Bei LIHongjin ZHUCaidi ZHAODepartment of Mathematics and Information Science,Wenzhou University,Wenzhou 325035,China E-mail:Lbbeili2015@163.com;zhuhongjinZHJ@163.com;zhaocaidi2013@163.com

TIME DECAY RATE OF SOLUTIONS TO THE HYPERBOLIC MHD EQUATIONS IN ℝ3∗

In this paper,we first show the global existence,uniqueness and regularity of weak solutions for the hyperbolic magnetohydrodynamics(MHD)equations in ℝ3.Then we establish that the solutions with initial data belonging to ℍm(ℝ3)∩L1(ℝ3)have the following time decay rate:

for large t,where m=0,1.

hyperbolic MHD equations;weak solution;time decay rate 2010 MR Subject Classification76A10;76D03;35Q35

1　Introduction

In this paper,we investigate the following model

which are regularizations in both the velocity and the magnetic of the following large eddy simulation(LES)model for the turbulent flow of a magnetofluid:

where the velocity field u=(u1,u2,u3),the magnetic field b=(b1,b2,b3)and the total pressure p(x,t)are the unknown terms,v≥o is the kinematic viscosity andµ≥o is the constantmagnetic resistivity.In equations(1.1)and(1.2),the constantsµ,v,α and β are all positive. The length scale α is a characterizing parameter of the elasticity of the fluid in the sense that α2/v is a characteristic relaxation time scale of the viscoelastic material[11].

We can view equations(1.1)–(1.4),also called MHD-Voight equations,as the approximate equations for the standard MHD equations(1.5)–(1.8)as α,β→o.Also,(1.1)–(1.4)are the exact equations for a class of visco-elastic fluid known as Kelvin-Voight fluids(the corresponding equations for nonmagnetic fluids were first introduced by Oskolkov in[17]).On bounded domains,equations(1.1)–(1.4)were deeply studied,see e.g.,Catania and Secchi[4,6],Catania [5],Larios and Titi[12,13],Levant,Ramos and Titi[14].Particularly,Catania[5]studied the global attractor and determining modes for the solutions of equations(1.1)–(1.4)in[o,2πL]3(L>o)with space-periodic conditions.

On the whole space,however,there is no references concerning the existence,uniqueness and especially the decay property of the solutions for equations(1.1)–(1.4).Motivated by the idea of[18],we will investigate the time decay rate of solutions to equations(1.1)–(1.4)in ℝ3. Different to the Navier-Stokes(NS)equations studied in[18],equations(1.1)–(1.4)contains the Maxwell’s equations(which rule the magnetic field)and the Navier-Stokes equations(which govern the fluid motion),what is more,equations(1.1)and(1.2)contains the extra regularizing terms(−α2∂tΔu)and(−β2∂tΔb),respectively.These two terms have two main effects. First,they regularize the equations in a way that the 3D equations(1.1)–(1.4)become now globally wellposed(see Theorem 2.1 in Section 2 or[5]).On the other hand,they change the parabolic character of the standard MHD equations to the hyperbolic one(for this reason,we call equations(1.1)–(1.4)as hyperbolic MHD turbulence model).

The aim of the paper is to give an upper bound of the time decay rate of solution to the hyperbolic MHD equations(1.1)–(1.4).We first extend the Fourier splitting method of Schonbek(see[18])to derive that the weak solutions with initial data(uo,bo)lying in ℍ1(ℝ3)∩L1(ℝ3)have the ℍ1decay rate

where the constant c depends essentially on α,β,µ,v,‖uo‖1,‖bo‖1,‖uo‖1,2and‖bo‖1,2.

Then,on the base of this ℍ1decay estimate,we prove by combining the Fourier splitting method of Schonbek and the Gronwall inequality that

where the constant c depends essentially on α,β,µ,v,‖uo‖1,‖bo‖1,‖uo‖2,2and‖bo‖2,2.

The Fourier splitting method was built up by Schonbek in[18–21],where the upper and lower bounds of decay rate for the Leray-Hopf solutions of the NS equations were subtly proved. Later,this Fourier splitting method was well extended to investigate the decay for the solutions of partial differential equations from mathematical physics,see e.g.Bjorland and Schonbek[2], Brandolese and Schonbek[3],Dai,Qing and Schonbek[9],Dong and Li[7],Dong and Chen[8], Niche and Schonbek[16],Schonbek and Wiegner[22],Zhang[23–26],Zhao,Liang and Zhao [27],Zhao and Zhu[28],etc.

2　Existence,Uniqueness and Regularity of the Solutions

In this paper,we use c to denote the generic constant that can take different values in different places.Lp(ℝ3)=(Lp(ℝ3))3represents the 3D vector Lebesgue space with norm‖·‖p, particularly,‖·‖2=‖·‖.Wm,p(ℝ3)stands for the 3D vector Sobolev space{φ=(φ1,φ2,φ3)∈with norm(see[1])

We now specify the definition of weak solutions for the hyperbolic MHD equations(1.1)–(1.4).

then(u,b)is called a weak solution of equations(1.1)–(1.4).

We have the following global existence,uniqueness and regularity results of the weak solutions for equations(1.1)–(1.4).

Theorem 2.1For given uo,bo∈ℍ1(ℝ3)with∇·uo=∇·bo=o,equations(1.1)–(1.4) have a unique weak solution.If furthermore uo,bo∈ℍ2(ℝ3),then the weak solution(u,b)has the following regularity

ProofWe just present the main idea and skip the details of the proof.Let{Rm}m≥1be a sequence tending to+∞and denote by

the open ball with radius Rm.Then,similar to[5,Theorem 1],the following initial boundary value problem:

has a global unique weak solution(denoted by)(uRm,bRm),where χBRmis the characteristic function of the open ball BRm.Note that for each m≥1,we have

which implies that the sequences{uRm}m≥1and{bRm}m≥1are uniformly(with respect to Rm)bounded in L∞(ℝ+;ℍ1(ℝ3)).Then,using the argument of[2,Theorem A.6],we can let Rm→∞and prove that the limiting function of the sequence{(uRm,bRm)}m≥1is a weak solution of equations(1.1)–(1.4).The uniqueness of the weak solutions can be proved by the similar way as[15,Theorem A.8].The regularity of the weak solutions is established through an inductive argument relying on energy methods(one can refer to[2,Theorem A.8]for a similar discussion on the viscous Camassa-Holm equations).

3　Upper Bound of the Time Decay Rate

This section is devoted to the proof of an upper bound of the time decay rate for the weak solutions.We first define the Fourier transform of an element u∈S′(ℝ3)as

For the Fourier transform of the weak solutions,we have the following estimate.

Lemma 3.1Assume uo,bo∈L1(ℝ3)∩ℍ1(ℝ3)and∇·uo=∇·bo=o.Let(u,b)be the solution of equations(1.1)–(1.4)with initial data(uo,bo),then

ProofApplying the Fourier transform to equations(1.1)and(1.2),we get

(3.3)and the formula of constant variation give

Since∇·u=∇·b=o,we have(u·∇)u=div(u⊗u)and(b·∇)b=div(b⊗b).Then

Now taking the divergence of equation(1.1),we obtain

Applying Fourier transform to(3.1o)yields

It follows from(3.8)–(3.9)and(3.11)that

We get(3.1)from(3.7)and(3.12).

Also since∇·u=∇·b=o,we have(u·∇)b=div(u⊗b)and(b·∇)u=div(b⊗u).Similar to(3.8),(3.9)and(3.12),we have

Applying the formula of constant variation to(3.4)and then using(3.13)–(3.15),we obtain (3.2).The proof is completed.□

We next use the Fourier splitting method to derive the ℍ1time decay rate of the weak solutions when the initial data belong to ℍ1(ℝ3)∩L1(ℝ3).We want to point out that although the proof of the following Theorem 3.1(also Theorem 3.2)is only formal,we can make it rigorous by working on the approximate solutions.See Remark 3.1(i)for details.

Theorem 3.1Let uo,bo∈L1(ℝ3)∩ℍ1(ℝ3),∇·uo=∇·bo=o,v>α2andµ>β2.Let (u,b)be the solution of equations(1.1)–(1.4)with initial data(uo,bo),then

where the constant c depends essentially on α,β,µ,v,

ProofTaking the inner product of equation(1.1)with u,and(1.2)with b in L2(ℝ3), respectively,and sum the resulting equalities up,we get

Applying the Plancherel’s theorem to(3.17)yields

where g(t)is a differentiable function of t satisfying

Now,multiplying(3.18)by g(t)gives

According to the definitions ofwe have

It then follows from(3.2o)and(3.21)that

Integrating(3.22)gives

Next,we estimate I1and I2(t).Denote

Note g(o)=1.Employing the Plancherel’s theorem,we get

To estimate I2(t),we now choose g(t)=(ln(e+t))3,t≥o.Then g(t)satisfies the conditions in(3.19).In fact,since v>α2andµ>β2,it is clear that

Also,with this g(t),we have

Set c1:=max.Then for any ξ∈Bξ(τ),we have

Using(3.1),(3.2)and the integration transform of sphere coordinates,we get

where we have also used the fact that

Combining(3.23)–(3.24)and(3.31),and using the Plancherel’s theorem,we get

Now we choose another g(t)=(1+t)2for t≥o,which also satisfies the conditions in(3.19). With this g(t),we have

Set c2=

Similarly to(3.31),we get

Note we have proved in(3.32)that

Hence,(3.23),(3.24)and(3.33)give for any t>o that

then(3.34)implies

By direct computations,

Thus,applying Gronwall inequality to(3.35)yields

Now,by(3.17)we get for any t>o that

By direct computations,

Consequently,(3.38)–(3.4o)give

The proof is completed.

We want to remark that it seem to hard to remove the restrict condition α2/v<1,β2/µ<1, although one can choose some more adequate function g(t)in the above proof.In fact,the ratio α2/v is a characteristic relaxation time scales of the viscoelastic material(see[11]).When α2/v<1/7 and β2/v<1/7,we can prove the higher order decay rate of the regular solutions with initial value in ℍ2(ℝ3)∩L1(ℝ3).The main arguments we will use are also the Fourier splitting method and the Gronwall inequality.

Lemma 3.2Assume uo,bo∈L1(ℝ3)∩ℍ1(ℝ3)and∇·uo=∇·bo=o=o.Let(u,b)be the solution of equations(1.1)–(1.4)with initial data(uo,bo),then

ProofBy Young’s inequality,we have

Taking the divergence of equation

Using(3.5)–(3.7),(3.41)–(3.43)and Schwartz inequality,we obtain

The proof is completed.

Theorem 3.2Let(u,b)be the solution of equations(1.1)–(1.4)corresponding to the initial data uo,bo∈ℍ2(ℝ3)∩L1(ℝ3)and∇·uo=∇·bo=o.Assume that α2/v<1/7,β2/v< 1/7.Then the solution(u,b)satisfies the time decay rate

for large t,where c depends essentially on α,β,v,µ,‖uo‖1,‖bo‖1,‖uo‖2,2and‖bo‖2,2.

ProofTaking the inner product of equation(1.1)with−Δu,and(1.2)with−Δb in L2, respectively,using the facts∇·Δu=∇·Δb=o,and then the H¨older and Gagliardo-Nirenberg inequalities,we get

We now choose t large enough such that c(1+t)−3/4

By the Plancherel’s Theorem,

We now denote by

and let B(ρ)=B(1)(ρ)∩B(2)(ρ)be the ball with radius ρ=zero,where f is a positive increasing function which will be specified to satisfy

Since ρ2·max{α2,β2}≤1,we get from(3.5o)–(3.52)that

Then Theorem 3.1,Lemma 3.2 and(3.53)give

Note that α2/v<1/7 and β2/µ<1/7.Then we choose f(t)=(1+t)3/2+2which satisfies (3.51)for any t>o.With this f(t),we have

Also,using this f(t)as an integrating factor,we get by(3.54)that

Letting r>t,integrating(3.55)for t over[o,r]and using the Lebesgue’s dominated convergence theorem and the Plancherel’s theorem,we obtain

Then the Gronwall inequality shows

Note δ=min{2v/α2,2µ/β2}.So(3.49)and(3.56)give

Gronwall inequality and(3.57)give

By direct computations,

Consequently,(3.58)–(3.6o)give

The proof is completed.

Remark 3.1The proofs of Theorems 3.1 and 3.2 are only formal.Because the integrations we used there might not be rigorous.However,one can first work on the approximate solutions and use the similar arguments of Theorems 3.1 and 3.2 to establish that the approximate solutions have the same decay rate as in Theorems 3.1 and 3.2.Then we can show that these approximate solutions converge strongly in L2(ℝ3×ℝ+)to a weak solution of equations (1.1)–(1.4),which makes the proofs of Theorems 3.1 and 3.2 rigorous.We refer to Schonbek [18–2o]for a similar argument for the 3D NS equations.

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∗April 20,2015;revised January 8,2016.Supported by NSFC(11271290)and GSPT of Zhejiang Province(2014R424062).†

Caidi ZHAO.