Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 17, pp. 1–10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu UNIFORM REGULARITY OF FULLY COMPRESSIBLE HALL-MHD SYSTEMS JISHAN FAN, YONG ZHOU Dedicated to Prof. Jiaxing Hong on his 80th birthday Abstract. In this article we study a fully compressible Hall-MHD system. These equations include shear viscosity, bulk viscosity of the flow, and heat conductivity and resistivity coefficients. We prove uniform regularity esti- mates. 1. Introduction In this article we consider the fully compressible Hall-MHD system [17], ∂tρ+ div(ρu) = 0, (1.1) ∂t(ρu) + div(ρu⊗ u) +∇p− µ∆u− (λ+ µ)∇div u = rot b× b+ ρ ∂w ∂t , (1.2) ∂t(ρe) + div(ρue)− div(κ∇θ) = µ 2 |∇u+∇uT |2 + λ(div u)2 + η| rot b|2, (1.3) ∂tb+ rot(b× u) + ξ rot ( rot b× b ρ ) = η∆b, (1.4) div b = 0 in T3 × (0,∞), (1.5) (ρ, u, θ, b)(·, 0) = (ρ0, u0, θ0, b0) in T3. (1.6) Here ρ, u, p, e, θ and b denote the density, velocity, pressure, internal energy, tem- perature, and magnetic field, respectively. The physical constants µ and λ are the shear viscosity and bulk viscosity of the flow and satisfy µ > 0 and λ + 2 3µ ≥ 0. κ > 0 is the heat conductivity. η > 0 is the resistivity coefficient. w is a given function. ξ is a Hall-constant. ∇uT is the transpose of the ∇u. For simplicity, we consider the case that the fluid is a polytropic ideal gas; that is e := CV θ, p := Rρθ with CV > 0 and R being the specific heat at constant volume and the generic gas constant, respectively. Applications of the Hall-MHD system cover a very wide range of physical objects, for example, magnetic reconnection in space plasmas, star formation, neutron stars, and geo-dynamo. For well-posedness, regularity and decay properties, and related 2010 Mathematics Subject Classification. 76W05, 35Q80, 70S15. Key words and phrases. Hall-MHD; uniform regularity; compressible. c©2021 Texas State University. Submitted November 10, 2020. Published March 21, 2021. 1 2 J. FAN, Y. ZHOU EJDE-2021/17 incompressible models, we refer to works [5, 6, 7, 10, 11, 20, 21, 22, 23] and references therein. When the Hall effect term rot ( rot b×b ρ ) is neglected, the system (1.1)-(1.5) reduces to the well-known fully compressible MHD system, which has been studied in [2, 4, 8, 9, 12, 13, 14]. The existence of local strong solution was proved by Fan-Yu [9]. Fan-Yu [8], Ducomet-Feireisl [4] and Hu-Wang [12, 13] established the global weak solutions. The low Mach number limit problem was studied by Jiang-Ju-Li-Xin [14] in R3 and Cui-Ou-Ren [2] in a bounded domain. Before stating our main results, we recall the existence of local smooth solutions to (1.1)-(1.6). Since the system (1.1)-(1.6) is parabolic-hyperbolic, we have the following result. Proposition 1.1 ([19]). Let ρ0, u0, θ0, b0 ∈ H3 and 1/C0 ≤ ρ0, θ0 for a positive constant C0. Then (1.1)-(1.6) has a unique smooth solution (ρ, u, θ, b) satisfying ρ ∈ C`([0, T );H3−`), u, θ, b ∈ C`([0, T );H3−2`), ` = 0, 1, and 1/C ≤ ρ, θ for some 0 < T ≤ ∞. The aim of this article is to prove uniform regularity estimates in (λ, µ, κ, η), as stated in the following theorem. Theorem 1.2. Let ξ2 ≤ Cη and w ∈ C([0, 1];H4), 0 < µ < 1, 0 < λ + µ < 1, 0 < η < 1, 0 < κ < 1, 0 < 1 C0 ≤ ρ0, θ0 ≤ C0, ρ0, u0, b0, θ0 ∈ H3(T3) with div b0 = 0 in T3. Let (ρ, u, b, θ) be the unique local smooth solutions to (1.1)-(1.5). Then ‖(ρ, u, b, θ)(·, t)‖H3 ≤ C in [0, T ] (1.7) holds for some positive constants C and T0 (≤ T ) independent of λ, µ, η and k. Remark 1.3. By the uniform estimates, one can easily take the limits of λ, µ, η and k to zero, hence we omit the details here. Our estimates are uniform in a with a := (λ, µ, η, k) while the ones in existence results of Hall-MHD depend on a. We define M(t) :=1 + ‖w‖C([0,1];H4) + sup 0≤τ≤t { ‖(ρ, u, b, θ)(·, τ)‖H3 + ‖∂tv(·, τ)‖L2 + ‖∂tθ(·, τ)‖L2 + ‖1 ρ (·, τ)‖L∞ + ‖1 θ (·, τ)‖L∞ } . (1.8) Here we note that v := u− w. Theorem 1.4. For any t ∈ [0, 1], we have M(t) ≤ C0(M0) exp(tC(M)) (1.9) for some nondecreasing continuous functions C0(·) and C(·). From (1.9) it follows that [1, 3, 16] M(t) ≤ C. (1.10) In the following proofs, we will use the bilinear commutator and product estimates due to Kato-Ponce [15], ‖Ds(fg)− fDsg‖Lp ≤ C(‖∇f‖Lp1 ‖Ds−1g‖Lq1 + ‖g‖Lp2 ‖Dsf‖Lq2 ), (1.11) ‖Ds(fg)‖Lp ≤ C(‖f‖Lp1 ‖Dsg‖Lq1 + ‖Dsf‖Lp2‖g‖Lq2 ), (1.12) EJDE-2021/17 FULLY COMPRESSIBLE HALL-MHD SYSTEMS 3 with s > 0 and 1 p = 1 p1 + 1 q1 = 1 p2 + 1 q2 . We only need to show Theorem 1.4, which is given in the next section. Our proof consists of two steps. In step 1, we give the lower order estimates and in step 2, we show the higher order estimates. 2. Proof of Theorem 1.4 Step 1. Lower order estimates. First, testing (1.1) by ρq−1, we see that 1 q d dt ∫ ρqdx = − ( 1− 1 q ) ∫ ρq div udx ≤ ‖div u‖L∞ ∫ ρqdx, and thus d dt ‖ρ‖Lq ≤ ‖div u‖L∞‖ρ‖Lq , which gives ‖ρ‖Lq ≤ ‖ρ0‖Lq exp (∫ t 0 ‖ div u‖L∞dτ ) . (2.1) In the limit as q → +∞, we obtain ‖ρ‖L∞ ≤ ‖ρ0‖L∞ exp(tC(M)). (2.2) It follows from (1.1) that ∂t 1 ρ + u · ∇1 ρ − 1 ρ div u = 0. (2.3) Testing (2.3) by ( 1 ρ )q−1 , we find that 1 q d dt ∫ (1 ρ )q dx = ( 1 + 1 q ) ∫ (1 ρ )q div udx ≤ ( 1 + 1 q ) ‖1 ρ ‖qLq‖ div u‖L∞ , and therefore d dt ‖1 ρ ‖Lq ≤ ( 1 + 1 q ) ‖1 ρ ‖Lq‖ div u‖L∞ , which gives ‖1 ρ ‖Lq ≤ ‖ 1 ρ0 ‖Lq exp (( 1 + 1 q ) ∫ t 0 ‖ div u‖L∞dτ ) and we have ‖1 ρ ‖L∞ ≤ ‖ 1 ρ0 ‖L∞ exp(tC(M)) (2.4) by letting q → +∞. Testing (1.3) by θq−1 and using (1.1) and denoting Q to be the right-hand side of (1.3), we obtain CV q d dt ∫ ρθqdx+ κ ∫ ∇θ · ∇θq−1dx = ∫ Qθq−1dx− ∫ pθq−1 div udx ≤ C(M)‖Q‖Lq‖ρ1/qθ‖q−1 Lq + C‖div u‖L∞‖ρ1/qθ‖qLq , and therefore d dt ‖ρ1/qθ‖Lq ≤ C(M)‖Q‖Lq + C‖ div u‖L∞‖ρ1/qθ‖Lq , 4 J. FAN, Y. ZHOU EJDE-2021/17 which, similarly to (2.2), implies ‖θ‖L∞ ≤ C0(M0) exp(tC(M)). (2.5) Multiplying (1.3) by 1 θ2 · 1 CV , we deduce that ρ∂t 1 θ + ρu · ∇1 θ + κ CV · 1 θ2 ∆θ = R CV ρ div u θ − Q θ2CV ≤ R CV ρ 1 θ div u. (2.6) Similarly to (2.5), testing (2.6) by ( 1 θ )q−1 , we have 1 q d dt ∫ ρ (1 θ )q dx ≤ R CV ∫ ρ (1 θ )q div udx ≤ R CV ‖ div u‖L∞ ∫ ρ (1 θ )q dx, and thus ‖1 θ ‖L∞ ≤ C0(M0) exp(tC(M)). (2.7) It is easy to verify that d dt ∫ |v|2dx = 2 ∫ v∂tvdx ≤ 2‖v‖L2‖∂tv‖L2 ≤ C(M), which implies ‖v‖L2 ≤ C0(M0) exp(tC(M)). (2.8) Testing (1.3) by b, we obtain 1 2 d dt ∫ |b|2dx+ η ∫ |∇b|2dx = − ∫ (u · ∇b− b · ∇u+ bdiv u)bdx = − ∫ (1 2 |b|2 div u− b · ∇u · b ) dx ≤ C‖∇u‖L∞‖b‖2L2 ≤ C(M), which leads to ‖b‖2L2 + η ∫ t 0 ∫ |∇b|2dxdτ ≤ C0(M0) exp(tC(M)). (2.9) Step 2. Higher order estimates. The equation (1.1)-(1.3) can be rewritten in the symmetric form θ ρ ∂tρ+ θ ρ u · ∇ρ+ θ div u = 0, (2.10) ρ∂tv + ρu · ∇v + ρ∇θ + θ∇ρ− µ∆v − (λ+ µ)∇ div v = r + rot b× b, (2.11) ρ θ ∂tθ + ρ θ u · ∇θ − κ θ ∆θ + ρdiv u = 1 θ Q, (2.12) where we have taken R = CV = 1 for simplicity, and r := µ∆w + (λ+ µ)∇ divw − ρu · ∇w. (2.13) EJDE-2021/17 FULLY COMPRESSIBLE HALL-MHD SYSTEMS 5 Taking D3 on (2.10), testing by D3ρ, using (1.1), (1.11) and (1.12), we obtain 1 2 d dt ∫ θ ρ (D3ρ)2dx+ ∫ θD3 div uD3ρdx = − ∫ [ D3 (θ ρ ∂tρ ) − θ ρ D3∂tρ ] D3ρdx − ∫ [ D3 (θ ρ u · ∇ρ ) − θ ρ u · ∇D3ρ ] D3ρdx − ∫ (D3(θ div u)− θD3 div u)D3ρdx + 1 2 ∫ ∂t (θ ρ ) (D3ρ)2dx− ∫ (θ ρ u · ∇D3ρ ) D3ρdx ≤ C ( ‖∇θ ρ ‖L∞‖D2∂tρ‖L2 + ‖∂tρ‖L∞‖D3 θ ρ ‖L2 ) ‖D3ρ‖L2 + C ( ‖∇θu ρ ‖L∞‖D3ρ‖L2 + ‖∇ρ‖L∞‖D3 θu ρ ‖L2 ) ‖D3ρ‖L2 + C(‖∇θ‖L∞‖D3u‖L2 + ‖∇u‖L∞‖D3θ‖L2)‖D3ρ‖L2 + C‖∂t θ ρ ‖L∞‖D3ρ‖2L2 + C‖∇θu ρ ‖L∞‖D3ρ‖2L2 ≤ C(M)(‖D2∂tρ‖L2 + ‖∂tρ‖L∞) + C(M) + C(M)‖∂t θ ρ ‖L∞ ≤ C(M) + κ 4 ∫ 1 θ (D4θ)2dx. (2.14) Here we have used the estimate [18]: ‖D3 1 ρ ‖L2 ≤ C(M)‖D3ρ‖L2 ≤ C(M). Applying D2 to (2.11), testing by D2∂tv, using (1.11) and (1.12), we obtain µ 2 d dt ∫ |D3v|2dx+ λ+ µ 2 d dt ∫ (D2 div v)2dx+ ∫ ρ|D2∂tv|2dx = − ∫ D2∇p ·D2∂tvdx− ∫ D2(ρu · ∇v) ·D2∂tvdx − ∫ [D2(ρ∂tv)− ρD2∂tv]D2∂tvdx+ ∫ D2r ·D2∂tvdx + ∫ D2 ( b · ∇b− 1 2 ∇|b|2 ) D2∂tvdx ≤ C‖D3p‖L2‖D2∂tv‖L2 + C‖ρ‖H2‖u‖H2‖v‖H3‖D2∂tv‖L2 + C(‖∇ρ‖L∞‖D∂tv‖L2 + ‖∂tv‖L∞‖D2ρ‖L2)‖D2∂tv‖L2 + ‖D2r‖L2‖D2∂tv‖L2 + ‖D2 ( b · ∇b− 1 2 ∇|b|2 ) ‖L2‖D2∂tv‖L2 ≤ C(M)‖D2∂tv‖L2 + C(M)(‖D∂tv‖L2 + ‖∂tv‖L∞)‖D2∂tv‖L2 ≤ C(M)‖D2∂tv‖L2 + C(M)(‖∂tv‖1/2L2 ‖D2∂tv‖1/2L2 + ‖∂tv‖L2 + ‖∂tv‖1/4L2 ‖D2∂tv‖3/4L2 )‖D2∂tv‖L2 6 J. FAN, Y. ZHOU EJDE-2021/17 ≤ C(M)‖D2∂tv‖L2 + C(M)(‖D2∂tv‖1/2L2 + ‖D2∂tv‖3/4L2 )‖D2∂tv‖L2 ≤ 1 2 ∫ ρ|D2∂tv|2dx+ C(M), which gives ∫ t 0 ∫ |D2∂tv|2dxdτ ≤ C0(M0) exp(tC(M)). (2.15) Applying D3 on (2.11), testing by D3v, using (1.1), (1.11) and (1.12), we have 1 2 d dt ∫ ρ|D3v|2dx+ µ ∫ |D4v|2dx+ (λ+ µ) ∫ (D3 div v)2dx + ∫ ρD3∇θ ·D3vdx+ ∫ θ∇D3ρ ·D3vdx+ ∫ (b×D3 rot b)D3vdx = − ∫ (D3(ρ∂tv)− ρD3∂tv)D3vdx− ∫ (D3(ρu · ∇v)− ρu · ∇D3v)D3vdx − ∫ (D3(ρ∇θ)− ρ∇D3θ)D3vdx− ∫ (D3(θ∇ρ)− θ∇D3ρ)D3vdx + ∫ D3rD3vdx− ∫ (D3(b× rot b)− b×D3 rot b)D3vdx ≤ C(‖∇ρ‖L∞‖D2∂tv‖L2 + ‖∂tv‖L∞‖D3ρ‖L2)‖D3v‖L2 + C(‖∇v‖L∞‖D3(ρu)‖L2 + ‖∇(ρu)‖L∞‖D3v‖L2)‖D3v‖L2 + C(‖∇ρ‖L∞‖D3θ‖L2 + ‖∇θ‖L∞‖D3ρ‖L2)‖D3v‖L2 + C(‖∇θ‖L∞‖D3ρ‖L2 + ‖∇ρ‖L∞‖D3θ‖L2)‖D3v‖L2 + C(M) + µ 16 ‖D4v‖2L2 + C‖∇b‖L∞‖D3b‖L2‖D3v‖L2 ≤ C(M) + C(M)(‖D2∂tv‖L2 + ‖∂tv‖L∞) + µ 16 ‖D4v‖2L2 ≤ C(M) + ‖D2∂tv‖2L2 + µ 16 ‖D4v‖2L2 . (2.16) Applying D3 on (1.4), testing by D3b, using (1.11) and (1.12), we have 1 2 d dt ∫ |D3b|2dx+ η ∫ |D4b|2dx+ ∫ (b×D3u)D3 rot bdx = − ∫ (D3(b× u)−D3b× u− b×D3u)D3 rot bdx − ∫ (D3b× u)D3 rot bdx+ ξ ∫ D3 ( b ρ × rot b ) D3 rot bdx = − ∫ rot(D3(b× u)−D3b× u− b×D3u)D3bdx+ ∫ (D3b×D3 rot b)udx = − ∫ rot(D3(b× u)−D3b× u− b×D3u)D3bdx + ∫ [1 2 ∇|D3b|2 − (D3b · ∇)D3b ] udx + ξ ∫ ( D3 ( b ρ × rot b ) − b ρ ×D3 rot b ) D3 rot bdx = − ∫ rot(D3(b× u)−D3b× u− b×D3u)D3bdx EJDE-2021/17 FULLY COMPRESSIBLE HALL-MHD SYSTEMS 7 − 1 2 ∫ |D3b|2 div udx+ ∫ D3b⊗D3b : ∇udx + ξ ∫ ( D3 ( b ρ × rot b ) − b ρ ×D3 rot b ) D3 rot bdx ≤ ‖ rot(D3(b× u)−D3b× u− b×D3u)‖L2‖D3b‖L2 + 1 2 ‖D3b‖2L2‖ div u‖L∞ + ‖D3b‖2L2‖∇u‖L∞ + C √ η ( ‖ b ρ ‖L∞‖D3b‖L2 + ‖∇b‖L∞‖D3 ( b ρ ) ‖L2 ) ‖D4b‖L2 ≤ C(M) + η 16 ‖D4b‖2L2 . Here we have used that a · ∇a+ a× rot a = 1 2∇|a| 2. Applying D2 on (1.3), testing by D2∂tθ, using (1.11) and (1.12), we have κ 2 d dt ∫ (D3θ)2dx+ ∫ ρ|D2∂tθ|2L2dx = − ∫ D2(pdiv u)D2∂tθdx− ∫ D2(ρu · ∇θ)D2∂tθdx − ∫ [D2(ρ∂tθ)− ρD2∂tθ]D 2∂tθdx+ ∫ D2Q ·D2∂tθdx with CV = 1) ≤ ‖D2(p div u)‖L2‖D2∂tθ‖L2 + ‖D2(ρu · ∇θ)‖L2‖D2∂tθ‖L2 + C(‖∇ρ‖L∞‖D∂tθ‖L2 + ‖∂tθ‖L∞‖D2ρ‖L2)‖D2∂tθ‖L2 + ‖D2Q‖L2‖D2∂tθ‖L2 ≤ C(M)‖D2∂tθ‖L2 + C(M)(‖D∂tθ‖L2 + ‖∂tθ‖L∞)‖D2∂tθ‖L2 ≤ C(M)‖D2∂tθ‖L2 + C(M)(‖∂tθ‖1/2L2 ‖D2∂tθ‖1/2L2 + ‖∂tθ‖L2 + ‖∂tθ‖1/4L2 ‖D2∂tθ‖3/4L2 )‖D2∂tθ‖L2 ≤ 1 2 ∫ ρ|D2∂tθ|2dx+ C(M), which leads to ∫ t 0 ∫ |D2∂tθ|2dxdτ ≤ C0(M0) exp(tC(M)). (2.17) Taking D3 on (2.12), testing by D3θ, using (1.11) and (1.12), we have 1 2 d dt ∫ ρ θ (D3θ)2dx+ κ ∫ 1 θ (D4θ)2dx+ ∫ ρD3 div uD3θdx = κ ∫ ∇θ θ2 D3θ∇D3θdx+ κ ∫ [ D3 (1 θ ∆θ ) − 1 θ ∆D3θ ] D3θdx − ∫ [ D3 (ρ θ ∂tθ ) − ρ θ D3∂tθ ] D3θdx+ 1 2 ∫ ∂t (ρ θ ) (D3θ)2dx − ∫ [ D3 (ρu θ ∇θ ) − ρu θ ∇D3θ ] D3θdx− ∫ ρu θ ∇D3θ ·D3θdx − ∫ (D3(ρ div u)− ρD3 div u)D3θdx+ ∫ D3 (Q θ ) D3θdx ≤ κ 8 ∫ 1 θ (D4θ)2dx+ C(M)‖∇θ‖2L∞‖D3θ‖2L2 8 J. FAN, Y. ZHOU EJDE-2021/17 + κC ( ‖∇θ θ2 ‖L∞‖D4θ‖L2 + ‖∆θ‖L∞‖D3 1 θ ‖L2 ) ‖D3θ‖L2 + C ( ‖∇ρ θ ‖L∞‖D2∂tθ‖L2 + ‖∂tθ‖L∞‖D3 ρ θ ‖L2 ) ‖D3θ‖L2 + C‖∂t ρ θ ‖L∞‖D3θ‖2L2 + C ( ‖∇ρu θ ‖L∞‖D3θ‖L2 + ‖∇θ‖L∞‖D3 ρu θ ‖L2 ) ‖D3θ‖L2 + C‖∇ρu θ ‖L∞‖D3θ‖2L2 + C(‖∇ρ‖L∞‖D3u‖L2 + ‖ div u‖L∞‖D3ρ‖L2)‖D3θ‖L2 + C ( ‖1 θ ‖L∞‖D3Q‖L2 + ‖Q‖L∞‖D3 1 θ ‖L2 ) ‖D3θ‖L2 ≤ κ 4 ∫ 1 θ (D4θ)2dx+ C(M) + C(M)(‖D2∂tθ‖L2 + ‖∂tθ‖L∞) + µ 2 ‖D4u‖2L2 + λ+ µ 2 ‖D3 div u‖2L2 + η 16 ‖D4b‖2L2 ≤ κ 4 ∫ 1 θ (D4θ)2dx+ ‖D2∂tθ‖2L2 + C(M) + µ 2 ‖D4v‖2L2 + λ+ µ 2 ‖D3 div v‖2L2 + η 16 ‖D4b‖2L2 . Summing (2.14), (2.16) and the above inequality, we arrive at 1 2 d dt ∫ (θ ρ (D3ρ)2 + ρ|D3v|2 + |D3b|2 + ρ θ (D3θ)2 ) dx+ µ 2 ∫ |D4v|2dx + λ+ µ 2 ∫ (D3 div v)2dx+ η 2 ∫ |D4b|2dx+ κ 2 ∫ 1 θ (D4θ)2dx ≤ − ∫ θD3 div uD3ρdx− ∫ ρD3∇θ ·D3udx− ∫ θD3∇ρ ·D3udx − ∫ ρD3 div uD3θdx+ ‖D2∂tu‖2L2 + ‖D2∂tθ‖2L2 + C(M) + ∫ (b×D3 rot b)D3wdx+ ∫ ρD3∇θ ·D3wdx+ ∫ θD3∇ρ ·D3wdx ≤ ∫ D3uD3ρ∇θdx+ ∫ D3uD3θ∇ρdx+ ‖D2∂tu‖2L2 + ‖D2∂tθ‖2L2 + C(M) + ∣∣ ∫ rot(b×D3w)D3bdx ∣∣+ ∣∣ ∫ D3θ div(ρD3w)dx ∣∣+ ∣∣ ∫ D3ρ div(θD3w)dx ∣∣ ≤ C(M) + ‖D2∂tu‖2L2 + ‖D2∂tθ‖2L2 . Here we have used that (b×D3 rot b) ·D3u+ (b×D3u) ·D3 rot b = 0. Using (2.15) and (2.17), we have ‖D3(ρ, u, b, θ)(·, t)‖L2 ≤ C0(M0) exp(tC(M)). (2.18) On the other hand, from (1.2) it follows that ‖∂tv‖L2 = ∥∥1 ρ ( b · ∇b− 1 2 ∇|b|2 + µ∆u+ (λ+ µ)∇ div u−∇p− ρu · ∇u )∥∥ L2 ≤ C0(M0) exp(tC(M)). (2.19) EJDE-2021/17 FULLY COMPRESSIBLE HALL-MHD SYSTEMS 9 Similarly, we have ‖∂tθ‖L2 ≤ C0(M0) exp(tC(M)). 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Zhou; Global well-posedness for the 3D incompressible Hall magnetohydrody- namic equations with Fujita-Kato type initial data. J. Math. Fluid Mech., 21 (2019), Paper No. 5, 16 pp. Jishan Fan Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China Email address: fanjishan@njfu.edu.cn Yong Zhou (corresponding author) School of Mathematics, Sun Yat-sen University, Zhuhai 519082, China. Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China Email address: zhouyong3@mail.sysu.edu.cn 1. Introduction 2. Proof of Theorem 1.4 Step 1 Step 2 Acknowledgments References