Convergence of a Singular Euler-Maxwell Approximation of the Incompressible Euler Equations

Abstract

This paper studies the Euler-Maxwell system which is a model of a collisionless plasma. By energy estimation and the curl-div decomposition of the gradient, we rigorously justify a singular approximation of the incompressible Euler equations via a quasi-neutral regime.

1. Introduction

We study a model of a collisionless plasma where the ions are supposed to be at rest and create a neutralizing background field. The dynamics of the compressible electrons for plasma physics in a uniform background of nonmoving ions with fixed unit density obey the (scaled) one-fluid Euler-Maxwell system which takes the form [1]𝜕𝜕_𝑡𝑛+div(𝑛𝑢)=0,(1.1)_𝑡𝛾𝜆₂𝑢+(𝑢⋅∇)𝑢=−(𝐸+𝛾𝑢×𝐵),(1.2)𝜕_𝑡𝐸𝜆−∇×𝐵=𝛾𝑛𝑢,𝛾𝜕_𝑡𝐵+∇×𝐸=0,(1.3)₂div𝐸=1−𝑛,div𝐵=0,(1.4) for (𝑥,𝑡)∈𝒯³×[0,𝑇], subject to initial conditions:(𝑛,𝑢,𝐸,𝐵)(𝑥,0)=𝑛₀,𝑢₀,𝐸₀,𝐵₀(𝑥).(1.5) Here, 𝑛,𝑢 denote the scaled macroscopic density and mean velocity vector of the electrons and 𝐸,𝐵 the scaled electric field and magnetic field. They are functions of a three-dimensional position vector 𝑥∈𝒯³ and of the time 𝑡>0, where 𝒯³=(ℝ/2𝜋ℤ)³ is the 3-dimensional torus. The fields 𝐸 and 𝐵 are coupled to the particles through the Maxwell equations and act on the particles via the Lorentz force 𝐸+𝛾𝑢×𝐵. In the system (1.1)–(1.4), 𝑗=𝑛𝑢 stand for the current densities for the particle. Equations (1.1)-(1.2) are the mass and momentum balance laws, respectively, while (1.3)-(1.4) are the Maxwell equations.

The dimensionless parameters 𝜆>0 and 𝛾>0 can be chosen independently on each other, according to the desired scaling. Physically, 𝜆 stands for the scaled Debye length, 𝛾 can be chosen to be proportional 1/𝑐, where 𝑐 is the speed of light. These parameters are small compared to the characteristic length of physical interest. Thus, regarding 𝜆 and 𝛾 as singular perturbation parameters, we can study the limits problem in the system (1.1)–(1.4) as these parameters tend to zero. Thus, the limit 𝜆→0 is called the quasineutral limit while the limit 𝛾→0 is called the nonrelativistic limit.

Note that the systems are pressureless, and the only force is due to Lorentz force 𝐸+𝛾𝑢×𝐵. The energy of systems (1.1)–(1.4) is given by 𝕰^𝜆=12_𝒯³𝑛|𝑢|²+𝜆²||𝐸||²+||𝐵||²𝑑𝑥.(1.6)

In the present paper, we will consider the combined quasineutral and nonrelativistic limit of Euler-Maxwell system in the following scaling case: 𝛾=𝜆²⟶0.(1.7)

Passing to the limits when 𝜆→0 and 𝛾→0 go to zero, it is easy to obtain, at least at a very formal level, the incompressible Euler equations of ideal fluids: 𝜕div𝑢⁰=0,𝑥∈𝒯³,𝑡>0,(1.8)_𝑡𝑢⁰+𝑢⁰⋅𝑢∇𝑢⁰+∇𝜙⁰=0,𝑥∈𝒯³,𝑡>0,(1.9)₀(𝑡=0)=𝑢⁰⁰,𝑥∈𝒯³.(1.10)

In these limits the plasma is expected to behave like an incompressible fluid, therefore governed by the incompressible Euler equation. The main aim of this note is to give a rigorous justification to this formal computation in the present paper.

Note that the above limit had been proven in [2] by an analysis of asymptotic expansions and a careful use of Modulated energy technique. In this paper, the case that the plasma is collisionless is considered. By energy estimation and the curl-div decomposition of the gradient, we obtain a convergence to the incompressible Euler system in 𝐻^𝑠 norm for any 𝑠 large enough under the assumption that the initial data are well prepared.

There have been a lot of interesting results about the topic on the quasineutral limit, for the readers to see [3–8] for isentropic Euler-Poisson equations and [9, 10] for nonisentropic Euler-Poisson equations and therein references. For example, by using the method of formal asymptotic expansions, Peng et al. in [9] studied the quasineutral limit for Cauchy problems of multidimensional nonisentropic Euler-Poisson equations for plasmas or semiconductors with prepared initial data. Li in [10] justified the convergence of the nonisentropic Euler-Poisson equation to the incompressible nonisentropic Euler type equation via the quasineutral limit and proved the local existence of smooth solutions to the limit equations by an iterative scheme. The Euler-Maxwell equations are more intricate than the Euler-Poisson equations for the complicated coupling of the Lorentz force. So there have been less studies on the Euler-Maxwell equations than the study on the Euler-Poisson equations. See [3, 6–8, 10–16] and the references therein. The first rigorously study of the Euler-Maxwell equations with extrarelaxation terms is due to Chen et al. [17], where a global existence result to weak solutions in one-dimensional case is established by the fractional step Godunov scheme together with a compensated compactness argument. Jerome [18] establishes a local smooth solution theory for the Cauchy problem of compressible Hydrodynamic-Maxwell systems via a modification of the classical semigroup-resolvent approach of Kato. Paper [19] has just been studied for the convergence of one-fluid isentropic Euler-Maxwell system to compressible Euler-Poisson system via the nonrelativistic limit.

We split the rest of the paper in two sections: the first give the main result of this paper, and the second is devoted to justify the convergence of Euler-Maxwell equations to incompressible Euler equations by using the the curl-div decomposition of the gradient and the 𝜆-weighted energy method.

1.1. Notations and Preliminary Results

Now we introduce some important notations and preliminary results that we will use in this paper as follows.(1)Throughout this paper, ∇=∇_𝑥 is the gradient, 𝛼=(𝛼₁,…,𝛼_𝑑) and 𝛽, and so forth are multi-index, and 𝐻^𝑠(𝒯³) denotes the standard Sobolev space in torus 𝒯³, which is defined by Fourier transform, namely, 𝑓∈𝐻^𝑠(𝒯³) if and only if ‖𝑓‖^2𝑠=(2𝜋)^𝑑_𝑘_∈𝒵𝑑1+||𝑘||²^𝑠||(ℱ𝑓)(𝑘)||²<+∞,(1.11)where (ℱ𝑓)(𝑘)=∫_𝒯3𝑓(𝑥)𝑒^{−𝑖𝑘𝑥}𝑑𝑥 is the Fourier transform of 𝑓∈𝐻^𝑠(𝒯³). Noting that if ∫_𝒯3𝑓(𝑥)𝑑𝑥=0, then ‖𝑓‖_𝐿2(𝒯³₎≤‖∇𝑓‖_𝐿2(𝒯³₎.(2) The following vector analysis formulas will be repeatedly used, see [1]: 𝑓⋅∇𝑔=(∇×d𝑔i)v×(𝑓𝑓×+𝑔∇)(=𝑓∇⋅𝑔×)−𝑓⋅∇𝑔𝑓−⋅𝑔∇,×∇𝑔𝑓⋅𝑓⋅,𝑔=₃(1.12)_𝑗=1𝑓⋅∇𝑓=(∇×𝑓)×𝑓+∇||𝑓||₂∇𝑓_𝑗𝑔_𝑗,(1.13)∇×(𝑓×𝑔)=𝑓div𝑔−𝑔div𝑓+(𝑔⋅∇2)𝑓−,(𝑓⋅∇)𝑔.((11..1145))

2. The Main Result

First, we recall the classical result on the existence of sufficiently regular solutions of the incompressible Euler equations (see [20]).

Proposition 2.1. Let 𝑢⁰⁰ be a divergence-free vector field on 𝒯³ and satisfy 𝑢⁰⁰∈𝐻^𝑠+2, 𝑠>3/2+2. Then there exist 0<𝑇≤∞, the maximal existence time, and a unique smooth solution (𝑢⁰,𝜙⁰) of the incompressible Euler equation (1.8)-(1.9) on [0,𝑇) satisfying that ∫_𝒯3𝜙⁰(𝑡,𝑥)𝑑𝑥=0, and, for any 𝑇₀<𝑇, the inequality sup_{0≤𝑡≤𝑇0}‖‖𝑢⁰‖‖_𝐻𝑠+2+‖‖𝜕_𝑡𝑢⁰‖‖_𝐻𝑠+1+‖‖∇𝜙⁰‖‖_𝐻𝑠+2+‖‖𝜕_𝑡∇𝜙⁰‖‖_𝐻𝑠+1≤𝐶𝑇₀(2.1) holds for some positive constant 𝐶(𝑇₀).

We will prove the following result.

Theorem 2.2. Let 𝑠≥3/2+2 with 𝑠∈ℕ. Assume that 𝑛₀=1,𝑢₀=𝑢⁰⁰+𝜆𝑢¹⁰ with div𝑢⁰⁰=0,𝑢⁰⁰∈𝐻^𝑠+2,𝑢¹⁰∈𝐻^𝑠,𝑠>3/2+2,‖𝑢¹⁰‖_𝑠≤𝑀₁ uniformly in 𝜆. Let 𝑇 be the maximal existence time of smooth solution (𝑢⁰,𝜙⁰) of the incompressible Euler equation (1.8)–(1.10). Then for any 𝑇₀<𝑇, there exist constants 𝜆₀(𝑇₀) and 𝑀(𝑇₀), depending only on 𝑇₀ and the initial data, such that Euler-Maxwell system (1.1)–(1.5) has a classical smooth solution (𝑛,𝑢,𝐸,𝐵), defined on [0,𝑇₀], satisfying ‖‖𝜆⁻²(𝑛−1)(𝑡,⋅)‖‖_𝑠−1+‖‖𝜆⁻¹𝑢−𝑢⁰(𝑡,⋅)‖‖_𝑠+𝜆²‖‖𝐸−∇𝜙⁰(𝑡,⋅)‖‖_𝑠+𝜆‖𝐵(𝑡,⋅)‖_𝑠≤𝑀𝑇₀,(2.2) for all 0<𝜆≤𝜆₀ and 0≤𝑡≤𝑇₀.

3. The Proof of Theorem 2.2

Set (𝑛,𝑢,𝐸,𝐵) to be the classical solutions to the problem (1.1)–(1.5), and set(𝑛,𝑢,𝐸,𝐵)=1+𝜆²𝑛¹−𝜆²Δ𝜙⁰,𝑢⁰+𝜆𝑢¹,∇𝜙⁰+𝐸¹,𝐵¹.(3.1) Obviously the vector (𝑛¹,𝑢¹,𝐸¹,𝐵¹) solves the error system 𝜕_𝑡𝑛¹+𝑢⋅∇𝑛¹+div𝑢¹𝜆+𝜆𝑛¹𝜕_𝑡−Δ𝜙⁰div𝑢¹=𝜕_𝑡+𝑢⋅∇Δ𝜙⁰,𝑢¹+(𝑢⋅∇)𝑢¹=−𝑢¹⋅∇𝑢⁰−𝐸¹𝜆−𝜆𝑢×𝐵¹𝜆₄,𝜕_𝑡𝐸¹−𝜆∇₂×𝐵¹=𝜆²𝑛𝑢−𝜆⁴𝜕_𝑡∇𝜙⁰,div𝐸¹=−𝑛¹,𝑛𝜕_𝑡𝐵¹+∇×𝐸¹=0,div𝐵¹=0,₁,𝑢¹,𝐸¹,𝐵¹||_𝑡=0=𝑛¹⁰,𝑢¹⁰,𝐸¹⁰,𝐵¹⁰(𝑥),(3.2) for 𝑥∈𝒯³,𝑡>0 and with the initial comparable condition div𝐸¹⁰=−𝑛¹⁰,div𝐵¹⁰=0.(3.3)

3.1. Reformulation of the System (3.2) with New Unknowns

Now we make the following change of unknowns: 𝑑¹,𝜔¹=div𝑢¹,∇×𝑢¹,(3.4) then one getsdiv𝑢=𝜆𝑑¹,∇×𝑢=𝜔=𝜔⁰+𝜆𝜔¹,(3.5) with 𝜔⁰=∇×𝑢⁰.

Taking the divergence of the second equation in (3.2) and noting that div𝐸¹=−𝑛¹, one gets 𝜕_𝑡𝑑¹+𝑢⋅∇𝑑¹−𝑛¹𝜆=−2𝐷𝑢⁰∶𝐷𝑢¹+𝜆𝐷𝑢¹∶𝐷𝑢¹−𝜆div𝑢×𝐵¹.(3.6)

Taking the curl of the second equation in (3.2), with the aid of the vector analysis formulas given in Section 1, by a direction calculation, we have 𝜕_𝑡∇×𝑢¹+∇×(𝑢⋅∇)𝑢¹+∇×𝑢¹⋅∇𝑢⁰=𝜆²𝜕_𝑡𝐵¹−𝜆∇×𝑢×𝐵¹.(3.7) Here we have used the error equation 𝜆²𝜕_𝑡𝐵¹+∇×𝐸¹=0 of the magnetic field in the error system (3.2). Since div𝐵¹=0, there exists a vector function 𝑏¹ such that𝐵¹=∇×𝑏¹.(3.8) Then, we have 𝜕_𝑡∇×𝑢¹−𝜆𝑏¹+∇×∇×𝑢¹−𝜆𝑏¹×𝑢=∇×∇𝑢⁰⋅𝑢¹−∇×𝑢¹⋅∇𝑢⁰.(3.9)

Next introduce the general vorticityΩ¹=𝜔¹−𝜆∇×𝑏¹=∇×𝑢¹−𝜆𝑏¹.(3.10) By using the vector analysis formulas (1.15), then it follows from the above that Ω¹ satisfies the following vorticity equation:𝜕_𝑡Ω¹+(𝑢⋅∇)Ω¹−Ω¹⋅∇𝑢−𝜆𝑑¹Ω¹−∇×∇𝑢⁰⋅𝑢¹+∇×∇𝑢¹⋅𝑢⁰=0.(3.11)

Therefore, we can rewrite the error system (3.2) as follows: 𝜕_𝑡𝑛¹+𝑢⋅∇𝑛¹+𝑑¹𝜆=𝜕^𝑡𝜕+𝑢⋅∇Δ𝜙⁰−𝜆𝑛¹−Δ𝜙⁰𝑑¹,(3.12)_𝑡𝑑¹+𝑢⋅∇𝑑¹−𝑛¹𝜆=−2𝐷𝑢⁰𝜕∶𝐷𝑢¹−𝜆𝐷𝑢¹∶𝐷𝑢¹−𝜆div𝑢×𝐵¹,(3.13)_𝑡Ω¹+(𝑢⋅∇)Ω¹𝜆−Ω¹⋅∇𝑢+𝜆𝑑¹Ω¹=∇×∇𝑢⁰⋅𝑢¹−∇×𝑢¹⋅∇𝑢⁰,(3.14)₄𝜕𝜆_𝑡𝐸¹−∇×𝐵¹=𝜆²𝑛𝑢,div𝐸¹=−𝑛¹,(3.15)₂𝑛𝜕_𝑡𝐵¹+∇×𝐸¹=0,div𝐵¹=0,(3.16)₁,𝑢¹,𝐸¹,𝐵¹||_𝑡=0=𝑛¹⁰,𝑢¹⁰,𝐸¹⁰,𝐵¹⁰(𝑥).(3.17)

In order to prove the Theorem 2.2, we introduce the Sobolev norms: ‖‖𝑊_𝜆‖⋅‖=‖⋅‖_𝐿2(𝒯³₎,(𝑡)‖‖_𝑠=‖‖𝑛¹(𝑡)‖‖^2𝑠−1+‖‖𝑢¹(𝑡)‖‖^2𝑠+𝜆²‖‖𝐸¹(𝑡)‖‖^2𝑠+𝜆‖‖𝐵¹(𝑡)‖‖^2𝑠^1/2.(3.18)

Next, one begins to estimate ‖𝑊^𝜆‖_𝑠. Our basic idea is to overcome the singularity which is caused by 𝑑¹/𝜆 and 𝑛¹/𝜆 by using the special structures between the first equation and the the second equation in the system (3.6).

3.2. Energy Estimates

Now we control (‖𝑛¹(𝑡)‖^2𝑠−1+‖𝑑¹(𝑡)‖^2𝑠−1+‖𝜔¹(𝑡)‖^2𝑠−1).

Lemma 3.1. Let 𝑠 be an integer with 𝑠>3/2+2. Then for any 0<𝑡<𝑇 with 𝑇∈(0,𝑇₀], one has ‖‖𝑛¹≤(𝑡𝐶)‖‖‖‖^2𝑠−1𝑛₁+‖‖𝑑¹(𝑡)‖‖^2𝑠−1+‖‖𝜔¹(𝑡)‖‖^2𝑠−1+𝐶_𝑡0(𝑡=0)‖‖^2𝑠−1+‖‖𝑑¹(𝑡=0)‖‖^2𝑠−1+‖‖𝜔¹(𝑡=0)‖‖^2𝑠−11+‖‖𝑊^𝜆(𝜏)‖‖^2𝑠+‖‖𝑊^𝜆(𝜏)‖‖^3𝑠𝑑𝜏.(3.19)

Proof. Let 𝛼∈ℕ³ with |𝛼|≤𝑠−1 and 𝑠>3/2+2. Taking 𝜕^𝛼𝑥 of (3.12) and multiplying the resulting equation by 𝜕^𝛼𝑥𝑛¹, by integration by parts, we have 𝑑𝑑𝑡‖‖𝜕^𝛼𝑥𝑛¹‖‖²=div𝑢𝜕^𝛼𝑥𝑛¹,𝜕^𝛼𝑥𝑛¹+2ℋ^𝛼(1),𝜕^𝛼𝑥𝑛¹−2𝜆𝜕^𝛼𝑥+2𝜕_𝛼𝑥𝑑¹,𝜕^𝛼𝑥𝑛¹𝜕_𝑡+𝑢⋅∇Δ𝜙⁰−𝜆𝑛¹−Δ𝜙⁰𝑑¹,𝜕^𝛼𝑥𝑛¹,(3.20) where the commutator ℋ^𝛼(1)=−𝜕^𝛼𝑥(𝑢⋅∇)𝑛¹−𝑢⋅𝜕^𝛼𝑥∇𝑛¹,(3.21) which can be estimated as follows: ‖‖ℋ^𝛼(1)‖‖=≤𝐶𝐶‖‖∇∇𝑢𝑢‖‖_𝐿^∞_𝑠−1‖‖𝜕^𝑥𝑠−2∇𝑛¹‖‖+𝐶‖‖∇𝑛¹‖‖_𝐿∞‖‖𝜕^𝑥𝑠−1𝑢‖‖≤𝐶1+𝜆‖‖𝑊‖‖𝜕^𝑥𝑠−2_𝜆∇𝑛¹‖‖+𝐶‖‖∇𝑛¹‖‖_𝑠−2‖‖𝜕^𝑥𝑠−1𝑢‖‖(𝑡)‖‖_𝑠‖‖𝑊^𝜆(𝑡)‖‖_𝑠.(3.22) Here we used the basic Moser-type calculus inequalities [21, 22]: ‖‖𝐷^𝛼𝑥(𝑓𝑔)−𝑓𝐷^𝛼𝑥𝑔‖‖_𝐿2≤𝐶_𝑠‖‖𝐷_𝑥𝑓‖‖_𝐿∞‖‖𝐷^𝑥𝑠−1𝑔‖‖_𝐿2+‖𝑔‖_𝐿^∞‖‖𝐷^𝑠𝑥𝑓‖‖_𝐿2,𝑠≥1.(3.23) Hence, by Cauchy-Schwartz's inequality and the Sobolev lemma, noting div𝑢=𝜆𝑑¹, one gets 2di𝜕v_𝛼𝑥𝑢𝜕^𝛼𝑥𝑛¹,𝜕^𝛼𝑥𝑛¹+2ℋ^𝛼(1),𝜕^𝛼𝑥𝑛^𝜆≤𝐶1+‖‖𝑊^𝜆(𝑡)‖‖^2𝑠+𝜆‖‖𝑊^𝜆(𝑡)‖‖^3𝑠,𝜕_𝑡+𝑢⋅∇Δ𝜙⁰−𝜆𝑛¹−Δ𝜙⁰𝑑¹,𝜕^𝛼𝑥𝑛¹≤𝐶1+‖‖𝑊^𝜆(𝑡)‖‖^2𝑠.(3.24) Combining (3.20) with (3.24), one obtains 𝑑𝑑𝑡‖‖𝜕^𝛼𝑥𝑛¹‖‖²≤𝐶1+‖‖𝑊^𝜆(𝑡)‖‖^2𝑠+𝜆‖‖𝑊^𝜆(𝑡)‖‖^3𝑠−2𝜆𝜕^𝛼𝑥𝑑¹,𝜕^𝛼𝑥𝑛¹.(3.25)
Next, Taking 𝜕^𝛼𝑥 of (3.13) and multiplying the resulting equation by 𝜕^𝛼𝑥𝑑¹, by integration by parts, we have 𝑑𝑑𝑡‖‖𝜕^𝛼𝑥𝑑¹‖‖²=div𝑢𝜕^𝛼𝑥𝑑¹,𝜕^𝛼𝑥𝑑¹+2ℋ^𝛼(2),𝜕^𝛼𝑥𝑑¹+2𝜆𝜕^𝛼𝑥−2𝜕_𝛼𝑥𝑑¹,𝜕^𝛼𝑥𝑛¹2𝐷𝑢⁰∶𝐷𝑢¹+𝜆𝐷𝑢¹∶𝐷𝑢¹,𝜕^𝛼𝑥𝑑¹+𝜆𝜕^𝛼𝑥∇×𝑢×𝐵¹,𝜕^𝛼𝑥𝑑¹,(3.26) where the commutator ℋ^𝛼(2)=−𝜕^𝛼𝑥(𝑢⋅∇)𝑢¹−𝑢⋅𝜕^𝛼𝑥∇𝑢¹,(3.27) which can be estimated by ‖‖ℋ^𝛼(2)‖‖≤=𝐶𝐶‖‖∇∇𝑢𝑢‖‖_{𝐿∞𝑠−1}‖‖𝜕^𝑥𝑠−2∇𝑢¹‖‖+𝐶‖‖∇𝑢¹‖‖_𝐿∞‖‖𝜕^𝑥𝑠−1𝑢‖‖≤𝐶1+𝜆‖‖𝑊‖‖𝜕^𝑥𝑠−2_𝜆∇𝑢¹‖‖+𝐶‖‖∇𝑢¹‖‖_𝑠−2‖‖𝜕^𝑥𝑠−1𝑢‖‖(𝑡)‖‖_𝑠‖‖𝑊^𝜆(𝑡)‖‖_𝑠.(3.28)
For the first, forth, and fifth terms, using the property of the approximate solution of incompressible Euler equations (1.8)–(1.10), Cauchy-Schwarz’s inequality, and the Sobolev lemma, we get −2𝜕_𝛼𝑥div𝑢𝜕^𝛼𝑥𝑑¹,𝜕^𝛼𝑥𝑑¹≤𝐶1+𝜆‖‖𝑊^𝜆(𝑡)‖‖_𝑠‖‖𝑊^𝜆(𝑡)‖‖^2𝑠,𝜆2𝐷𝜕_𝛼𝑥𝑢⁰∶𝐷𝑢¹+𝜆𝐷𝑢¹∶𝐷𝑢¹,𝜕^𝛼𝑥𝑑¹≤𝐶1+𝜆‖‖𝑊^𝜆(𝑡)‖‖_𝑠‖‖𝑊^𝜆(𝑡)‖‖^2𝑠,∇×𝑢×𝐵¹,𝜕^𝛼𝑥𝑑¹≤𝐶1+𝜆‖‖𝑊^𝜆(𝑡)‖‖_𝑠‖‖𝑊^𝜆(𝑡)‖‖^2𝑠.(3.29)
For the second term, using the estimate (3.28) for ℋ^𝛼(2), we obtain 2ℋ^𝛼(2),𝜕^𝛼𝑥𝑑¹≤𝐶1+𝜆‖‖𝑊^𝜆(𝑡)‖‖_𝑠‖‖𝑊^𝜆(𝑡)‖‖^2𝑠.(3.30)
Combining (3.26) with (3.29)–(3.30), we have 𝑑𝑑𝑡‖‖𝜕^𝛼𝑥𝑑¹‖‖²≤𝐶1+𝜆‖‖𝑊^𝜆(𝑡)‖‖_𝑠‖‖𝑊^𝜆(𝑡)‖‖^2𝑠+2𝜆𝜕^𝛼𝑥𝑑¹,𝜕^𝛼𝑥𝑛¹.(3.31)
From (3.25) and (3.31), we can get 𝑑𝑑𝑡‖‖𝜕^𝛼𝑥𝑛¹‖‖²+‖‖𝜕^𝛼𝑥𝑑¹‖‖²≤𝐶1+‖‖𝑊^𝜆(𝑡)‖‖^2𝑠+𝜆‖‖𝑊^𝜆(𝑡)‖‖^3𝑠,(3.32) which leads to ‖‖𝑛¹(𝑡)‖‖^2𝑠−1+‖‖𝑑¹(𝑡)‖‖^2𝑠−1≤+𝐶𝐶‖‖_𝑡0𝑛¹(𝑡=0)‖‖^2𝑠−1+‖‖𝑑¹(𝑡=0)‖‖^2𝑠−11+‖‖𝑊^𝜆(𝜏)‖‖^2𝑠+𝜆‖‖𝑊^𝜆(𝜏)‖‖^3𝑠𝑑𝜏.(3.33)
Taking Taking 𝜕^𝛼𝑥 on (3.14) and multiplying the resulting equation by 𝜕^𝛼𝑥, by integration by parts, we have 𝑑𝑑𝑡‖‖𝜕^𝛼𝑥Ω¹‖‖²=+d2iv𝜕𝑢_𝛼𝑥𝜕^𝛼𝑥Ω¹,𝜕^𝛼𝑥Ω¹−2𝜕^𝛼𝑥Ω¹⋅∇𝑢−𝜆𝑑¹Ω¹,𝜕^𝛼𝑥Ω¹∇×∇𝑢⁰⋅𝑢¹−∇×𝑢¹⋅∇𝑢⁰,𝜕^𝛼𝑥Ω¹+2ℋ^𝛼(3),𝜕^𝛼𝑥Ω¹,(3.34) where the commutator ℋ^𝛼(3)=−𝜕^𝛼𝑥(𝑢⋅∇)Ω¹−𝑢⋅𝜕^𝛼𝑥∇Ω¹,(3.35) which can be estimated by ‖‖ℋ^𝛼(3)‖‖=≤𝐶𝐶‖‖∇∇𝑢𝑢‖‖_{𝐿∞𝑠−1}‖‖𝜕^𝑥𝑠−2∇Ω¹‖‖+𝐶‖‖∇𝑢¹‖‖_𝐿∞‖‖𝜕^𝑥𝑠−1Ω‖‖≤𝐶1+𝜆‖‖𝑊‖‖𝜕^𝑥𝑠−2_𝜆∇Ω¹‖‖+𝐶‖‖∇Ω¹‖‖_𝑠−2‖‖𝜕^𝑥𝑠−1𝑢‖‖(𝑡)‖‖_𝑠‖‖Ω¹(𝑡)‖‖_𝑠−1.(3.36) Using the estimates (3.36) for ℋ^𝛼(3), we have, with the aid of Cauchy-Schwarz's inequality and the Sobolev lemma, −2𝜕_𝛼𝑥div𝑢𝜕^𝛼𝑥Ω¹,𝜕^𝛼𝑥Ω¹≤𝐶1+𝜆‖‖𝑊^𝜆(𝑡)‖‖_𝑠‖‖Ω¹(𝑡)‖‖^2𝑠−1,2𝜕_𝛼𝑥Ω¹⋅∇𝑢−𝜆𝑑¹Ω¹,𝜕^𝛼𝑥Ω¹≤𝐶1+𝜆‖‖𝑊^𝜆(𝑡)‖‖_𝑠‖‖Ω¹(𝑡)‖‖^2𝑠−1,∇×∇𝑢⁰2⋅𝑢ℋ¹−_𝛼(3)∇×𝑢¹⋅∇𝑢⁰,𝜕^𝛼𝑥Ω¹≤𝐶1+‖‖𝑊^𝜆(𝑡)‖‖_𝑠‖‖Ω¹(𝑡)‖‖^2𝑠−1,,𝜕^𝛼𝑥Ω¹≤𝐶1+𝜆‖‖𝑊^𝜆(𝑡)‖‖_𝑠‖‖Ω¹(𝑡)‖‖^2𝑠−1.(3.37) Combining (3.34) with (3.37), we get 𝑑𝑑𝑡‖‖𝜕^𝛼𝑥Ω¹‖‖²≤𝐶1+‖‖𝑊^𝜆(𝑡)‖‖_𝑠‖‖Ω¹(𝑡)‖‖^2𝑠−1,(3.38) which yields, for any 0<𝑡<𝑇, ‖‖Ω¹‖‖^2𝑠−1≤‖‖Ω¹(𝑡=0)‖‖^2𝑠−1+𝐶^𝑡01+‖‖𝑊^𝜆(𝜏)‖‖_𝑠‖‖Ω¹(𝜏)‖‖^2𝑠−1𝑑𝜏.(3.39) Noting the definition of Ω¹ (see (3.10)), we obtain ‖‖‖‖Ω𝜔¹₁‖‖^2𝑠−1≤2‖‖Ω¹‖‖^2𝑠−1+2𝜆²‖‖𝐵¹‖‖^2𝑠−1,‖‖^2𝑠−1≤2‖‖𝜔¹‖‖^2𝑠−1+2𝜆²‖‖𝐵¹‖‖^2𝑠−1.(3.40) Then (3.39)–(3.40) give the estimate ‖‖𝜔¹‖‖^2𝑠−1≤𝐶‖‖Ω¹(𝑡=0)‖‖^2𝑠−1+𝜆²‖‖𝐵¹(𝑡=0)‖‖^2𝑠−1+𝐶𝜆²‖‖𝐵¹‖‖^2𝑠−1+𝐶^𝑡01+‖‖𝑊^𝜆(𝜏)‖‖_𝑠‖‖𝜔¹(𝜏)‖‖^2𝑠−1+𝜆²‖‖𝐵¹‖‖^2𝑠−1𝑑𝜏.(3.41)
So, from (3.33) and (3.41) we complete the proof of Lemma 3.2.

Next, we obtain the high order energy estimates on the electric-magnetic field as follows.

Lemma 3.2. Let 𝑠 be an integer with 𝑠>3/2+2. Then for any 0<𝑡<𝑇 with 𝑇∈(0,𝑇₀], one has 𝜆⁴‖‖𝐸¹(𝑡)‖‖^2𝑠+𝜆²‖‖𝐵¹(𝑡)‖‖^2𝑠≤𝐶𝜆⁴‖‖𝐸¹(𝑡=0)‖‖^2𝑠+𝜆²‖‖𝐵¹(𝑡=0)‖‖^2𝑠+𝐶_𝑡01+‖‖𝑊^𝜆(𝜏)‖‖^2𝑠+‖‖𝑊^𝜆(𝜏)‖‖^3𝑠𝑑𝜏.(3.42)

Proof. Let 𝛽∈ℕ³ with |𝛽|≤𝑠 and 𝑠>3/2+2. Taking 𝜕^𝛽𝑥 of (3.15) and then taking the 𝐿² inner product of the resulting equation with 𝜕^𝛽𝑥𝐸¹, and taking 𝜕^𝛽𝑥 of (3.16) and then taking the 𝐿² inner product of the resulting equation with 𝜕^𝛽𝑥𝐵¹, and summing them up, we get 𝑑𝑑𝑡𝜆⁴‖‖𝜕^𝛽𝑥𝐸¹‖‖²+𝜆²‖‖𝜕^𝛽𝑥𝐵¹‖‖²=2𝜆₂≤𝐶1𝜕+^𝛽𝑥‖‖𝑊1_𝜆−𝜆²Δ𝜙⁰𝑢,𝜕^𝛽𝑥𝐸¹+2𝜆⁴𝜕^𝛽𝑥𝑛¹𝑢,𝜕^𝛽𝑥𝐸¹(𝑡)‖‖^2𝑠−2𝜆⁴𝑢𝜕^𝛽𝑥div𝐸¹,𝜕^𝛽𝑥𝐸¹−2𝜆⁴ℋ^𝛽,𝜕^𝛽𝑥𝐸¹,(3.43) where the commutator is defined by ℋ_𝛽=𝜕^𝛽𝑥𝑢div𝐸¹−𝑢𝜕^𝛽𝑥div𝐸¹,(3.44) which can be estimated as follows 𝐻_𝛽≤≤𝐶𝐶‖‖∇∇𝑢𝑢‖‖_𝐿^∞_𝑠−1‖‖𝜕^𝑥𝑠−1div𝐸¹‖‖+‖‖div𝐸¹‖‖_𝐿∞‖‖𝜕^𝑠𝑥𝑢‖‖=𝐶1+𝜆‖‖𝑊‖‖_𝜆𝜕^𝑥𝑠−1div𝐸¹‖‖+‖‖div𝐸¹‖‖_𝑠−1‖‖𝜕^𝑠𝑥𝑢‖‖(𝑡)‖‖_𝑠‖‖𝐸¹‖‖_𝑠.(3.45) Here we have used Sobolev lemma, 𝑠>3/2+2 and the vector analysis formula (1.12).
With the aid of H𝑜̈lder's inequality, we have, using the estimate (3.45) of the commutator ℋ_𝛽, that −2𝜆⁴ℋ^𝛽,𝜕^𝛽𝑥𝐸¹≤𝐶1+𝜆‖‖𝑊^𝜆(𝑡)‖‖_𝑠‖‖𝑊^𝜆(𝑡)‖‖^2𝑠.(3.46) By integration by parts and Sobolev lemma, −2𝜆⁴𝑢𝜕^𝛽𝑥div𝐸¹,𝜕^𝛽𝑥𝐸¹≤𝐶1+𝜆‖‖𝑊^𝜆(𝑡)‖‖_𝑠‖‖𝑊^𝜆(𝑡)‖‖^2𝑠.(3.47) Combining (3.43) with (3.46) and (3.47), one gets 𝑑𝑑𝑡𝜆⁴‖‖𝜕^𝛽𝑥𝐸¹(𝑡)‖‖²+𝜆²‖‖𝜕^𝛽𝑥𝐵¹(𝑡)‖‖²≤𝐶1+‖‖𝑊^𝜆(𝑡)‖‖^2𝑠+𝜆‖‖𝑊^𝜆(𝑡)‖‖^3𝑠,(3.48) which yields (3.42).

3.3. The End of Proof of Theorem 2.2

Now, we introduce an 𝜆-weighted Sobolev type energy function ℰ^𝜆(𝑡)=‖‖𝑊^𝜆(𝑡)‖‖^2𝑠.(3.49) Then it follows from the estimates (3.19), (3.42), and the curl-divergence decomposition inequality ‖‖∇𝑢¹‖‖_𝑠−1≤𝐶‖‖𝑑¹‖‖_𝑠−1+‖‖Ω¹‖‖_𝑠−1(3.50) that there exists an 𝜆₀>0, depending only upon 𝑇₀, such that, for any 0<𝜆≤𝜆₀ and any 0<𝑡<𝑇, ℰ^𝜆(𝑡)≤𝐶ℰ^𝜆(𝑡)(𝑡=0)+𝐶^𝑡01+ℰ^𝜆(𝜏)+ℰ^𝜆(𝜏)^3/2𝑑𝜏.(3.51) Since ℰ^𝜆(𝑡=0)≤𝑀₀ for some positive constant, now applying Gronwall's inequality to (3.51), one can conclude that there exists an 𝜆₀ sufficiently small such that for any 𝜆≤𝜆₀ and 0<𝑡<𝑇, ℰ^𝜆(𝑡)≤𝑀₀,(3.52) which gives the desired priori estimate ||‖‖𝑊^𝜆(𝑡)‖‖||_𝑠,𝑇0≤𝑀𝑇₀.(3.53)

Finally, we can return to the problem (1.1)–(1.4) to conclude our Theorem 2.2. In fact, the assumptions in Theorem 2.2 imply that the estimate (3.53) holds. Thus, it follows from the estimate (3.53) and the transform (3.1) that the estimate (2.2) holds. The proof of Theorem 2.2 is complete.

Acknowledgment

The author is grateful for the support from the National Science Foundation of China (Grant no. 10771099) and the Beijing Science Foundation (Grant no. 1082001).