONS 3
We mention that when taking b = 0 in the system (1.1)-(1.4), it is reduced tothe full compressible Navier-Stokes system and a lot of regularity criteria can befound in [5, 20, 23] and the references cited therein.The remainder of this paper is devoted to the proof of Theorem 1.1. We givesome preliminaries in section 2 and present the proof of Theorem 1.1 in section 3.Below we shall use the letter C to denote the positive constant which may changefrom line to line. 2. Preliminaries
First, we consider the boundary value problem for the Lam´e operator L LU , µ ∆ U + ( µ + λ ) ∇ div U = F in Ω ,U ( x ) = 0 on ∂ Ω . (2.1)Here U = ( U , U , U ) , F = ( F , F , F ). It is well known that the system (2.1) isa strongly elliptic system, thus there exists a unique weak solution U ∈ H (Ω) for F ∈ W − , (Ω). Lemma 2.1.
Let q ∈ (1 , ∞ ) and U be a solution of (2.1) . There exists a constant C depending only on λ, µ, q and Ω such that the following estimates hold:(1) if F ∈ L q (Ω) , then k U k W ,q (Ω) ≤ C k F k L q (Ω) ; (2.2) (2) if F ∈ W − ,q (Ω) (that is, F = div f with f = ( f ij ) × , f ij ∈ L q (Ω)) , then k U k W ,q (Ω) ≤ C k f k L q (Ω) ; (2.3) (3) if F = div f with f ij = ∂ k h kij and h kij ∈ W ,q (Ω) for i, j, k = 1 , , , then k U k L q (Ω) ≤ C k h k L q (Ω) . (2.4) Proof.
The estimates (2.2) and (2.3) are classical for strongly elliptic systems, seefor example [2]. The estimate (2.4) can be proved by a duality argument with thehelp of (2.2). (cid:3)
We need an endpoint estimate for L in the case q = ∞ . Let BM O (Ω) stand forthe John-Nirenberg space of bounded mean oscillation whose norm is defined by k f k BMO (Ω) := k f k L (Ω) + [ f ] BMO , with [ f ] BMO (Ω) := sup x ∈ Ω ,r ∈ (0 ,d ) | Ω r ( x ) | Z Ω r ( x ) | f ( y ) − f Ω r ( x ) | dy,f Ω r ( x ) := 1 | Ω r ( x ) | Z Ω r ( x ) f ( y ) dy. Here Ω r ( x ) := B r ( x ) ∩ Ω , B r ( x ) is a ball with center x and radius r, d is the diameterof Ω and | Ω r ( x ) | denotes the Lebesque measure of Ω r ( x ). J.-S. FAN, F.-C. LI, AND G. NAKAMURA
Lemma 2.2 ( [1])) . If F = div f with f = ( f ij ) × , f ij ∈ L ∞ (Ω) ∩ L (Ω) , then ∇ U ∈ BM O (Ω) and there exists a constant C depending only on λ, µ and Ω suchthat k∇ U k BMO (Ω) ≤ C ( k f k L ∞ (Ω) + k f k L (Ω) ) . (2.5)Let us conclude this section by recalling a variant of the Brezis-Waigner inequal-ity [3]. Lemma 2.3 ( [21]) . Let Ω be a bounded Lipschitz domain in R and f ∈ W ,q with < q < ∞ . There exists a constant C depending on q and the Lipschitz propertyof Ω such that k f k L ∞ (Ω) ≤ C (1 + k f k BMO (Ω) ln( e + k∇ f k L q (Ω) )) . (2.6) Lemma 2.4 ( [13]) . Let b be a solution to the Poisson equation − ∆ b = f in Ω with the boundary condition b · n = 0 , rot b × n = 0 on ∂ Ω . Then there holds k∇ b k L p ≤ C k f k L p + C k∇ b k L with < p < ∞ . (2.7)In the following proofs, we will use the Poincar´e inequality [17]: k b k L ≤ C ( k div b k L + k rot b k L ) (2.8)for any b ∈ H (Ω) with b · n = 0 or b × n = 0 on ∂ Ω.We will also use the inequality [22]: k∇ b k L ≤ C ( k div b k L + k rot b k L ) (2.9)for any b ∈ H (Ω) with b · n = 0 or b × n = 0 on ∂ Ω.3.
Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1, we only need to show apriori estimates. For simplicity, we will take ν = C V = R = 1.Testing (1.2) by u , (1.4) by b , summing up the results and using (1.1) and (1.10),we see that12 ddt Z ( ρ | u | + | b | ) dx + Z ( µ |∇ u | + ( λ + µ )(div u ) + | rot b | ) dx = − Z u ∇ pdx = Z p div udx ≤ λ + µ Z (div u ) dx + C, which gives Z ( ρ | u | + | b | ) dx + Z T Z ( |∇ u | + | rot b | ) dxdt ≤ C. (3.1) EGULARITY CRITERION FOR FULL MHD EQUATIONS 5
Integrating (1.3) over Ω × (0 , t ) and using (1.10) and (3.1), we find that Z ρθdx ≤ C. (3.2)By the same calculations as that in [12], we get Z ρ | u | dx + Z T Z |∇ u | | u | dxdt ≤ C. (3.3)We define v ∈ H satisfying Lv := µ ∆ v + ( λ + µ ) ∇ div v = ∇ p, (3.4)and w := u − v . Thanks to Lemma 2.1, for any 1 < r < ∞ , there hold k∇ v k L r (Ω) ≤ C k p k L r (Ω) , k∇ v k L r (Ω) ≤ C k∇ p k L r (Ω) . (3.5)It is easy to see that w satisfies µ ∆ w + ( λ + µ ) ∇ div w = ρ ˙ u − rot b × b, (3.6)Then it follows from Lemma 2.1 that k∇ w k L (Ω) ≤ C k ρ ˙ u k L (Ω) + C k rot b × b k L (Ω) . (3.7)Let E be the specific energy defined by E := θ + | u | . Then ∂ t (cid:18) ρE + | b | (cid:19) + div ( ρEu + pu + | b | u )= 12 µ ∆ | u | + µ div ( u · ∇ u ) + λ div ( u div u ) + div (( u · b ) b ) − div (rot b × b ) . (3.8)Testing (1.2) by u t and using (1.1) and denoting ˙ f := f t + u · ∇ f , we deducethat 12 ddt Z ( µ |∇ u | + ( λ + µ )(div u ) ) dx + Z ρ | ˙ u | dx = Z ρ ˙ u · ( u · ∇ ) udx + Z (cid:20)(cid:18) p + 12 | b | (cid:19) I − ( b ⊗ b ) (cid:21) : ∇ u t dx ≤ Z ρ | ˙ u | dx + C Z | u | |∇ u | dx + ddt Z (cid:20)(cid:18) p + 12 | b | (cid:19) I − ( b ⊗ b ) (cid:21) : ∇ udx − Z p t div udx − Z (cid:20) | b | I − ( b ⊗ b ) (cid:21) t : ∇ udx. (3.9)We remark that − Z p t div vdx = Z v ∇ p t dx = Z v ( µ ∆ v t + ( λ + µ ) ∇ div v t ) dx = − ddt Z ( µ |∇ v | + ( λ + µ )(div v ) ) dx. (3.10) J.-S. FAN, F.-C. LI, AND G. NAKAMURA
And according to (3.8) and (1.1), − Z p t div wdx = − Z (cid:18) ρE + 12 | b | (cid:19) t div wdx + Z (cid:18) ρ | u | (cid:19) t div wdx + Z ( b · b t )div wdx = − Z (cid:18) ρEu + pu + | b | u − µ ∇| u | − µ ( u · ∇ ) u − λu div u − ( u · b ) b + rot b × b (cid:19) ∇ div wdx − Z div ( ρu ) | u | div wdx + Z ρu t u div wdx + Z bb t div wdx = − Z (cid:18) ρθu + | b | u − µ ∇| u | − µ ( u · ∇ ) u − λu div u − ( u · b ) b + rot b × b (cid:19) ∇ div wdx + Z ρ ˙ uu div wdx + Z bb t div wdx ≤ C Z ρ | u | dx + C k u k L + C Z | u | |∇ u | dx + C k∇ b k L + δ k∇ w k L + δ k√ ρ ˙ u k L + C k√ ρu k L k∇ w k L + C k∇ w k L + δ k b t k L ≤ C + C k∇ u k L + C Z | u | |∇ u | dx + C k∇ b k L + Cδ k√ ρ ˙ u k L + δ k√ ρ ˙ u k L + C k∇ w k L k∇ w k L + C k∇ w k L + δ k b t k L ≤ C + C k∇ u k L + C Z | u | |∇ u | dx + C k∇ b k L + Cδ k√ ρ ˙ u k L + δ k√ ρ ˙ u k L + δ k b t k L (3.11)for any small 0 < δ , δ and δ . Here we have used the Gagliardo-Nirenberg in-equality k∇ w k L ≤ C k∇ w k L k∇ w k L + C k∇ w k L and k∇ w k L ≤ k∇ u k L + k∇ v k L ≤ C + k∇ u k L . Observing that the last term of (3.9) can be bounded as − Z (cid:20) | b | I − ( b ⊗ b ) (cid:21) t : ∇ udx ≤ δ k b t k L + C k∇ u k L . (3.12)On the other hand, testing (1.4) by b t − ∆ b , we get ddt Z | rot b | dx + Z ( | b t | + | ∆ b | ) dx = Z | rot ( b × u ) | dx ≤ C Z |∇ u | dx + C k u k L k∇ b k L ≤ C k∇ u k L + C k u k L k∇ b k L + 12 k ∆ b k L + C k u k L k∇ b k L . (3.13)Here we have used the inequality k∇ b k L ≤ C k ∆ b k L + C k∇ b k L (3.14) EGULARITY CRITERION FOR FULL MHD EQUATIONS 7 and the Gagliardo-Nirenberg inequality k∇ b k L ≤ C k∇ b k L k∇ b k L + C k∇ b k L . (3.15)Inserting (3.10), (3.11) and (3.12) into (3.9) and combining (3.13) and choosing δ , δ and δ suitably small and using the Gronwall inequality, we havesup ≤ t ≤ T Z ( |∇ u | + |∇ b | ) dx + Z T Z ( |√ ρu t | + | b t | + |∇ b | ) dxdt ≤ C. (3.16)Now we are in a position to give a high order regularity estimates of the solutions.The calculations were motivated by [20]. First of all, we rewrite the equation (1.2)as ρ ˙ u + ∇ p − Lu = g := rot b × b to find that ρ ˙ u t + ρu · ∇ ˙ u + ∇ p t + div ( ∇ p ⊗ u )= µ [∆ u t + div (∆ u ⊗ u )] + ( λ + µ )[ ∇ div u t + div ( ∇ div u ⊗ u )] + g t + div ( g ⊗ u ) . Testing the above equation by ˙ u and using (1.1), we have12 ddt Z ρ | ˙ u | dx − µ Z ˙ u [∆ u t + div (∆ u ⊗ u )] dx − ( λ + µ ) Z ˙ u [ ∇ div u t + div ( ∇ div u ⊗ u )] dx = Z ( p t div ˙ u + ( u · ∇ ) ˙ u · ∇ p ) dx + Z ( g t + div ( g ⊗ u )) ˙ udx. (3.17)As in [20], one can estimate the second and third terms in above equation as follows. − Z ˙ u [∆ u t + div (∆ u ⊗ u )] dx ≥ Z (cid:18) |∇ ˙ u | − C |∇ u | (cid:19) dx, and − Z ˙ u [ ∇ div u t + div ( ∇ div u ⊗ u )] dx ≥ Z (cid:18)
12 (div ˙ u ) − |∇ ˙ u | − C |∇ u | (cid:19) dx. Since p := ρθ , we rewrite (1.3) as follows, p t + div ( pu ) + p div u = µ |∇ u + ∇ u t | + λ (div u ) + | rot b | . (3.18)Using (3.18), as in [12, 20], one can estimate the fourth term in (3.17) as follows. Z ( p t div ˙ u + ( u · ∇ ) ˙ u · ∇ p ) dx ≤ C + C k∇ u k L + C k∇ b k L + µ k∇ ˙ u k L . (3.19)Using b · ∇ b + b × rot b = ∇| b | , and (3.16), we bound the last term of (3.17) asfollows. Z ( g t + div ( g ⊗ u )) ˙ udx = Z (cid:20) div (cid:18) | b | I − b ⊗ b (cid:19) + div ( g ⊗ u ) (cid:21) ˙ udx J.-S. FAN, F.-C. LI, AND G. NAKAMURA = − Z (cid:18) | b | I − b ⊗ b + g ⊗ u (cid:19) : ∇ ˙ udx ≤ C ( k b k L + k b k L ∞ k rot b k L k u k L ) k∇ ˙ u k L ≤ C (1 + k∇ b k L ) k∇ ˙ u k L ≤ µ k∇ ˙ u k L + C k∇ b k L + C. Inserting the those estimates into (3.16) and using k∇ b k L ≤ C k b k L ∞ k b k H , k∇ u k L ≤ k∇ u k L k∇ u k L ≤ C ( k∇ v k L + k∇ w k L ) ≤ C (1 + k√ ρ ˙ u k L ) , We have 12 ddt Z ρ | ˙ u | dx + µ Z |∇ ˙ u | dx + ( λ + µ ) Z (div ˙ u ) dx ≤ C (1 + k∇ u k L + k b k H ) ≤ C + C k√ ρ ˙ u k L + C k b k H , which gives k√ ρ ˙ u k L ∞ (0 ,T ; L ) + k ˙ u k L (0 ,T ; H ) ≤ C. (3.20)By the same calculations as in [12], it is easy to verify thatsup ≤ t ≤ T k∇ w k H + Z T ( k∇ w k L p + k∇ w k L ∞ ) dt ≤ C with any 2 ≤ p ≤ . (3.21)Applying ∂ t to (1.4), testing the result by b t , using (3.3) and (3.20), we have12 ddt Z | b t | dx + Z | rot b t | dx = − Z ∂ t ( b × u )rot b t dx = − Z ( b t × u + b × ˙ u − b × ( u · ∇ ) u )rot b t dx ≤ ( k b t k L k u k L + k b k L k ˙ u k L + k b k L ∞ k u · ∇ u k L ) k rot b t k L ≤ C ( k b t k L + k ˙ u k L + k u · ∇ u k L ) k rot b t k L ≤ k rot b t k L + C k b t k L + C k ˙ u k L + C k u · ∇ u k L , which implies k b t k L ∞ (0 ,T ; L ) + k b t k L (0 ,T ; H ) ≤ C. (3.22)This and (1.4) and (3.16) lead to k b k L ∞ (0 ,T ; H ) + k b k L (0 ,T ; W , ) ≤ C, (3.23)where we used k u k L ∞ (0 ,T ; W , ) ≤ k v k L ∞ (0 ,T ; W , ) + k w k L ∞ (0 ,T ; W , ) ≤ C. Direct calculations show that ddt k∇ ρ k L q ≤ C (1 + k∇ u k L ∞ ) k∇ ρ k L q + C k∇ u k L q , (3.24) EGULARITY CRITERION FOR FULL MHD EQUATIONS 9 and ddt k∇ p k L q ≤ C (1 + k∇ u k L ∞ )( k∇ p k L q + k∇ u k L q ) + C k∇ b k L ∞ k∇ b k L q . (3.25)We bound the last term of (3.25) as follows. k∇ b k L ∞ k∇ b k L q ≤ C (1 + k∇ b k L q ) k∇ b k L q ≤ C + C k∇ b k L q . (3.26)As in [12], it is easy to prove that k∇ ρ k L ∞ (0 ,T ; L q ) + k∇ p k L ∞ (0 ,T ; L q ) ≤ C, (3.27) k∇ u k L (0 ,T ; L ∞ ) + k u k L ∞ (0 ,T ; H ) ≤ C. (3.28)This completes the proof. (cid:3) Acknowledgements:
Fan is supported by NSFC (Grant No. 11171154). Li issupported partially by NSFC (Grant No. 11271184) and PAPD.
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