Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 11, pp. 1–16. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu FREE BOUNDARY VALUE PROBLEM FOR COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS HUIHUI KONG, RUXU LIAN Abstract. In this article we consider a free boundary value problem for barotropic compressible magnetohydrodynamic equations with density-dependent viscosity coefficients. Under certain assumptions imposed on the initial data, there exists a unique global strong solution which is strictly positive after a fi- nite time. Furthermore, the free boundaries propagate along the particle path and the domain expands outwards at an algebraic rate. 1. Introduction The three-dimensional barotropic compressible magnetohydrodynamic equation with density-dependent viscosity coefficients read ρt + div(ρU) = 0, (ρU)t + div(ρU⊗U) +∇P (ρ) = (∇×H)×H + µ∆U + (µ+ λ(ρ))∇ divU, Ht −∇× (U×H) = −∇× (ν∇×H), divH = 0, (x, t) ∈ R3 × [0, T ], (1.1) where ρ(x, t) > 0, U(x, t) and P (ρ) = ργ(γ > 1) stand for the flow density, velocity and pressure respectively. H(x, t) is the magnetic field with x = (x, y, z). The shear viscosity coefficient µ > 0 is a positive constant, and the bulk viscosity coefficient is λ(ρ) = ρβ with β > 0. The constant ν > 0 is the resistivity coefficient which is inversely proportional to the electrical conductivity constant. In this article we focus on the free boundary value problem for one-dimensional barotropic compressible Magnetohydrodynamic equations with density-dependent viscosity coefficients. The existence, regularity and dynamical behavior of global strong solution will be discussed. For γ > 1 and β > 0, we show that the free boundary value problem with regular initial data admits a unique global strong solution which is strictly positive from a finite time and decays pointwise to zero at an algebraic time-rate. also the domain expands outwards at an algebraic rate. See Theorem 2.1. The rest of this article is arranged as follows. In section 2, the main results about existence and dynamical behavior of global strong solutions for compressible 2010 Mathematics Subject Classification. 35A01, 35R35, 35Q35. Key words and phrases. Magnetohydrodynamic equations; free boundary value problem; density-dependent viscosity coefficient; strong solution. c©2020 Texas State University. Submitted September 25, 2019. Published January 23, 2020. 1 2 H. KONG, R. LIAN EJDE-2020/11 Magnetohydrodynamic equations are stated. In section 3, a priori estimates will be given. Then, the main results are proven in section 4. 2. Main results Consider a magnetic flow which is moving in the x-direction and uniform in the transverse direction (y, z) under a planar magnetic field. Let ρ(x, t) = ρ(x, t), U(x, t) = (u(x, t), 0, 0) and H(x, t) = (0, H2(x, t), H3(x, t)). This means the lon- gitudinal velocity is u(x, t) and the transverse velocity is (0, 0). The longitudinal magnetic field is 0 and the transverse magnetic field is (H2(x, t), H3(x, t)). We assume H2(x, t) = H(x, t), H3(x, t) = kH(x, t) and the constant k is in [0,+∞). We investigate the existence and dynamics of of a global solution of the free boundary value problem for the planar magnetohydrodynamic equations with density- dependent viscosity coefficient, ρt + (ρu)x = 0, x ∈ (a(t), b(t)), t > 0, (ρu)t + (ρu2)x + (ργ)x = −(1 + k2)HHx + ((2µ+ ρβ)ux)x, x ∈ (a(t), b(t)), t > 0, Ht + (uH)x = νHxx, x ∈ (a(t), b(t)), t > 0, (ργ − (2µ+ ρβ)ux)(a(t), t) = 0, (ργ − (2µ+ ρβ)ux)(b(t), t) = 0, t > 0, H(a(t), t) = H(b(t), t) = 0, t > 0, (ρ, u,H)(x, 0) = (ρ0, u0, H0), x ∈ [a0, b0], (2.1) where x = a(t) and x = b(t) are the free boundaries defined by d dt a(t) = u(a(t), t), a(0) = a0, d dt b(t) = u(b(t), t), b(0) = b0, t > 0. (2.2) The initial data satisfies inf [a0,b0] ρ0 ≥ ρ > 0, ρ0 ∈ L1([a0, b0]), ρ0x ∈ L2([a0, b0]), (ργ0 − (2µ+ ρβ0 )u0x)(a0) = 0, (ργ0 − (2µ+ ρβ0 )u0x)(b0) = 0, H0(a0) = H0(b0) = 0, u0 ∈ H2([a0, b0]), H0 ∈ H1([a0, b0]), (2.3) where ρ is a positive constant. Without loss of generality, the total initial mass can be renormalized to be one. By the conservation of mass,∫ b(t) a(t) ρ(x, t)dx = ∫ b0 a0 ρ0(x)dx := 1. (2.4) Then, we have the existence of a global solution and time-asymptotical behavior of strong solution as follows. EJDE-2020/11 COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS 3 Theorem 2.1. Let γ > 1, β > 0 and T > 0. Assume that the initial data satisfies (2.3). Then, there exists a global strong solution (ρ, u,H, a, b) to (2.1) satisfying (ρ, u) ∈ C0([0, T ]× [a(t), b(t)]), cT ≤ ρ ∈ L∞(0, T ;H1([a(t), b(t)])), ρt ∈ L∞(0, T ;L2([a(t), b(t)])), u ∈ L∞(0, T ;H2([a(t), b(t)])) ∩ L2(0, T ;H3([a(t), b(t)])), ut ∈ L∞(0, T ;L2([a(t), b(t)])) ∩ L2(0, T ;H1([a(t), b(t)])), H ∈ L∞(0, T ;H1([a(t), b(t)])), a(t), b(t) ∈ H2([0, T ]), ργ − (2µ+ ρβ)ux ∈ C0([0, T ]× ([a(t), b(t)])), (2.5) where cT > 0 is a constant depending on time. As β > 0, the domain expands outwards in time as DM (t) := sup τ∈[0,t] (b(τ)− a(τ)) ≥ { C(1 + t) γ−1 γ , 1 < γ < 2, C(1 + t)1/γ(1 + ln(1 + t))−1/γ , γ ≥ 2 . (2.6) Furthermore, b(t)− a(t) ≥ C(1 + t)1/γ(1 + ln(1 + t))−1/γ , γ ≥ 2, (2.7) where C > 0 is a constant independent of time. In particular, if 0 < β ≤ 1, we have∫ b(t) a(t) ργ(x, t)dx+ ∫ b(t) a(t) H2(x, t)dx ≤ C(1 + t)−η, 0 < β ≤ 1. (2.8) where 0 < η ≤ min{γ − 1, β} denotes a positive constant. Remark 2.2. If H3(x, t) 6= kH2(x, t), the system becomes ρt + (ρu)x = 0, (ρu)t + (ρu2)x + (ργ)x = −H2H2x −H3H3x + ((2µ+ ρβ)ux)x, H2t + (uH2)x = νH2xx, H3t + (uH3)x = νH3xx, (2.9) using the some method as in Theorem 2.1, the well-posedness of the solutions to the free boundary value problem with initial finite mass can also be proved. Remark 2.3. conditions (2.6) and (2.7) imply that as time approaches infinity, the lower bound approaches infinity. 3. A priori estimates In this section, we deduce a priori estimates for the solution (ρ, u,H) to the (2.1). 4 H. KONG, R. LIAN EJDE-2020/11 Lemma 3.1. Under the assumptions of Theorem 2.1, for every strong solution (ρ, u,H) of (2.1) satisfies∫ b(t) a(t) (1 2 ρu2 + 1 + k2 2 H2 + 1 γ − 1 ργ ) dx+ ∫ t 0 ∫ b(s) a(s) (2µ+ ρβ)u2x dx ds + ν ∫ t 0 ∫ b(s) a(s) H2 x dx ds = ∫ b0 a0 (1 2 ρ0u 2 0 + 1 + k2 2 H2 0 + 1 γ − 1 ργ0 ) dx, t ∈ [0, T ]. (3.1) Proof. Taking the product of (2.1)2 and (2.1)3 with u and H respectively, and integrating on [a(t), b(t)], we have d dt ∫ b(t) a(t) (1 2 ρu2 + 1 + k2 2 H2 + 1 γ − 1 ργ ) dx+ ∫ b(t) a(t) (2µ+ ρβ)u2xdx + ν ∫ b(t) a(t) H2 xdx = 0, (3.2) which leads to (3.1) after the integrating with respect to t ∈ [0, T ]. � Lemma 3.2. Under the assumptions of Theorem 2.1, we have cT ≤ ρ(x, t) ≤ C, (x, t) ∈ [a(t), b(t)]× [0, T ], T > 0, (3.3) where C is a positive constant independent of time, and cT is also a positive constant but dependent of time. Proof. Firstly, denote the effective viscous flux by F = (2µ+ λ(ρ))ux − ργ − 1 + k2 2 H2. (3.4) Then, we can rewrite (2.1)2 as ρu̇ = Fx, (3.5) where u̇ = ut + uux. Define ζ(x, t) = ∫ x a(t) ρu(x, t)dx, (3.6) η(x, t) = ρu2(x, t)− ρu2(a(t), t). (3.7) Integrating (2.1)2 from a(t) to x, and using (3.6) and (3.7), we have ζt + η − F = −ρu2(a(t), t). (3.8) Define θ(ρ) = ∫ ρ 1 2µ+ λ(s) s ds = 2µ ln ρ+ 1 β (ρβ − 1). (3.9) Multiplying (2.1)1 by θ′(ρ), we have θ(ρ)t + uθ(ρ)x + (2µ+ λ(ρ))ux = 0, (3.10) which together with (3.9) gives θ(ρ)t + uθ(ρ)x + F + ργ + 1 + k2 2 H2 = 0. (3.11) EJDE-2020/11 COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS 5 Using that η − uζx = −ρu2(a(t), t), (3.12) we obtain (ζ + θ(ρ))t + u(ζ + θ(ρ))x + ργ + 1 + k2 2 H2 = 0. (3.13) Define the particle path X(t̃;x, t) through the point (x, t) ∈ [a(t), b(t)] as d dt̃ X(t̃;x, t) = u(X(t̃;x, t), t̃), X(t̃;x, t)|t̃=t = x, (3.14) which together with (3.13) gives d dt̃ (ζ + θ(ρ))(X(t̃;x, t), t̃) + ργ(X(t̃;x, t), t̃) + 1 + k2 2 H2(X(t̃;x, t), t̃) = 0. (3.15) Integrating (3.15) over [0, t], we have 2µ ln ρ(x, t) ρ0(X0) + 1 β (ρβ(x, t)− ρβ0 (X0)) + ζ(x, t)− ζ(X0, 0) + ∫ t 0 ργ(x, s)ds+ 1 + k2 2 ∫ t 0 H2(x, s)ds = 0, (3.16) where X0 = X(t̃;x, t)|t̃=0 ∈ [a0, b0]. Using (3.1) and Hölder’s inequality, we have |ζ(x, t)| = ∣∣ ∫ x a(t) ρudx ∣∣ = ∣∣ ∫ x a(t) √ ρ √ ρudx ∣∣ ≤ (∫ b(t) a(t) ρdx )1/2(∫ b(t) a(t) ρu2dx )1/2 ≤ C, (3.17) where C is a positive constants independent of time. Then, if ρ > 1, we obtain sup t∈[0,T ] ‖ρ‖L∞([a(t),b(t)]) ≤ C. (3.18) If ρ ≤ 1, then 2µ ln 1 ρ ≤ CT + 1 + k2 2 ∫ t 0 ‖H‖2L∞ds ≤ CT , (3.19) from which, we obtain the positive lower bound of the density ρ(x, t) ≥ cT . (3.20) � Remark 3.3. From (3.1) and (3.3), we obtain that b(t)− a(t) = b0 − a0 + ∫ t 0 (b′(s)− a′(s))ds ≤ b0 − a0 + C(1 + t)1/2 (∫ t 0 (b′(s)2 + a′(s)2)ds )1/2 , (3.21) and ∫ t 0 (b′(s)2 + a′(s)2)ds = ∫ t 0 (u(b(s), s)2 + u(a(s), s)2)ds ≤ 2 ∫ t 0 ‖u‖2L∞dx ≤ CT , (3.22) 6 H. KONG, R. LIAN EJDE-2020/11 where CT is the positive constant depending on time. Lemma 3.4. Under the assumptions of Theorem 2.1, we have ∫ b(t) a(t) H2 xdx+ ∫ t 0 ∫ b(s) a(s) H2 xx dx ds ≤ CT , t ∈ [0, T ], (3.23) where CT is a positive constant depending on time. Proof. Multiplying (2.1)3 by Hxx and integrating on (a(t), b(t)), we obtain 1 2 d dt ∫ b(t) a(t) H2 xdx+ ν ∫ b(t) a(t) H2 xxdx = 1 2 ∫ b(t) a(t) uxH 2 xdx+ ∫ b(t) a(t) uxHHxxdx+ 2 ∫ b(t) a(t) uHxHxxdx ≤ ν 2 ∫ b(t) a(t) H2 xxdx+ CT ∫ b(t) a(t) H2 xdx ∫ b(t) a(t) u2xdx + C ( ‖H‖2L∞ ∫ b(t) a(t) u2xdx+ ‖u‖2L∞ ∫ b(t) a(t) H2 xdx ) ≤ ν 2 ∫ b(t) a(t) H2 xxdx+ CT ∫ b(t) a(t) H2 xdx ( 1 + ∫ b(t) a(t) u2xdx ) + C ∫ b(t) a(t) H2dx ∫ b(t) a(t) u2xdx ≤ ν 2 ∫ b(t) a(t) H2 xxdx+ CT (1 + ∫ b(t) a(t) u2xdx) ( 1 + ∫ b(t) a(t) H2 xdx ) , (3.24) which together with Gronwall’s inequality gives (3.23). � Lemma 3.5. Under the assumptions of Theorem 2.1, we have ∫ b(t) a(t) F 2dx+ ∫ t 0 ∫ b(s) a(s) ρu̇2 dx ds ≤ CT , t ∈ [0, T ], (3.25) where CT is a positive constant depending on time. Proof. After a straight calculation, we deduce that (u̇)x = utx + uuxx + u2x = (F + ργ + 1+k2 2 H2 2µ+ ρβ ) t + u (F + ργ + 1+k2 2 H2 2µ+ ρβ ) x + u2x = Dt ( F 2µ+ ρβ ) +Dt ( ργ 2µ+ ρβ ) + 1 2 Dt ( H2 2µ+ ρβ ) + u2x, (3.26) EJDE-2020/11 COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS 7 where Dtf = ft + ufx. Multiplying (3.26) by F , we have 1 2 d dt ∫ b(t) a(t) F 2 2µ+ ρβ dx+ ∫ b(t) a(t) ρu̇2dx = 1 2 ∫ b(t) a(t) F 2ux ( ρ( 1 2µ+ ρβ )′ − 1 2µ+ ρβ ) dx + ∫ b(t) a(t) Fux ( ρ( ργ 2µ+ ρβ )′ − ργ 2µ+ ρβ ) dx − (1 + k2) ∫ b(t) a(t) FH(Ht + uHx) 2µ+ ρβ dx + 1 + k2 2 ∫ b(t) a(t) FH2ux ( ρ( 1 2µ+ ρβ )′ − 1 2µ+ ρβ ) dx := I1 + I2 + I3 + I4. (3.27) Using (3.1), (3.3) and (3.23), we deduce that I1 ≤ CT ‖F‖2L4‖ux‖L2 ≤ CT ‖F‖2L∞‖ux‖L2 ≤ CT ‖Fx‖L2‖F‖L2‖ux‖L2 ≤ 1 2 ‖√ρu̇‖2L2 + CT ‖ F√ 2µ+ ρβ ‖2L2‖ux‖2L2 , (3.28) I2 ≤ CT ‖F‖L2‖ux‖L2 ≤ CT ‖ F√ 2µ+ ρβ ‖2L2 + CT ‖ux‖2L2 , (3.29) I3 ≤ CT ‖FHt‖L1 + CT ‖FuHx‖L1 ≤ CT ‖F‖2L2 + CT ‖Ht‖2L2 + CT ‖F‖2L2 + CT ‖uHx‖2L2 ≤ CT ‖ F√ 2µ+ ρβ ‖2L2 + CT ‖Ht‖2L2 + CT (‖u‖2L2 + ‖ux‖2L2)‖Hx‖2L2 ≤ CT ‖ F√ 2µ+ ρβ ‖2L2 + CT ‖Ht‖2L2 + CT ‖ux‖2L2 , (3.30) I4 ≤ CT ‖F‖L2‖ux‖L2 ≤ CT ‖ F√ 2µ+ ρβ ‖2L2 + CT ‖ux‖2L2 . (3.31) Substituting (3.28)-(3.31) into (3.27), we have 1 2 d dt ∫ b(t) a(t) F 2 2µ+ ρβ dx+ ∫ b(t) a(t) ρu̇2dx ≤ CT (1 + ‖ F√ 2µ+ ρβ ‖2L2)(1 + ‖ux‖2L2) + CT ‖Ht‖2L2 , (3.32) which together with Gronwall’s inequality gives (3.25). � Lemma 3.6. Under the assumptions of Theorem 2.1, we have∫ T 0 ‖ux‖2L∞([a(t),b(t)])dt ≤ CT , (3.33) where CT is a positive constant depending on time. 8 H. KONG, R. LIAN EJDE-2020/11 Proof. From (3.1), (3.3) and (3.23), we have ‖ux‖L∞ ≤ C‖(2µ+ ρβ)ux − ργ − 1 + k2 2 H2‖L∞ + C‖ργ‖L∞ + C‖H2‖L∞ ≤ C‖(2µ+ ρβ)ux − ργ − 1 + k2 2 H2‖1/2L2 × ∥∥((2µ+ ρβ)ux − ργ − 1 + k2 2 H2)x ∥∥1/2 L2 + CT ≤ CT (‖ √ 2µ+ ρβux‖L2 + 1)1/2(‖√ρut‖L2 + ‖ux‖L∞‖ √ ρu‖L2)1/2 + CT ≤ CT ( ‖ √ 2µ+ ρβux‖L2 + 1 )1/2‖√ρut‖1/2L2 + (‖ √ 2µ+ ρβux‖L2 + 1)1/2‖ux‖1/2L∞ + CT ≤ 1 2 ‖ux‖L∞ + CT ‖ √ 2µ+ ρβux‖L2 + CT ‖ √ ρut‖L2 + CT . (3.34) Then, it holds that ‖ux‖2L∞ ≤ CT ‖ √ 2µ+ ρβux‖2L2 + CT ‖ √ ρut‖2L2 + CT , (3.35) which implies (3.33) after the integration with respect to t . � Lemma 3.7. Under the assumptions of Theorem 2.1, we have∫ b(t) a(t) ρ2xdx ≤ CT , (3.36) where CT is a positive constant depending on time. Proof. Differentiating (2.1)1 with respect to x, we have ρtx + ρxxu+ 2ρxux + ρuxx = 0. (3.37) Multiplying (3.37) by ρx, it holds 1 2 d dt ∫ b(t) a(t) ρ2xdx = −3 2 ∫ b(t) a(t) ρ2xuxdx− ∫ b(t) a(t) ρρxuxxdx ≤ C‖ux‖L∞‖ρx‖2L2 + C‖ρ‖L∞‖ρx‖L2‖uxx‖L2 . (3.38) Using (3.34) and that ‖uxx‖L2 ≤ C(‖ρut‖L2 + ‖ρuux‖L2 + ‖(ργ)x‖L2 + ‖(H2)x‖L2 + ‖(ρβ)xux‖L2) ≤ C(‖√ρut‖L2 + ‖ux‖L∞ + ‖(ργ)x‖L2 + ‖(H2)x‖L2 + ‖ux‖L∞‖(ρβ)x‖L2) ≤ CT (1 + ‖√ρut‖L2 + ‖(ργ)x‖L2 + (1 + ‖√ρut‖L2)‖(ρβ)x‖L2), (3.39) EJDE-2020/11 COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS 9 we obtain d dt ∫ b(t) a(t) ρ2xdx ≤ CT (1 + ‖√ρut‖L2)‖ρx‖2L2 + CT ( 1 + ‖√ρut‖L2 + ‖(ργ)x‖L2 + ( 1 + ‖√ρut‖L2 ) ‖(ρβ)x‖L2 ) ≤ CT (1 + ‖√ρut‖2L2)‖ρx‖2L2 + CT (‖(ργ)x‖2L2 + ‖(ρβ)x‖2L2) + CT (1 + ‖√ρut‖2L2) ≤ CT (1 + ‖√ρut‖2L2)‖ρx‖2L2 + CT (1 + ‖√ρut‖2L2), (3.40) which together with Gronwall’s inequality gives (3.36). � Lemma 3.8. Under the assumptions of Theorem 2.1, we have∫ b(t) a(t) ρu̇2dx+ ∫ t 0 ∫ b(s) a(s) (2µ+ ρβ)(u̇)2x dx ds ≤ CT , t ∈ [0, T ], (3.41) where CT is a positive constant depending on time. Proof. Differentiating (3.5) with respect to t, we have ρu̇t + ρtu̇ = Fxt. (3.42) Multiplying (3.42) by u̇, we have 1 2 d dt ∫ b(t) a(t) ρu̇2dx+ ∫ b(t) a(t) (2µ+ ρβ)(u̇)2xdx = (ρuu̇)(a(t), t)− ∫ b(t) a(t) ρuu̇(u̇)xdx− β ∫ b(t) a(t) ρβ−1ρtux(u̇)xdx + ∫ b(t) a(t) (2µ+ ρβ)u2x(u̇)xdx+ ∫ b(t) a(t) (2µ+ ρβ)uuxx(u̇)xdx + γ ∫ b(t) a(t) ργ−1ρt(u̇)xdx+ ∫ b(t) a(t) HHt(u̇)xdx := J1 + J2 + J3 + J4 + J5 + J6 + J7. (3.43) Using (3.1), (3.3), (3.23), (3.33) and (3.36), we have J1 + J2 ≤ CT ∫ b(t) a(t) ρu̇2dx+ 1 8 ∫ b(t) a(t) (2µ+ ρβ)(u̇)2xdx, (3.44) J3 = β ∫ b(t) a(t) ρβ−1ρxuux(u̇)xdx+ β ∫ b(t) a(t) ρβ−1ρu2x(u̇)xdx ≤ CT ‖ux‖2L∞ (∫ b(t) a(t) ρ2xdx+ ∫ b(t) a(t) u2xdx ) + 1 8 ∫ b(t) a(t) (2µ+ ρβ)(u̇)2xdx ≤ CT ‖ux‖2L∞ + 1 8 ∫ b(t) a(t) (2µ+ ρβ)(u̇)2xdx, (3.45) 10 H. KONG, R. LIAN EJDE-2020/11 J4 ≤ CT ‖ux‖2L∞ ∫ b(t) a(t) u2xdx+ 1 8 ∫ b(t) a(t) (2µ+ ρβ)(u̇)2xdx ≤ CT ‖ux‖2L∞ + 1 8 ∫ b(t) a(t) (2µ+ ρβ)(u̇)2xdx, (3.46) J5 ≤ CT ∫ b(t) a(t) u2xxdx+ 1 8 ∫ b(t) a(t) (2µ+ ρβ)(u̇)2xdx ≤ CT + CT ∫ b(t) a(t) ρu2tdx+ 1 8 ∫ b(t) a(t) (2µ+ ρβ)(u̇)2xdx, (3.47) J6 = −γ ∫ b(t) a(t) ργ−1ρxu(u̇)xdx− γ ∫ b(t) a(t) ργ−1ρux(u̇)xdx ≤ CT (∫ b(t) a(t) ρ2xdx+ ∫ b(t) a(t) u2xdx ) + 1 8 ∫ b(t) a(t) (2µ+ ρβ)(u̇)2xdx ≤ CT + 1 8 ∫ b(t) a(t) (2µ+ ρβ)(u̇)2xdx, (3.48) J7 ≤ CT ∫ b(t) a(t) H2 t dx+ 1 8 ∫ b(t) a(t) (2µ+ ρβ)(u̇)2xdx. (3.49) Then d dt ∫ b(t) a(t) ρu̇2dx+ 1 4 ∫ b(t) a(t) (2µ+ ρβ)(u̇)2xdx ≤ CT + CT ∫ b(t) a(t) ρu̇2dx+ CT ‖ux‖2L∞ + CT ∫ b(t) a(t) ρu2tdx+ CT ∫ b(t) a(t) H2 t dx ≤ CT + CT ∫ b(t) a(t) ρu̇2dx+ CT ∫ b(t) a(t) H2 xxdx + CT ∫ b(t) a(t) u2H2 xdx+ CT ∫ b(t) a(t) H2u2xdx ≤ CT + CT ∫ b(t) a(t) ρu̇2dx+ CT ∫ b(t) a(t) H2 xxdx+ CT ‖u‖2L∞ ∫ b(t) a(t) H2 xdx + CT ‖H‖2L∞ ∫ b(t) a(t) u2xdx ≤ CT + CT ∫ b(t) a(t) ρu̇2dx+ CT ∫ b(t) a(t) H2 xxdx + C (∫ b(t) a(t) u2dx+ ∫ b(t) a(t) u2xdx )∫ b(t) a(t) H2 xdx + C (∫ b(t) a(t) H2dx+ ∫ b(t) a(t) H2 xdx )∫ b(t) a(t) u2xdx ≤ CT + CT ∫ b(t) a(t) ρu̇2dx+ CT ∫ b(t) a(t) H2 xxdx EJDE-2020/11 COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS 11 + CT ( 1 + ∫ b(t) a(t) u2xdx )( 1 + ∫ b(t) a(t) H2 xdx ) , which together with (3.23), (3.25), (3.33), (3.35) and Gronwall’s inequality gives (3.41). � 4. Proof of the main results Proof of Theorem 2.1. The existence of a global strong solution to (2.1) is estab- lished in terms of the short time existence carried out as in [6], the uniform a-priori estimates and the analysis of regularities which indeed follow from Lemmas 3.1-3.8. Next, we will give the large time behaviors of the strong solution to the free boundary value problem (2.1) as follows. Firstly, we prove (2.6). Define the follow- ing energy functional L(t) := ∫ b(t) a(t) (x− (1 + t)u)2ρdx+ 2 γ − 1 (1 + t)2 ∫ b(t) a(t) ργdx + (1 + k2)(1 + t)2 ∫ b(t) a(t) H2dx. (4.1) After a direct calculation, we have L′(t) = 2(3− γ) γ − 1 (1 + t) ∫ b(t) a(t) ργdx+ (1 + k2)(1 + t) ∫ b(t) a(t) H2dx + 2(1 + t) ∫ b(t) a(t) (2µ+ ρβ)uxdx− 2(1 + t)2 ∫ b(t) a(t) (2µ+ ρβ)u2xdx − 2ν(1 + k2)(1 + t)2 ∫ b(t) a(t) H2 xdx ≤ 2(3− γ) γ − 1 (1 + t) ∫ b(t) a(t) ργdx+ (1 + k2)(1 + t) ∫ b(t) a(t) H2dx + (1 + t) ∫ b(t) a(t) (2µ+ ρβ)dx, (4.2) where C is a positive constant independent of time. If γ ≥ 3, we deduce from (4.2) that L′(t) ≤ 1 1 + t L(t) + C(b(t)− a(t)), (4.3) which leads to L(t) ≤ C(1 + t){1 + ∫ t 0 b(τ)− a(τ) 1 + τ dτ}. (4.4) Thus, we have∫ b(t) a(t) ργdx ≤ C(1 + t)−1{1 + ∫ t 0 b(τ)− a(τ) 1 + τ dτ}, γ ≥ 3. (4.5) Similarly, from (4.2), we have L′(t) ≤ 2(2− γ) γ − 1 (1 + t) ∫ b(t) a(t) ργdx+ L(t) 1 + t + C(b(t)− a(t)) 12 H. KONG, R. LIAN EJDE-2020/11 ≤ { L(t) 1+t + C(b(t)− a(t)), 2 ≤ γ < 3, (3− γ)L(t)1+t + C(b(t)− a(t)), 1 < γ < 2, which together with Gronwall’s inequality yields∫ b(t) a(t) ργdx ≤ CL(t) (1 + t)2 ≤ { C(1 + t)−1{1 + ∫ t 0 b(τ)−a(τ) 1+τ dτ}, 2 ≤ γ < 3, C(1 + t)1−γ{1 + ∫ t 0 b(τ)−a(τ) (1+τ)3−γ dτ}, 1 < γ < 2. (4.6) Note that 1 = ∫ b0 a0 ρ0dx = ∫ b(t) a(t) ρdx ≤ (b(t)− a(t)) γ−1 γ (∫ b(t) a(t) ργdx )1/γ , (4.7) which, combined with (4.5) and (4.6), implies (b(t)− a(t))γ−1{1 + ∫ t 0 b(τ)− a(τ) 1 + τ dτ} ≥ C(1 + t), γ ≥ 2, (b(t)− a(t))γ−1{1 + ∫ t 0 b(τ)− a(τ) (1 + τ)3−γ dτ} ≥ C(1 + t)γ−1, 1 < γ < 2. (4.8) From (4.8), we can obtain DM (t) := sup τ∈[0,t] (b(τ)− a(τ)) ≥ { C(1 + t) 1 γ (1 + ln(1 + t))−1/γ , γ ≥ 2, C(1 + t) γ−1 γ , 1 < γ < 2. (4.9) Next, we prove (2.7). Using a similar argument as to (4.2), we obtain L′(t) ≤ 2(3− γ) γ − 1 (1 + t) ∫ b(t) a(t) ργdx+ (1 + k2)(1 + t) ∫ b(t) a(t) H2dx + ∫ b(t) a(t) ρβdx+ 4µ(1 + t) ∫ b(t) a(t) uxdx ≤ 2(3− γ) γ − 1 (1 + t) ∫ b(t) a(t) ργdx+ (1 + k2)(1 + t) ∫ b(t) a(t) H2dx + 4µ(1 + t) d dt (a(t)− b(t)) + C ≤ L(t) 1 + t + 4µ(1 + t) d dt (a(t)− b(t)) + C, if γ ≥ 2 and β ≥ 1, (4.10) which implies L(t) ≤ C(1+t)(b(t)−a(t)+1+ln(1+t)) ≤ C(1+t)(b(t)−a(t))(1+ln(1+t)). (4.11) Thus, we have∫ b(t) a(t) ργdx ≤ C(1 + t)−1(b(t)− a(t))(1 + ln(1 + t)), γ ≥ 2, (4.12) which, combined with (4.7), leads to b(t)− a(t) ≥ C(1 + t)1/γ(1 + ln(1 + t))−1/γ , γ ≥ 2. (4.13) EJDE-2020/11 COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS 13 Finally, to prove (2.8), we define the Lagrange coordinates transformation ξ = ∫ x a(t) ρ(y, t)dy, τ = t. (4.14) Since the conservation of total mass holds, the boundaries x = a(t) and x = b(t) are transformed into ξ = 0 and ξ = 1 respectively. The domain [a(t), b(t)] is transformed into [0, 1]. The FBVP (2.1) is reformed into ρτ + ρ2uξ = 0, ξ ∈ (0, 1), τ > 0, uτ + (ργ)ξ = −(1 + k2)HHξ + (ρ(2µ+ ρβ)uξ)ξ, ξ ∈ (0, 1), τ > 0, Hτ + ρHuξ = νρ(ρHξ)ξ, ξ ∈ (0, 1), τ > 0, (ργ − ρ(2µ+ ρβ)uξ)(0, τ) = 0, (ργ − ρ(2µ+ ρβ)uξ)(1, τ) = 0, τ ≥ 0, H(0, τ) = H(1, τ) = 0, τ ≥ 0, (ρ0, u0, H0) = (ρ0, u0, H0)(ξ), ξ ∈ [0, 1], (4.15) where the initial data satisfies inf [0,1] ρ0 ≥ ρ > 0, ρ0 ∈ H1([0, 1]), u0 ∈ H2([0, 1]), H0 ∈ H1([0, 1]), (ργ0 − ρ0(2µ+ ρβ0 )u0x)(0) = 0, (ργ0 − ρ0(2µ+ ρβ0 )u0x)(1) = 0, (4.16) and the consistencies between initial data and boundary conditions hold. From (4.15)2 we find that d dτ ∫ 1 0 u(ξ, τ)dξ = 0, (4.17) and without loss of generality, we can renormalize ∫ 1 0 u0(ξ)dξ to be zero, then we denote w = u− 1 1 + τ ∫ ξ 0 1 ρ dζ + 1 1 + τ ∫ 1 0 ∫ ξ 0 1 ρ dζdξ. (4.18) Applying (4.17), we obtain wξ = uξ − 1 (1 + τ)ρ = (1 ρ ) τ − 1 (1 + τ)ρ , (4.19) wτ + w 1 + τ = uτ . (4.20) 14 H. KONG, R. LIAN EJDE-2020/11 Then, the system (4.15) becomes ρτ + ρ2wξ + ρ 1 + τ = 0, wτ + w 1 + τ + (ργ)ξ = −(1 + k2)HHξ + (ρ(2µ+ ρβ)wξ + ρβ 1 + τ )ξ, Hτ + ρH(wξ + 1 (1 + τ)ρ ) = νρ(ρHξ)ξ, (ργ − ρ(2µ+ ρβ)(wξ + 1 (1 + τ)ρ ))(0, τ) = 0, (ργ − ρ(2µ+ ρβ)(wξ + 1 (1 + τ)ρ ))(1, τ) = 0, H(0, τ) = H(1, τ) = 0, τ ∈ [0, T ], (ρ0, H0, w0) = (ρ0, H0, u0 − ∫ ξ 0 1 ρ0 dζ + ∫ 1 0 ∫ ξ 0 1 ρ0 dζdξ)(ξ). (4.21) Multiplying (4.21)2 by w and (4.21)3 by H ρ , integrating the result equations over (0, 1), after a straightforward calculation, we have 1 2 d dτ ∫ 1 0 w2dξ + 1 + k2 2 d dτ ∫ 1 0 H2 ρ dξ + 1 1 + τ ∫ 1 0 w2dξ + ∫ 1 0 ρ(2µ+ ρβ)w2 ξdξ + ν(1 + k2) ∫ 1 0 ρH2 ξ dξ + 1 + k2 2(1 + τ) ∫ 1 0 H2 ρ dξ = − 1 1 + τ ∫ 1 0 ρβwξdξ + ∫ 1 0 ργwξdξ. (4.22) For 0 < β < 1, from (4.19) it holds − 1 1 + τ ∫ 1 0 ρβwξdξ = − 1 1 + τ ∫ 1 0 ρβ {(1 ρ ) τ − 1 (1 + τ)ρ } dξ = 1 (β − 1)(1 + τ) ∫ 1 0 (ρβ−1)τdξ + 1 (1 + τ)2 ∫ 1 0 ρβ−1dξ, (4.23) and ∫ 1 0 ργwξdξ = ∫ 1 0 ργ {(1 ρ ) τ − 1 (1 + τ)ρ } dξ = 1 1− γ ∫ 1 0 (ργ−1)τdξ − 1 1 + τ ∫ 1 0 ργ−1dξ, (4.24) which together with (4.22) leads to 1 2 d dτ ∫ 1 0 w2dξ + 1 γ − 1 d dτ ∫ 1 0 ργ−1dξ + 1 + k2 2 d dτ ∫ 1 0 H2 ρ dξ + 1 (1− β)(1 + τ) d dτ ∫ 1 0 ρβ−1dξ + 1 1 + τ ∫ 1 0 w2dξ + 1 1 + τ ∫ 1 0 ργ−1dξ + ∫ 1 0 ρ(2µ+ ρβ)w2 ξdξ + ν(1 + k2) ∫ 1 0 ρH2 ξ dξ + 1 + k2 2(1 + τ) ∫ 1 0 H2 ρ dξ − 1 (1 + τ)2 ∫ 1 0 ρβ−1dξ = 0. (4.25) EJDE-2020/11 COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS 15 Multiplying (4.25) by (1 + τ)η for some 0 < η < 1 to be determined later, we have d dτ ∫ 1 0 ( (1 + τ)η 2 w2 + (1 + τ)η γ − 1 ργ−1 + (1 + k2)(1 + τ)η 2 H2 ρ + (1 + τ)η−1 1− β ρβ−1 ) dξ + (1− η 2 )(1 + τ)η−1 ∫ 1 0 w2dξ + γ − 1− η γ − 1 (1 + τ)γ−1 ∫ 1 0 ργ−1dξ + (1 + τ)η ∫ 1 0 ρ(2µ+ ρβ)w2 ξdξ + ν(1 + k2)(1 + τ)η ∫ 1 0 ρH2 ξ dξ + 1 2 (1 + k2)(1− η)(1 + τ)η−1 ∫ 1 0 H2 ρ dξ + β − η 1− β (1 + τ)η−2 ∫ 1 0 ρβ−1dξ = 0. (4.26) If 0 < η ≤ min{γ − 1, β}, from(4.26), we obtain∫ 1 0 ργ−1dξ + ∫ 1 0 H2 ρ dξ ≤ C(1 + τ)−η. (4.27) For β = 1 and 0 < η ≤ min{γ − 1, 1}, from (4.22) it holds d dτ ∫ 1 0 ( (1 + τ)η 2 w2 + (1 + τ)η γ − 1 ργ−1 + (1 + k2)(1 + τ)η 2 H2 ρ ) dξ + (1− η 2 )(1 + τ)η−1 ∫ 1 0 w2dξ + γ − 1− η γ − 1 (1 + τ)η−1 ∫ 1 0 ργ−1dξ + (1 + τ)η ∫ 1 0 ρ(2µ+ ρ)w2 ξdξ + ν(1 + k2)(1 + τ)η ∫ 1 0 ρH2 ξ dξ + 1 2 (1 + k2)(1− η)(1 + τ)η−1 ∫ 1 0 H2 ρ dξ = d dτ ( (1 + τ)η−1 ∫ 1 0 ln ρdξ ) + (1− η)(1 + τ)η−2 ∫ 1 0 ln ρdξ + (1 + τ)η−2. 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Zhu; Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), no. 2, 329-363. Huihui Kong School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China Email address: konghuihuiking@126.com Ruxu Lian College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China. Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China Email address: lianruxu@mail.iap.ac.cn 1. Introduction 2. Main results 3. A priori estimates 4. Proof of the main results Acknowledgements References