Shibin SU(蘇仕斌)Xiaokui ZHAO(趙小奎)

School of Mathematical Sciences,Xiamen University,Xiamen 361005,China

E-mail:19020151153423@stu.xmu.edu.cn;zhaoxiaokui@126.com

Abstract The initial boundary value problem of the one-dimensional magneto-hydrodynamics system,when the viscosity,thermal conductivity,and magnetic diffusion coefficients are general smooth functions of temperature,is considered in this article.A unique global classical solution is shown to exist uniquely and converge to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent γ.The initial data can be large if γ is sufficiently close to 1.

Key words MHD system;global well-posedness;temperature-dependent viscosity

1 Introduction

As it is well known,the motion of a conducting fluid in an electromagnetic field is governed by the equations of magnetohydrodynamics(MHD),which is a coupled system of the induction equation of the magnetic field and the Navier-Stokes equations of fluid dynamics(see also[1,4–6,8,16,32])in Rd:

where ρ,U,P=Rρθ,H,and θ are unknown density,velocity field,pressure,magnetic field,and absolute temperature.R>0 is the specific gas constant.And the operator S is defined by

where?U?is the transpose matrix of?U and Idis the d×d identity matrix.The shear viscosity coefficient λ,the bulk viscosity coefficient ν,the thermal conductivity coefficient κ,and the magnetic diffusion coefficient σ are prescribed through constitutive relations as functions of the density and temperature satisfying ν >0,κ >0,σ >0,and 2ν +dλ >0.The total energy E is defined by

where e=cVθ is specific internal energy,cv=R/(γ ? 1)and γ >1 are the specific heat at constant volume and the adiabatic exponent,respectively.

Due to its physical importance,complexity,rich phenomena,and mathematical challenges,there have been extensive studies on MHD by many physicists and mathematicians.The issues of well-posedness and dynamical behaviors of MHD system are rather complicated to investigate because of the strong coupling and interplay interaction between the fluid motion and the magnetic field.In spite of these,there is much recent important progress on the mathematical analysis on these topics for the MHD system;refer,for example,to[9,10,12–15,19–22,25–29,39,41,42,44–46]and the references therein.Among them,we brie fly recall the results concerned with the compressible MHD equations in multi-dimension.Kawashima[26]obtained the global existence of smooth solutions to the general electro-magneto- fluid equations in two dimensions when the initial data are small perturbations of a given constant state.Umeda,Kawashima,and Shizuta[39]studied the global existence and time decay rate of smooth solutions to the linearized two-dimensional compressible MHD equations.The optimal decay estimates of classical solutions to the compressible MHD system were obtained by Zhang and Zhao[45]when the initial data are close to a non-vacuum equilibrium.The local strong solutions to the compressible MHD with large initial data were obtained,by Vol’pert and Khudiaev[41]as the initial density is strictly positive and by Fan and Yu[15]as the initial density may contain vacuum,respectively.Hu and Wang[19,22]and Fan and Yu[14]proved the global existence of renormalized solutions to the compressible MHD equations for general large initial data.In[44],Zhang,Jiang,and Xie considered a MHD model describing the screw pinch problem in plasma physics and showed the global existence of weak solutions with cylindrical symmetry.For σ=0,we obtained the 1-D globe strong solutions to the compressible MHD with large initial data[46].However,the decay rate of the solutions can not be arrived in[46].Motivated by[23,36,38],we establish the long time behavior of the strong solutions when γ ? 1 is small enough in this article.

By means of the Chapman-Enskog expansion,the compressible Navier-Stokes system is the first order approximation of the Boltzmann equation[7,11,17,40],and the transport coefficients μ and κ depend solely on the temperature.In view of the above,we are interested in the case where the transport coefficients μ,κ,and σ are smooth functions of the temperature.More specifically,suppose that

Similar to that in[2,18,24,30,31],on the basis of the specific choice of dependent variables with y∈[0,1]?R and t∈R+:

we consider the simplest compressible,viscous,heat-conducting,and resistive MHD equations for ideal polytropic fluids in dimension one:

Clearly,the magnetic field obeys the divergence constraint divH=0 because of the special dependent variables.System(1.4)is supplemented with the initial data:

and the boundary conditions:

To state the main result,let x be the Lagrangian space variable,t be the time variable,andthe specific volume.Then,system(1.4)withμ=μ(θ),κ=κ(θ),and σ=σ(θ)becomes

The initial and boundary conditions are

NotationsFor the convenience,we defineWe will also use A.B to denote the statement that A≤CB for some absolute constant C>0,which may be different on different lines.

The main results in this article are stated as follows.

Theorem 1.1Suppose that the coefficientsμ,κ,and σ satisfy(1.3).Let the initial data(v0,u0,b0,θ0)be compatible with the boundary conditions(1.9)and satisfy

where S0and V0are positive constants independent of γ ?1.Then,there exist constants ε0>0 and C1>0,which depend only on S0and V0,such that if γ ?1 ≤ ε0,then the initial boundary value problem(1.7)–(1.9)has a unique global solution(v,u,b,θ) ∈ C([0,∞),H3([0,1]))satisfying

and

The rest of this article is arranged as follows.Section 2 is devoted to deriving the necessary a priori estimates on strong solutions that are needed to extend the local solution to all time.The main results in Theorem 1.1 are proved in Section 3.

2 A Priori Estimates

To prove Theorem 1.1,we first define the following set of functions for which the solutions to problem(1.7)–(1.9)will be sought:

for constants M,N,t1,and t2(t1≤t2),where

As(v,u,b,θ)∈ X(0,T;M,N),it follows from Sobolev’s inequality that

Keeping(2.2)–(2.3)in mind,we have the following estimates.

Lemma 2.1(Basic energy estimate) Under the conditions listed in Lemma 3.1,suppose that the local solution(v(t,x),u(t,x),b(t,x),θ(t,x))constructed in Lemma 3.1 has been extended to the time step t=T;then for 0≤t≤T,we have the followings:

where φ(z):=z ?lnz?1.

ProofBy(2.2)and(1.7)1,(2.4)–(2.5)are immediately obtained by choosing γ?1 suitable small.Multiplying(1.7)1–(1.7)4by R(1? v?1),u,(1? θ?1),and b,respectively,(2.6)can be obtained similarly as[36].The details of the proof is omitted for simplicity. ?

Lemma 2.2Assume that the conditions listed in Lemma 2.1 hold,γ?1 is chosen sufficiently small such that

then

ProofBy virtue of chain rule and(1.7)1-(1.7)2,one has

Taking the inner product(2.9)withon[0,t]×[0,1],we have

Multiplying(1.7)4byand integrating over[0,t]× [0,1],byone has

By means of Cauchy-Schwarz’s and H?lder’s inequality,Lemma 2.1,and,we obtain

Adding(2.10)and(2.11),putting(2.12)into it,and by Lemma 2.1 and(2.7),one can derive that

By Cauchy-Schwarz’s and H?lder’s inequality,and(2.7),we have

Putting the above inequality into(2.13)and taking ε1and ε suitable small,Lemma 2.2 is thus completed. ?

By 0≤x≤1,we have the following upper and lower bounds of v.

Lemma 2.3Under the conditions listed in Lemma 2.2,we can deduce that

holds for all(t,x)∈ [0,T]×[0,1].

ProofBy Lemma 2.1,one has

Hence,by mean value theorem and Lemma 2.2,we have

That is,

Combined with Lemma 2.1–2.3,we have

Corollary 2.4Under the conditions listed in Lemma 2.2,we can deduce that

holds for all(t,x)∈ [0,T]×[0,1].

The estimates on first-order derivatives are established in the following lemmas.Before estimating the first-order derivative of velocity,we have the following L4-norm estimate of the magnetic field.

Lemma 2.5Under the conditions listed in Lemma 2.2,we can deduce that

holds for all(t,x)∈ [0,T]×[0,1].

ProofTaking the inner product(1.7)4with 4b3,by Cauchy-Schwarz’s inequality andwe have

By Gronwall’s inequality,Lemma 2.1,and Lemma 2.3,the proof of Lemma 2.5 is thus completed.?

Lemma 2.6Under the conditions listed in Lemma 2.2,we can deduce that

holds for all(t,x)∈ [0,T]×[0,1].

ProofMultiplying(1.7)2by 2uxx,integrating over[0,t]×[0,1],byand integration by parts,we have

By Cauchy-Schwarz’s inequality,one obtains

By the upper and lower bound of θ and v listed in Lemma 2.1 and Lemma 2.3,one can derive

Hence,by Cauchy-Schwarz’s inequality,Lemma 2.2,and kθxk∞≤ (γ ? 1)1/4N,it follows that

Inserting(2.24)and(2.26)into(2.23),by Lemma 2.1,Lemma 2.3–Lemma 2.5,and(1.7)1,the proof of Lemma 2.6 is thus completed. ?

Next,we derive the first-order estimate of magnetic field.

Lemma 2.7Under the conditions listed in Lemma 2.2,we can deduce that

holds for all(t,x)∈ [0,T]×[0,1].

ProofBy virtue of(1.7)4and(1.7)1,one has

It follows from the above equality,,and integration by part that

By Cauchy-Schwarz’s inequality,(2.3),and Lemmas 2.1–2.2,we have

Inserting(2.30)–(2.31)into(2.29),(2.27)is immediately obtained after choosing ε suitable small.?

The first-order estimate of temperature is established in the following lemma.

Lemma 2.8Under the conditions listed in Lemma 2.2,we can deduce that

holds for all(t,x)∈ [0,T]×[0,1].

ProofMultiplying(1.7)3by θxx,integrating over[0,1]on x,by(2.3),one can deduce

By Lemmas 2.1–2.7 and the above inequality,Lemma 2.8 can be established. ?

Corollary 2.9Under the conditions listed in Lemma 2.2,we can deduce that

holds for all(t,x)∈ [0,T]×[0,1].

ProofThanks to Lemmas 2.6–2.7 and Sobolev’s inequality,we obtain

By means of(1.7)2,one has

By(2.35),H?lder’s inequality,and Corollary 2.4,one can deduce

Similarly,by(1.7)3,one obtains

By(2.3),H?lder’s inequality,and Lemmas 2.6–2.7,we have

By virtue of(2.37)–(2.39)and Lemmas 2.1–2.8,the proof of this corollary is completed. ?

The estimates on second-order derivatives are established in the following lemmas.

Lemma 2.10Under the conditions listed in Lemma 2.2,we can deduce that

holds for all(t,x)∈ [0,T]×[0,1].

ProofApplying?tto(1.7)2,multiplying by ut,and integrating over[0,1],we have

Hence,integrating the above inequality over[0,T],we have

By(2.36),one has

By virtue of(2.42)–(2.43)and Lemma 2.6,we complete the proof of Lemma 2.10. ?

Lemma 2.11Under the conditions listed in Lemma 2.2,we can deduce that

holds for all(t,x)∈ [0,T]×[0,1].

ProofApplying?tto(1.7)4,multiplying by bt,and integrating over[0,1];by integration by parts Cauchy-Schwarz’s inequality,we deduce

By means of(2.3),Corollary 2.9,and Lemmas 2.1–2.8,we have

By(1.7)4and Lemmas 2.6–2.8,one can derive

By(2.46)–(2.47),the proof of Lemma 2.11 is completed. ?

Lemma 2.12Under the conditions listed in Lemma 2.2,we can deduce that

holds for all(t,x)∈ [0,T]×[0,1].

ProofApplying ?tto(1.7)3,multiplying by cVθt,and integrating over[0,t]× [0,1];by Cauchy-Schwarz’s inequality,(2.3),Lemma 2.1,Lemma 2.6,and Lemmas 2.10–2.11,we have

By means of(1.7)3and Lemmas 2.1–2.11,we have

The proof of Lemma 2.12 is completed.?

Lemma 2.13Under the conditions listed in Lemma 2.2,we can deduce that

holds for all(t,x)∈ [0,T]×[0,1].

Integrating the above equality over[0,t]× [0,1],then by Cauchy-Schwarz’s inequality,one has

By virtue of Lemmas 2.1–2.11,we can deduce

Putting(2.54)–(2.56)into(2.53),and by means of

and Lemmas 2.1–2.11,one can derive

Thanks to Lemma 2.13,we have the following result.

Lemma 2.14Under the conditions listed in Lemma 2.2,we can deduce that

holds for all(t,x)∈ [0,T]×[0,1].

ProofBy(1.7)3,we have

By means of γ ? 1<1 and Lemmas 2.1–2.13,we have

By(1.7)2,one has

which combined with Lemmas 2.1–2.13 yields

By(1.7)4,one has

which means that

Next,we estimate the third-order derivatives.

Lemma 2.15Under the conditions listed in Lemma 2.2,for all(t,x)∈ [0,T]×[0,1],it hold that

ProofFirst of all,we need know that

Applying the operator?tto(1.7)2,multiplying by uxxt,integrating over[0,t]×[0,1],then by Cauchy-Schwarz’s inequality and Lemmas 2.1–2.14,we have

Applying the operator?tto(1.7)4,multiplying by bxxt,integrating over[0,t]× [0,1],then by Cauchy-Schwarz’s inequality and Lemmas 2.1–2.14,we have

Applying the operator ?tto(1.7)3,multiplying by θxxt,integrating over[0,t]× [0,1],and by Cauchy-Schwarz’s inequality and Lemmas 2.1–2.14,we have

Hence,similar to(2.61),(2.63),and(2.65),we have

Applying the operator ?xxto(2.9),multiplying byintegrating over[0,t]×[0,1],then by Cauchy-Schwarz’s inequality and Lemmas 2.1–2.14,we have

Similar to(2.57),by the above inequality,one has

Similar to Lemma 2.14,we have

The proof of this lemma is thus completed by(2.69)–(2.72)and(2.74)–(2.75).

3 The Proof of Theorem 1.1

In this section,we will complete the proof of Theorem 1.1 by combining the a priori bounds obtained in Section 2 and the continuation argument.Under the assumptions given in Theorems 1.1,we can get the following local existence result like[3,36].

Lemma 3.1(Local existence) Let the initial data(v0,u0,b0,θ0)∈ H3([0,1])satisfy the followings:

for some positive constants λ1,λ2,Λ,and S,which are independent of γ ?1.Then,there exists a positive constant T0=T0(λ1,λ2,Λ,S),which depends only on λ1,λ2,Λ,and S,such that the initial boundary value problem(1.7)-(1.9)has a unique solution(v,u,b,θ)∈ X(0,T0;2λ1,2Λ).

According to(1.10)–(1.11),there exists a(γ ? 1)-independent positive constant C0,such that

Let ε1,Cj(j=1,···,10)be chosen in Section 2.We assume that γ ? 1 ≤ ε0with

where

Applying Lemma 3.1,we can find a positive constant t1=T0(V0,V0,C0,S0)such that there exists a unique solution(v,u,b,θ) ∈ X(0,t1;2V0,2C0)to the initial boundary value problem(1.7)–(1.9).

As γ ?1 ≤ δ1,we can apply Corollary 2.4,Corollary 2.9,and Lemmas 2.2–2.15 with T=t1to deduce that there exists a positive constant C1(γ),depending only on V0,S0,and γ,such that for each t∈ [0,t1],the local solution(v,u,b,θ)satisfies

Repeating the above procedure,we can then extend the solution(v,u,b,θ),step by step,to a global one provided that γ ?1 ≤ ε0.Furthermore,

where C2(γ)is some positive constant depending on γ,S0,and V0.It follows from(3.5)that the time asymptotic behavior(1.13)is obtained immediately.