On non-resistive limit of 1D MHD equations with no vacuum at infinity

Zilai Li; Huaqiao Wang; Yulin Ye

doi:10.1515/anona-2021-0209

Open Access Published by De Gruyter November 29, 2021

On non-resistive limit of 1D MHD equations with no vacuum at infinity

Zilai Li , Huaqiao Wang and Yulin Ye

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2021-0209

Abstract

In this paper, the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity is considered, but the initial vacuum can be permitted inside the region. By deriving a priori ν (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for the compressible resistive MHD equations is also established.

Keywords: 1D compressible MHD equations; Cauchy problem; global strong solutions; non-resistive limit

MSC 2010: 35D35; 35Q35; 76N10; 76W05

1 Introduction and Main Results

Compressible magnetohydrodynamics (MHD) is used to describe the macroscopic behavior of the electrically conducting fluid in a magnetic field. The application of MHD has a very wide of physical objects from liquid metals to cosmic plasmas. The system of the resistive MHD equations has the form:

ρt+(ρu)x=0,(ρu)t+(ρu2+P(ρ)+12b2)x=(μux)x,bt+(ub)x=νbxx, (1.1)

in ℝ × [0, ∞). Here, ρ, u, P(ρ) and b denote the density, velocity, pressure and magnetic field, respectively. μ > 0 is the viscosity coefficient, the constant ν > 0 is the resistivity coefficient acting as the magnetic diffusion coefficient of the magnetic field. In this paper, we consider the isentropic compressible MHD equations in which the equation of the state has the form

P(ρ)=Rργ,γ>1.

For simplicity, we set R = 1. We focus on the initial condition:

(ρ,u,b)|t=0=(ρ0(x),u0(x),b0(x))→(ρ¯,0,b¯) as |x|→+∞, (1.2)

where ρ̄ and b̄ are both non-zero constants. Note that we can always normalize ρ̄ such that ρ̄ ≥ 1.

However, it is well known that the resistivity coefficient ν is inversely proportional to the electrical conductivity, therefore it is more reasonable to ignore the magnetic diffusion which means ν = 0, when the conducting fluid considered is of highly conductivity, for example the ideal conductors. So instead of equations (1.1), when there is no resistivity, the system reduces to the so called compressible, isentropic, viscous and non-resistive MHD equations:

ρ~t+(ρ~u~)x=0,(ρ~u~)t+(ρ~u~2+P(ρ~)+12b~2)x=(μu~x)x,b~t+(u~b~)x=0, (1.3)

in ℝ × [0, +∞), with the following initial condition:

(ρ~,u~,b~)|t=0=(ρ~0,u~0,b~0)→(ρ¯,0,b¯) as |x|→+∞. (1.4)

Because of the tight interaction between the dynamic motion and the magnetic field, the presence of strong nonlinearities, rich phenomena and mathematical challenges, many physicists and mathematicians are attracted to study in this field. Before stating our main theorems, we briefly recall some previous known results on compressible MHD equations. Firstly, we begin with the MHD equations with magnetic diffusion. For one-dimensional case, Vol'pert-Hudjaev [22] proved the local existence and uniqueness of strong solutions to the Cauchy problem and Kawashima-Okada [13] obtained the global smooth solutions with small initial data. For large initial data and the density containing vacuum, Ye-Li [27] proved the global existence of strong solutions to the 1D Cauchy problem with vacuum at infinity. When considering the full MHD equations and the heat conductivity depends on the temperature θ, Chen-Wang [3] studied the free boundary value problem and established the existence, uniqueness and Lipschitz dependence of strong solutions. Recently, Fan-Huang-Li [6] obtained the global strong solutions to the initial boundary value problem to the planner MHD equations with temperature-dependent heat conductivity. Later, with the effect of self-gravitation as well as the influence of radiation on the dynamics at high temperature regimes taken into account, Zhang-Xie [29] obtained the global strong solutions to the initial boundary value problem for the nonlinear planner MHD equations. For multi-dimensional MHD equations, Lv-Shi-Xu [19] considered the 2-D isentropic MHD equations and proved the global existence of classical solutions provided that the initial energy is small, where the decay rates of the solutions were also obtained. Vol'pert-Hudjaev [22] and Fan-Yu [5] obtained the local classical solution to the 3-D compressible MHD equations with the initial density is strictly positive or could contain vacuum, respectively. Hu-Wang [9] derived the global weak solutions to the 3-D compressible MHD equations with large initial data. Recently, Li-Xu-Zhang [14] established the global existence of classical solution of 3-D MHD equations with small energy but possibly large oscillations. Later, the result was improved by Hong-Hou-Peng-Zhu [8] just provided ((γ−1)19+ν−14)E0 is suitably small. When the resistivity is zero, then the magnetic equation is reduced from the heat-type equation to the hyperbolic-type equation, the problem becomes more challenging, hence the results are few. Kawashima [12] obtained the classical solutions to 3-D MHD equations when the initial data are of small perturbations in H³-norm and away from vacuum. Xu-Zhang [23] proved a blow-up criterion of strong solutions for 3-D isentropic MHD equations with vacuum. Fan-Hu [4] established the global strong solutions to the initial boundary value problem of 1-D heat-conducting MHD equations. With more general heat-conductivity, Zhang-Zhao [31] established the global strong solutions and also obtained the non-resistivity limits of the solutions in L²-norm. Li-Jiang [17] obtained the global solutions to the Cauchy problem of 1D heat-conductive MHD equations of viscous non-resistive gas under the frame work of Lagrangian coordinates. Jiang-Zhang [10] obtained the non-resistive limit of the strong solution and the “magnetic boundary layer” estimates to the initial boundary value problem of 1-D isentropic MHD equations as the resistivity ν → 0. Yu [28] obtained the global existence of strong solutions to the initial boundary value problem of 1-D isentropic MHD equations. For the Cauchy problem with large initial data and vacuum, Li-Wang-Ye [16] established the global well-posedness of strong solutions to the 1D isentropic MHD equations with vacuum at infinity, that is ρ̄ = 0. However, for the Cauchy problem with no vacuum at infinity, the global well-posedness of strong solutions and the non-resistive limits when the resistivity coefficient ν → 0 are still unknown. The goal of this paper is trying to answer these problems.

Now we give some comments on the analysis of this paper. The non-resistive limit of global strong solutions to 1D MHD equations (1.1)-(1.2) can be obtained by global uniform a priori estimates which are independent of resistivity ν. Thus, to obtain the a priori ν (resistivity coefficient)-independent estimates, some of the main new difficulties will be encountered due to the initial density and initial magnetic which approach non-zero constants at infinity.

It turns out that the key issue in this paper is to derive upper bound for the density, magnetic field and the time-dependent higher norm estimates which are independent of resistivity ν. This is achieved by modifying upper bound estimate for the density developed in [25] and [27] in the theory of Cauchy problem with vacuum and initial magnetic field approaching zero at infinity. However, in comparison with the Cauchy problem with vacuum at infinity in [25] and [16], some new difficulties will be encountered. The first difficulty lies in no integrability for the density itself just from the elementary energy estimate (see Lemma 3.1), which is required when deriving the upper bound of the density. To overcome this difficulty, we use the technique of mathematical frequency decomposition to divide the momentum ξ into two parts:

ξ=∫−∞xρudy=∫−∞xρ−ρ¯ρudy+ρ¯∫−∞xρudy=ξ1+ξ2.

It is crucial to obtain the upper bound of ξ₁ and ξ₂. For getting ∥ξ₁∥_L^∞, we use the technique of mathematical frequency decomposition to get the estimate of ∥ρ−ρ¯∥L2 by the elementary energy estimates, and then using Hölder's inequality, we can obtain the upper bound of ξ₁ (see (3.30)). For obtaining ∥ξ_{2∥_L^∞}, due to the Sobolev embedding theory, we need some L^p̃ integrability of ξ₂. However, we can not obtain it just from ∥ξ_2x∥_L² directly, because the Poincáre type inequality is no longer valid in the whole space ℝ. To overcome this difficulty, we use the Caffarelli-Kohn-Nirenberg weighted inequality and the Gagliardo-Nirenberg inequality to obtain the upper bound of ξ₂ (see (3.31)). It is worth noting that the non-resistive MHD equation (1.3) looks similar to the compressible model for gas and liquid two-phase fluids. Hence, in some previous results, it is technically assumed that ρ̃, b̃ ≥ 0 and 0 ≤ b~ρ~ < ∞ in (1.3), which implies the magnetic field b̃ is bounded provided the density ρ̃ is bounded. However, this is not physical and realistic in magnetohydrodynamics. Moreover, compared with that for the Cauchy problem with vacuum at infinity in [16], since the magnetic field b̃ → b̄, as |x| → +∞, the method of getting the upper bound of magnetic field b in [16] can not be used here anymore. So another difficulty is how to get the uniform (independent of ν) upper bound of the magnetic field b without the assumption similar as that in two-phase fluids. To overcome this difficulty, we will make full use of the structure of the momentum equation and effective viscous flux (see Lemma 3.4 and Lemma 3.5).

Notations: We denote the material derivative of u and effective viscous flux by

u˙≜ut+uuxandF≜μux−P(ρ)−P(ρ¯)+b2−b¯22,

and define potential energy by

Φ(ρ)=ρ∫ρ¯ρP(s)−P(ρ¯)s2ds=1γ−1ργ−ρ¯γ−γρ¯γ−1(ρ−ρ¯).

Sometimes we write ∫_ℝf(x) dx as ∫ f(x) for simplicity.

Now, we state the main result of this paper.

Theorem 1.1

Suppose that the initial data (ρ₀, u₀, b₀)(x) satisfies

ρ0−ρ¯∈H1(R),b0−b¯∈H1(R),u0∈H2(R),12ρ0u02+Φ(ρ0)+(b0−b¯)22|x|α∈L1(R) (1.5)

for some α ∈ (1, 2], and the compatibility condition

μu0x−P(ρ0)−12b02x=ρ0g(x),x∈R (1.6)

with some g ∈ L²(ℝ). Then for each fixed ν > 0, there exist a positive constant C and a unique global strong solution (ρ, u, b) to the Cauchy problem (1.1)-(1.2) such that

0≤ρ(x,t)≤C,∀(x,t)∈R×[0,T], (1.7)

sup0≤t≤T12ρu2+Φ(ρ)+(b−b¯)22(1+|x|α)L1(t)+∫0Tμux(1+|x|α2)L22+νbx(1+|x|α2)L22dt≤C, (1.8)

and

sup0≤t≤T∥u∥L2+∥uxx∥L2+∥ux∥L2+∥bx∥L2+∥ρx∥L2+∥ρu˙∥L2(t)+∫0T∥uxt∥L22+ν∥bxx∥L22dt≤C. (1.9)

Moreover, as ν → 0, we have

(ρ,u,b)→(ρ~,u~,b~) strongly in L∞(0,T;L2),νbx→0,ux→u~x strongly in L2(0,T;L2), (1.10)

and

sup0≤t≤T∥ρ−ρ~∥L22+∥u−u~∥L22+∥b−b~∥L22+∫0Tμ∥(u−u~)x∥L22dt≤Cν, (1.11)

where C is a positive constant independent of ν.

Remark 1.1

In Theorem 1.1, we do not need the artificial assumption similarly as that in two-phase fluids. Moreover, if ignoring the magnetic field, then MHD system reduces to the compressible Navier-Stokes equations. So, Theorem 1.1 can be seen as an extension of that in [25].

Remark 1.2

In Theorem 1.1, we give that the global strong solution of resistive MHD equation (1.1)-(1.2) converges to that of non-resistive MHD equation (1.3)-(1.4) in L²-norm as ν → 0, moreover, the convergence rates are also justified.

The rest of the paper is organized as follows. In Section 2, we recall some preliminary lemmas which will be used later. Section 3 is devoted to establishing global ν-independent estimates for (1.1) and (1.2), which will be used to justify the non-resistive limit. Section 5 is devoted to proving Theorem 1.1.

2 Preliminaries

In this section, we will recall some known facts and elementary inequalities that will be used frequently later.

Lemma 2.1

(Gagliardo-Nirenberg inequality [7, 20]). For any f ∈ W^1,m(ℝ) ∩ L^r(ℝ), there exists some generic constant C > 0 which may depend on q, r such that

∥f∥Lq≤C∥f∥Lr1−θ∥∇f∥Lmθ, (2.1)

where θ=(1r−1q)(1r−1m+1)−1, if m > 1, then q ∈ [r, ∞].

The following Caffarelli-Kohn-Nirenberg weighted inequality is the key to deal with the Cauchy problem in this paper.

Lemma 2.2

(Caffarelli-Kohn-Nirenberg weighted inequality [2). ∀ h ∈ C0∞ (ℝ), it holds that

∥|x|κh∥r≤C∥|x|α|∂xh|∥pθ∥|x|βh∥q1−θ (2.2)

where 1≤p,q<∞,0<r<∞,0≤θ≤1,1p+α>0,1q+β>0,1r+κ>0 and satisfy

1r+κ=θ1p+α−1+(1−θ)1q+β, (2.3)

and

κ=θσ+(1−θ)β

with 0 ≤ α − σ if θ > 0 and 0 ≤ α − σ ≤ 1 if θ > 0 and 1p+α−1=1r+κ.

Proof

The proof can be found in [2]. Here, we omit the details. □

Remark 2.1

The lemma 2.2 is also valid for any h ∈ Dα,βp,q (ℝ), where Dα,βp,q (ℝ) is the completion of the space of smooth compactly supported functions with the norm ∫R|x|α|∂xh|pdx1p+∫R|x|β|h|qdx1q.

By direct calculation, the potential energy Φ(ρ) has the following properties:

Lemma 2.3

Observing the function of the potential energy Φ(ρ), we will easily find the following properties for positive constants c₁, c₂, C₁, C₂:

if 0 ≤ ρ ≤ 2ρ̄, c₁(ρ-ρ̄)² ≤ Φ(ρ) ≤ c₂(ρ-ρ̄)²;
if ρ > 2ρ̄, ρ^γ−ρ̄^γ ≤ C₁(ρ-ρ̄)^γ ≤ C₂ Φ(ρ).

3 Global ν-independent estimates for (1.1) and (1.2)

The main purpose of this section is to derive the global ν-independent a priori estimates of the solutions (ρ, u, b) to the system (1.1) and (1.2), which is used to justify the non-resistive limit. To do this, before going any further, we first let the initial density have lower bound δ > 0 and get the global ν-independent a priori estimates of the smooth solutions (ρ, u, b), which is uniform of δ. Then taking δ → 0⁺, we will get what we want.

Due to the initial density approaches no vacuum at infinity lim|x|→+∞ρ0(x)=ρ¯>0, then there exists a large number M > 0 such that if |x| ≥ M, ρ₀(x) ≥ ρ¯2 . For any 0 < δ < ρ¯2 , we define

ρ0δ(x)=ρ0(x)+δ,if|x|≤M,ρ0(x)+δs(x),ifM≤|x|≤M+1,ρ0(x),if|x|≥M+1, (3.1)

where s(x) = s(|x|) is a smooth and decreasing function satisfying s(x) = 1 if |x| ≤ M and s(x) = 0 if |x| ≥ M + 1. Clearly, we have

ρ0δ−ρ¯,P(ρ0δ)−P(ρ¯)→(ρ0−ρ¯,P(ρ0)−P(ρ¯))inH1(R),

and

Φ(ρ0δ)(1+|x|α)→Φ(ρ0)(1+|x|α)inL1(R),asδ→0+.

To approximate the initial velocity, we define u0δ as

u0δ=u~0δ,|x|<M+1,u0,|x|≥M+1, (3.2)

where u~0δ is the unique solution to the following elliptic equation:

(μu~0xδ)x=P(ρ0δ)x+12(b02)x+ρ0g(x),inΩM:={x| |x|<M+1},u~0δ||x|=M+1=u0. (3.3)

Since ρ0δ−ρ¯,P(ρ0δ)−P(ρ¯)∈H1(R),b0∈H1(R) and ρ0g∈L2(R), by the elliptic theory, (1.6) and (3.3), we have

∥u~0δ∥H2(ΩM)≤C∥P(ρ0δ)x∥L2+∥(b02)x∥L2+∥ρ0g∥L2+1≤C∥P(ρ0δ)x∥L2+∥b0∥L∞∥b0x∥L2+∥ρ0∥L∞∥g∥L2+1≤C. (3.4)

From the compatibility conditions (1.6) and (3.3), it follows that

μ(u~0δ−u0)xx=P(ρ0δ)−P(ρ0)x,inΩM,(u~0δ−u0)||x|=M+1=0, (3.5)

which yields

u~0δ−u0∈H01(ΩM)∩H2(ΩM),

and

∥u~0δ−u0∥H2(ΩM)≤C∥(P(ρ0δ)−P(ρ0))x∥L2≤Cδ→0,asδ→0+. (3.6)

Furthermore

ρ0δu0δ(1+|x|α2)→ρ0u0(1+|x|α2)inL2(R),asδ→0+.

For any T ∈ (0, ∞), let (ρ^δ, u^δ, b^δ) with positive ρ^δ be the smooth solution to (1.1)-(1.2). Without confusion, we still denote the solution by (ρ, u, b) instead of (ρ^δ, u^δ, b^δ) to simplify the presentation. Throughout this section, we denote C to be a generic constant which is uniform of δ for any δ ∈ (0, 1) and may depend on ρ₀, u₀, b₀, γ, μ, T and some other known constants but independent of ν.

First of all, we can prove the following elementary energy estimates.

Lemma 3.1

Let (ρ, u, b) be a smooth solution of (1.1) and (1.2). Then for any T > 0, it holds that

sup0≤t≤T12ρu2+Φ(ρ)+(b−b¯)22L1(R)+∫0Tμ∥ux∥L22+ν∥bx∥L22dt≤C. (3.7)

Proof

By the definition of the potential energy Φ, and using (1.1)₁, we deduce

Φt+(uΦ)x+(ργ−ρ¯γ)ux=0. (3.8)

Adding the equation (1.1)₂ multiplied by u, (1.1)₃ multiplied by b - b̄ into (3.8), and then integrating the resulting equation over ℝ × [0, T] with respect to the variables x and t, we have

∫12ρu2+Φ(ρ)+(b−b¯)22dx+∫0Tμ∥ux∥L22+ν∥bx∥L22dt≤∫12ρ0u2+Φ(ρ0)+(b0−b¯)22dx≤C∥ρ0∥L∞∥u0∥L22+C(∥ρ0−ρ¯∥L22+∥b0−b¯∥L22)≤C(∥ρ0−ρ¯∥H12+ρ¯)(∥u0∥L22+1)+C∥b0−b¯∥L22≤C.

Then the proof of Lemma 3.1 is completed. □

To obtain the upper bound of the density ρ, we need the following weighted energy estimates.

Lemma 3.2

Let (ρ, u, b) be a smooth solution of (1.1) and (1.2). Then for any T > 0 and some index 1 < α ≤ 2, it holds that

sup0≤t≤T12ρu2+Φ(ρ)+(b−b¯)22|x|αL1(R)+∫0Tμ∥ux|x|α2∥L22+ν∥bx|x|α2∥L22dt≤C. (3.9)

Proof

Multiplying the equation (1.1)₂ by u|x|^α and integrating the resulting equation over ℝ with respect to x, we have

12ddt∫ρu2|x|α+μ∫ux2|x|α=12∫ρu3α|x|α−2x−μ∫α|x|α−2xuux−∫P(ρ)+b22xu|x|α. (3.10)

It follows from the integration by parts that

−∫P(ρ)+b22xu|x|α=−∫P(ρ)−P(ρ¯)x+bbxu|x|α=−∫P(ρ)−P(ρ¯)x+(b−b¯)(b−b¯)x+b¯(b−b¯)xu|x|α=∫P(ρ)−P(ρ¯)+(b−b¯)22+b¯(b−b¯)ux|x|α+uα|x|α−2x. (3.11)

Then, combining (3.10) and (3.11) gives

12ddt∫ρu2|x|α+μ∫ux2|x|α=12∫ρu3α|x|α−2x−μ∫α|x|α−2xuux+∫P(ρ)−P(ρ¯)+(b−b¯)22+b¯(b−b¯)ux|x|α+uα|x|α−2x. (3.12)

To deal with the last term on the right-hand side of (3.12), first, multiplying (3.8) by |x|^α and integrating over ℝ with respect to x, yields

ddt∫Φ(ρ)|x|α−∫uΦ(ρ)α|x|α−2x+∫(ργ−ρ¯γ)|x|αux=0, (3.13)

and then multiplying the equation (1.1)₃ by (b-b̄)|x|^α and integrating over ℝ with respect to x, we have

12ddt∫(b−b¯)2|x|α+12∫(b−b¯)2xu|x|α+∫(b−b¯)2ux|x|α+b¯∫(b−b¯)ux|x|α=ν∫bxx(b−b¯)|x|α,

which together with the integration by parts implies that

12ddt∫(b−b¯)2|x|α+ν∫bx2|x|α=−∫(b−b¯)22(ux|x|α−uα|x|α−2x)−b¯∫(b−b¯)ux|x|α−ν∫bx(b−b¯)α|x|α−2x. (3.14)

Now, putting (3.13) and (3.14) into (3.12), we have

ddt∫12ρu2+Φ(ρ)+(b−b¯)22|x|αdx+μ∫ux2|x|α+ν∫bx2|x|α=∫12ρu2+Φ(ρ)+(b−b¯)2uα|x|α−2x+∫(ργ−ρ¯γ)uα|x|α−2x−μ∫α|x|α−2xuux+b¯∫(b−b¯)uα|x|α−2x−ν∫(b−b¯)bxα|x|α−2x=:I1+I2+I3+I4+I5. (3.15)

Next, we estimate the terms I₁-I₅ as follows:

I1≤∫12ρu2+Φ(ρ)+(b−b¯)22|u||x|α−1≤∫12ρu2+Φ(ρ)+(b−b¯)22|x|αα−1α12ρu2+Φ(ρ)+(b−b¯)221α|u|≤C∥u∥L∞12ρu2+Φ(ρ)+(b−b¯)22|x|αL11−1α12ρu2+Φ(ρ)+(b−b¯)22L11α≤C(1+∥ux∥L2)12ρu2+Φ(ρ)+(b−b¯)22|x|αL1+1, (3.16)

where α > 1 and we have used Hölder's inequality, Lemma 3.1 and the following facts:

ρ¯∫u2=∫(ρ¯−ρ+ρ)u2≤∫(ρ¯−ρ)|{0≤ρ≤2ρ¯}+(ρ¯−ρ)|{ρ>2ρ¯}u2+∫ρu2≤C∥(ρ−ρ¯)|{0≤ρ≤2ρ¯}∥L2∥u∥L42+∥(ρ−ρ¯)|{ρ>2ρ¯}∥Lγ∥u∥L2γγ−12+1≤C∥Φ(ρ)∥L112∥u∥L234∥ux∥L2142+∥Φ(ρ)∥L11γ∥u∥L21−12γ∥ux∥L212γ2+1≤ε∥u∥L22+C1+∥ux∥L22,

where we have used the properties of the potential energy Φ in Lemma 2.3. This together with the Gagliardo-Nirenberg inequality implies that

∥u∥L22≤C(1+∥ux∥L22), (3.17)

and

∥u∥L∞≤C∥u∥L212∥ux∥L212≤C(1+∥ux∥L2). (3.18)

Applying the property of Φ(ρ), Hölder's inequality, the Caffarelli-Kohn-Nirenberg weighted inequality (2.3), Lemma 3.1, Young's inequality and (3.18), we obtain

I2=∫(ργ−ρ¯γ)|{0≤ρ≤2ρ¯}+(ργ−ρ¯γ)|{ρ>2ρ¯}uα|x|α−2x≤C∫|ρ−ρ¯||{0≤ρ≤2ρ¯}+(ρ−ρ¯)γ|{ρ>2ρ¯}|u||x|α−1≤C∫Φ12(ρ)|{0≤ρ≤2ρ¯}+Φ(ρ)|{ρ>2ρ¯}|u||x|α−1≤C∫Φ12(ρ)|{0≤ρ≤2ρ¯}|x|α2|u||x|α2−1+C∫Φ(ρ)|{ρ>2ρ¯}|x|αα−1αΦ1α(ρ)|u|≤C∥Φ(ρ)|x|α∥L112|x|α2−1uL2+∥Φ(ρ)|x|α∥L11−1α∥Φ(ρ)∥L11α∥u∥L∞≤C(1+∥ux∥L22)(1+∥Φ(ρ)|x|α∥L1), (3.19)

where we have used the fact: for 1 < α ≤ 2,

∥|x|α2−1u∥L2≤C∥u∥L2α2∥ux∥L21−α2≤C∥u∥L2+∥ux∥L2≤C(1+∥ux∥L2). (3.20)

Similarly, using the Caffarelli-Kohn-Nirenberg weighted inequality (2.3), we have

I3≤C∥|x|α2ux∥L2∥|x|α2−1u∥L2≤ε∥|x|α2ux∥L22+C1+∥ux∥L22. (3.21)

For the term I₄, it follows from the Hölder's inequality and the Caffarelli-Kohn-Nirenberg weighted inequality that

I4≤C(b−b¯)|x|α2L2|x|α2−1uL2≤C(b−b¯)|x|α2L22+∥ux∥L22+1. (3.22)

Similarly, using the Caffarelli-Kohn-Nirenberg weighted inequality (2.3), one has

I5≤Cν∥|x|α2bx∥L2∥(b−b¯)|x|α2−1∥L2≤Cν∥|x|α2bx∥L2∥b−b¯∥L2α2∥bx∥L21−α2≤εν∥|x|α2bx∥L22+Cν1+∥bx∥L22, (3.23)

where 1 < α ≤ 2.

Then, substituting (3.16), (3.19), (3.21), (3.22) and (3.23) into (3.15), choosing ε > 0 small enough, one deduces

ddt∫12ρu2+Φ(ρ)+(b−b¯)22|x|α+μ∫|x|αux2+ν∫|x|αbx2≤C(1+∥ux∥L22)12ρu2+Φ(ρ)+(b−b¯)22|x|αL1+C(1+∥ux∥L22+ν∥bx∥L22).

This together with Gronwall's inequality, (1.5) and Lemma 3.1 gives

∫12ρu2+Φ(ρ)+(b−b¯)22|x|α+μ∫0T∫|x|αux2+ν∫0T∫|x|αbx2≤C(T).

Then, the proof of Lemma 3.2 is completed. □

The upper bound of the density ρ can be shown in the similar manner as that in [25]. However, for completeness of the paper, we give the details here.

Lemma 3.3

Let (ρ, u, b) be a smooth solution of (1.1) and (1.2). Then for any T > 0, it holds that

0<δe−C(T)≤ρ(x,t)≤C,forall(x,t)∈R×[0,T],

and

sup0≤t≤T∥ρ−ρ¯∥L2(R)≤C.

Proof

Let ξ=∫−∞x ρ udy, then the momentum equation (1.1)₂ can be rewritten as

ξtx+ρu2+P(ρ)−P(ρ¯)+b2−b¯22x=(μux)x.

Integrating the above equality with respect to x over (-∞, x) yields that

ξt+ρu2+P(ρ)−P(ρ¯)+b2−b¯22=μux, (3.24)

which together with (1.1)₁ gives

ξt+ρu2+P(ρ)−P(ρ¯)+b2−b¯22+μρt+uρxρ=0. (3.25)

Next, we define the particle trajectory X(x, t) as follows:

dX(x,t)dt=u(X(x,t),t),X(x,0)=x, (3.26)

which implies

dξdt=ξt+uξx=ξt+ρu2.

Then combining the above equality and (3.25), we infer that

ddt(ξ+μln⁡ρ)X(x,t),t+P(ρ)+b22X(x,t),t−P(ρ¯)+b¯22=0, (3.27)

which together with P(ρ)+b22X(x,t),t≥0 yields

ddt(ξ+μln⁡ρ)X(x,t),t≤P(ρ¯)+b¯22X(x,t),t≤C. (3.28)

Thus, integrating it over [0, T] with respect to t, we have

(ξ+μln⁡ρ)X(x,t),t≤(ξ+μln⁡ρ)(x,0)+C.

By direct calculation, we obtain

ln⁡ρ≤1μ(ξ0+μln⁡ρ0+C−ξ)≤1μξ0+μln⁡ρ0+C−∫−∞xρudy≤1μξ0+μln⁡ρ0+C−∫−∞xρu(ρ−ρ¯)dy−ρ¯∫−∞xρudy≤1μ(ξ0+μln⁡ρ0+C−ξ1−ξ2). (3.29)

Firstly, it follows from Lemma 2.3 and Lemma 3.1 that ∥ξ₁∥_L^∞ can be estimated as

|ξ1|=|∫−∞xρu(ρ−ρ¯)dy|≤C∥ρu∥L2∥(ρ−ρ¯)|{0≤ρ≤2ρ¯}∥L2+∥(ρ−ρ¯)|{ρ>2ρ¯}∥L2≤C∥ρu∥L2∥(ρ−ρ¯)|{0≤ρ≤2ρ¯}∥L2+∥ρ−ρ¯|{ρ>2ρ¯}∥L2≤C∥ρu∥L2∥(ρ−ρ¯)|{0≤ρ≤2ρ¯}∥L2+∥(ρ−ρ¯){ρ>2ρ¯}∥L112≤C∥ρu∥L2∥(ρ−ρ¯)|{0≤ρ≤2ρ¯}∥L2+∥(ρ−ρ¯)γ|{ρ>2ρ¯}∥L112≤C∥ρu∥L2∥Φ(ρ)∥L112+∥Φ(ρ)∥L112≤C. (3.30)

Secondly, using the Gagliardo-Nirenberg inequality, the Caffarelli-Kohn-Nirenberg weighted inequality, Lemma 3.1 and Lemma 3.2, we can bound ∥ξ₂∥_L^∞ as

∥ξ2∥L∞≤C∥ξ2∥Lp~p~p~+2∥ξ2x∥L22p~+2≤C(∥ξ2x∥L2η∥|x|κξ2∥Lq1−η)p~p~+2∥ξ2x∥L22p~+2≤C(∥ξ2x∥L2η∥|x|α2ξ2x∥L21−η)p~p~+2∥ξ2x∥L22p~+2≤C∥ξ2x∥L21−(1−η)p~p~+2∥|x|α2ξ2x∥L2(1−η)p~p~+2=C∥ξ2x∥L21−1α∥|x|α2ξ2x∥L21α≤C(∥ρu∥L21−1α∥|x|α2ρu∥L21α)≤C. (3.31)

Here the indexes 1 ≤ p̃ < ∞, q > 1, η ∈ (0, 1) and satisfy

1p~=(12−1)η+1q+κ(1−η),1q+κ=12+α2−1>0,

which gives

α>1,η=p~(α−1)−2αp~>0⇒p~>2α−1. (3.32)

Similarly as that for (3.30) and (3.31), we can obtain

|ξ0|≤C. (3.33)

Then, substituting (3.30), (3.31) and (3.33) into (3.28), we have

ln⁡ρ≤C,

which gives

ρ≤C.

On the other hand, if ρ₀ ≥ δ > 0, then combining ρ ≤ C and integrating (3.27) over [0, T], we have

|μ(ln⁡ρ(X(x,t),t)−ln⁡ρ(x,0))|=|−ξ(X(x,t),t)+ξ(x,0)−∫0TP(ρ)+b22−P(ρ¯)+b¯22(X(x,t),t)dt|≤C(T),

|ln⁡ρ(X(x,t),t)−ln⁡ρ(x,0)|≤C(T)⇒ln⁡ρ(X(x,t),t)−ln⁡ρ(x,0)≥−C(T),

which implies

ln⁡ρ≥ln⁡ρ0−C(T)⇒ρ≥ρ0e−C(T)≥δe−C(T)>0.

It should be noted that we have used the ∥b∥_{L^∞(0,T;L^∞)} in advance to obtain the lower bound of the density, which will be proved in the later Lemma 3.5. However, this makes sense, because we will only use the upper bound of the density to prove the upper bound of the magnetic field b in the Lemma 3.5. Hence, we put the proof ahead of time just for the sake of convenience.

Furthermore, it follows from Lemma 2.3 and Lemma 3.1 that

∥ρ−ρ¯∥L2=(ρ−ρ¯)2|{ρ≤2ρ¯}L112+(ρ−ρ¯)|{ρ≥2ρ¯}L2≤Φ(ρ)L112+(ρ−ρ¯)|{ρ≥2ρ¯}L2≤C+(ρ−ρ¯)|{ρ≥2ρ¯}L2. (3.34)

To deal with the last term on the right-hand side of (3.34), we discuss it in the following two cases:

if 1 < γ ≤ 2, by Lemma 2.3, Lemma 3.1 and the fact ρ ≤ C, it holds

∥(ρ−ρ¯)|{ρ≥2ρ¯}∥L2(R)≤∥ρ−ρ¯∥Lγ(R)γ2∥ρ−ρ¯∥L∞(R)1−γ2≤∥Φ(ρ)∥L1(R)12∥ρ−ρ¯∥L∞(R)1−γ2≤C(T). (3.35)
if γ > 2, using the fact that ρ−ρ¯>ρ¯⇒1(ρ−ρ¯)γ−2<1ρ¯γ−2 and Lemma 3.1, we have

∥(ρ−ρ¯)|{ρ≥2ρ¯}∥L2(R)≤(ρ−ρ¯)2|{ρ≥2ρ¯}L112≤(ρ−ρ¯)γ|{ρ≥2ρ¯}1(ρ−ρ¯)γ−2|{ρ≥2ρ¯}L112≤(ρ−ρ¯)γ|{ρ≥2ρ¯}1ρ¯γ−2L112≤1ρ¯γ−2Φ(ρ)L112≤C. (3.36)

Thus, combining (3.35), (3.36) and (3.34), we can obtain ∥ρ-ρ̄∥_L² ≤ C. This completes the proof of Lemma 3.3. □

Lemma 3.4

Let (ρ, u, b) be a smooth solution of (1.1)-(1.2). Then for any T > 0, it holds that

sup0≤t≤T(μ∥ux∥L22+ν∥bx∥L22+∥b−b¯∥L44)+∫0T∥b−b¯∥L66+μν∥(b−b¯)bx∥L22dt+∫0Tν2∥bxx∥L22dt+∫0T∥ρu˙∥L22dt≤C,

and

sup0≤t≤T(∥u∥L2+∥u∥L∞)≤C.

Proof

The proof of Lemma 3.4 will be divided into four steps.

Multiplying the equation (1.1)₂ by u̇ and integrating the resulting equation over ℝ with respect to x yields

μ2ddt∫ux2+∫ρu˙2=−μ∫ux(uux)x−∫P(ρ)x(ut+uux)−∫b22x(ut+uux)=:J1+J2+J3. (3.37)

Firstly, by integration by parts, we find

J1=−μ∫ux3−μ∫uux22x=−μ∫ux3+μ∫ux22⋅ux=−μ2∫ux3. (3.38)

Similarly, we have

J2=−∫P(ρ)−P(ρ¯)x(ut+uux)=ddt∫P(ρ)−P(ρ¯)ux−∫P(ρ)−P(ρ¯)t+P(ρ)−P(ρ¯)xuux=ddt∫P(ρ)−P(ρ¯)ux+γ∫ργux2, (3.39)

and

J3=−∫b(b−b¯)x(ut+uux)=−∫(b−b¯)22x(ut+uux)−b¯∫(b−b¯)x(ut+uux)=ddt∫(b−b¯)22+b¯(b−b¯)ux−∫(b−b¯)22t+(b−b¯)22xuux−b¯∫(b−b¯)t+u(b−b¯)xux=ddt∫(b−b¯)22+b¯(b−b¯)ux−∫(b−b¯)+b¯(νbxx−bux)ux, (3.40)

where we have used the following facts:

(P(ρ)−P(ρ¯))t+u(P(ρ)−P(ρ¯))x+γργux=0,(b−b¯)22t+u(b−b¯)22x+b(b−b¯)ux=νbxx(b−b¯),

and

(b−b¯)t+u(b−b¯)x+bux=νbxx.

Substituting (3.38)-(3.40) into (3.37), using Hölder's and Cauchy-Schwarz's inequalities, we have

ddt∫μ2ux2−P(ρ)−P(ρ¯)ux−(b−b¯)22ux−b¯(b−b¯)uxdx+∫ρu˙2=−μ2∫ux3+γ∫ργux2−ν∫bxx(b−b¯)ux+∫(b−b¯)2ux2+2b¯∫(b−b¯)ux2−b¯∫νbxxux+b¯2∫ux2≤C(∥ux∥L33+∥ux∥L22+ν∥bxx∥L2∥b−b¯∥L6∥ux∥L3+∥ux∥L32∥b−b¯∥L62+∥ux∥L32∥b−b¯∥L3+ν∥bxx∥L2∥ux∥L2)≤εν2∥bxx∥L22+C∥ux∥L33+∥b−b¯∥L66+(∥b−b¯∥L212∥b−b¯∥L612)3+∥ux∥L22≤εν2∥bxx∥L22+C∥ux∥L33+∥b−b¯∥L66+∥ux∥L22+1. (3.41)

Now, we estimate the term ∥ux∥L33 on the right-hand side of (3.41). Employing the effective viscous flux, momentum equation, Lemma 3.1 and Lemma 3.3, we have

∥F∥L3≤∥F∥L256∥Fx∥L216≤Cμux−P(ρ)−P(ρ¯)−(b−b¯)22+b¯(b−b¯)L256∥ρu˙∥L216≤C∥ux∥L2+∥ρ−ρ¯∥L2+∥b−b¯∥L42+∥b−b¯∥L256∥ρu˙∥L216≤C∥ux∥L2+∥b−b¯∥L212∥b−b¯∥L632+156∥ρu˙∥L216≤C∥ux∥L2+∥b−b¯∥L632+156∥ρu˙∥L216. (3.42)

Moreover, we can obtain

∥ux∥L3=F+P(ρ)−P(ρ¯)+b2−b¯22μL3≤C∥F∥L3+∥ρ−ρ¯∥L3+(b−b¯)22+b¯(b−b¯)L3≤C∥F∥L3+∥ρ−ρ¯∥L223∥ρ−ρ¯∥L∞13+∥b−b¯∥L62+∥b−b¯∥L3≤C∥ux∥L2+∥b−b¯∥L632+156∥ρu˙∥L216+∥b−b¯∥L62+1, (3.43)

which implies

∥ux∥L33≤C∥ux∥L2+∥b−b¯∥L632+156∥ρu˙∥L2163+∥b−b¯∥L66+1≤ε∥ρu˙∥L22+C∥ux∥L24+∥b−b¯∥L66+1. (3.44)

Then, substituting (3.44) into (3.41) and choosing ε sufficiently small, we have

ddt∫μ2ux2−P(ρ)−P(ρ¯)ux−(b−b¯)22ux−b¯(b−b¯)ux+∫ρu˙2≤εν2∥bxx∥L22+ε∥ρu˙∥L22+C∥ux∥L24+∥b−b¯∥L66+1. (3.45)
To control the term ∥b−b¯∥L66 on the right-hand side of (3.45), we rewrite the magnetic field equation as

(b−b¯)t+u(b−b¯)x+(b−b¯)ux+b¯ux=νbxx.

Then multiplying the above equation by (b-b̄)³ and integrating it over ℝ with respect to x, we have

14ddt∫(b−b¯)4+3ν∫(b−b¯)2bx2=−∫34(b−b¯)4+b¯(b−b¯)3ux=−∫34(b−b¯)4+b¯(b−b¯)3F+P(ρ)−P(ρ¯)+b2−b¯22μ=−∫34(b−b¯)4+b¯(b−b¯)3F+P(ρ)−P(ρ¯)+(b−b¯)22+b¯(b−b¯)μ=−34μ∫(b−b¯)4F+P(ρ)−P(ρ¯)−b¯μ∫(b−b¯)3F+P(ρ)−P(ρ¯)−38μ∫(b−b¯)6−5b¯4μ∫(b−b¯)5−b¯2μ∫(b−b¯)4,

which gives

14ddt∫(b−b¯)4+38μ∫(b−b¯)6+b¯2μ∫(b−b¯)4+3ν∫(b−b¯)2bx2=−34μ∫(b−b¯)4F+P(ρ)−P(ρ¯)−b¯μ∫(b−b¯)3F+P(ρ)−P(ρ¯)−5b¯4μ∫(b−b¯)5=:K1+K2+K3. (3.46)

Next, we estimate K₁-K₃ term by term. Using (3.42), Lemma 3.3 and Young's inequality, we have

K1≤C∥b−b¯∥L64∥F∥L3+∥P(ρ)−P(ρ¯)∥L3≤C∥b−b¯∥L64∥ux∥L2+∥b−b¯∥L632+156∥ρu˙∥L216+∥ρ−ρ¯∥L3≤ε∥b−b¯∥L66+ε∥ρu˙∥L22+C(1+∥ux∥L24). (3.47)

Similarly, we can get

K2≤C∥b−b¯∥L63∥F∥L2+∥P(ρ)−P(ρ¯)∥L2≤C∥b−b¯∥L63∥ux∥L2+∥ρ−ρ¯∥L2+∥b−b¯∥L42+∥b−b¯∥L2≤C∥b−b¯∥L63∥ux∥L2+∥ρ−ρ¯∥L2+∥b−b¯∥L212∥b−b¯∥L632+∥b−b¯∥L2≤ε∥b−b¯∥L66+C∥ux∥L24+1, (3.48)

and

K3≤C∥b−b¯∥L55≤C∥b−b¯∥L212∥b−b¯∥L692≤ε∥b−b¯∥L66+C. (3.49)

Then, substituting (3.47)-(3.49) into (3.46) and choosing ε > 0 small enough yields

14ddt∫(b−b¯)4+38μ∫(b−b¯)6+b¯2μ∫(b−b¯)4+3ν∫(b−b¯)2bx2≤ε∥ρu˙∥L22+C1+∥ux∥L24. (3.50)
To control the term ν2∥bxx∥L22 on the right hand-side of (3.45), we multiply the equation (1.1)₃ by ν b_xx and integrate it over ℝ to get

ν2ddt∫bx2+ν2∫bxx2=ν∫bxxubx+ν∫bxx(b−b¯)ux+b¯ν∫bxxux=−ν2∫bx2ux+ν∫bxx(b−b¯)ux+b¯ν∫bxxux=:H1+H2+H3. (3.51)

For the term H₁, by the effective viscous flux, it shows that

H1=−ν2∫bx2F+P(ρ)−P(ρ¯)+b2−b¯22μ≤−ν2μ∫bx2F+ν2μ∫bx2P(ρ¯)+b¯22≤C1+∥F∥L∞ν∥bx∥L22≤C1+∥ρu˙∥L2+∥ux∥L2+∥b−b¯∥L632ν∥bx∥L22≤ε∥ρu˙∥L22+∥b−b¯∥L66+C1+∥ux∥L22+ν∥bx∥L22ν∥bx∥L22, (3.52)

where we have used the following inequality:

∥F∥L∞≤C∥F∥L212∥Fx∥L212≤C∥ux∥L2+∥ρ−ρ¯∥L2+∥b−b¯∥L4+∥b−b¯∥L212∥ρu˙∥L212≤C1+∥ρu˙∥L2+∥ux∥L2+∥b−b¯∥L632. (3.53)

For the terms H₂ and H₃, using Hölder's inequality, Cauchy's inequality and (3.44), we have

H2≤ν∥bxx∥L2∥b−b¯∥L6∥ux∥L3≤εν2∥bxx∥L22+C∥b−b¯∥L66+∥ux∥L33≤εν2∥bxx∥L22+ε∥ρu˙∥L22+C1+∥ux∥L24+∥b−b¯∥L66, (3.54)

and

H3≤Cν∥bxx∥L2∥ux∥L2≤εν2∥bxx∥L22+C∥ux∥L22. (3.55)

Thus, substituting (3.52), (3.54) and (3.55) into (3.51) and choosing ε > 0 small enough, we have

ν2ddt∫bx2+ν2∫bxx2≤2ε∥ρu˙∥L22+C1+∥ux∥L22+ν∥bx∥L22ν∥bx∥L22+C1+∥ux∥L24+∥b−b¯∥L66. (3.56)
Adding the equation (3.50) multiplied by 8μ3(2C+1) and (3.45) into (3.56), we have

ddt∫μ2ux2+ν2bx2+2μ(2C+1)3(b−b¯)4−ψ(ρ,u,b)+∫ρu˙2+ν2∫bxx2+∫(b−b¯)6+8b¯2(2C+1)3(b−b¯)4+8μν(2C+1)(b−b¯)2bx2≤C+C1+∥ux∥L22+ν∥bx∥L22∥ux∥L22+ν∥bx∥L22, (3.57)

where ψ(ρ,u,b)=P(ρ)−P(ρ¯)ux+(b−b¯)22ux+b¯(b−b¯)ux. Integrating (3.57) over [0, T] with respect to t, we obtain

∫μux2+νbx2+(b−b¯)4+∫0T∫ρu˙2+(b−b¯)6+μν(b−b¯)2bx2+ν2bxx2≤C+C∫0T1+∥ux∥L22+ν∥bx∥L22∥ux∥L22+ν∥bx∥L22+∫ψ(ρ,u,b)−∫ψ(ρ0,u0,b0). (3.58)

By Lemma 3.1, Lemma 3.3 and Cauchy-Schwarz's inequality, we can obtain

∫ψ(ρ,u,b)−∫ψ(ρ0,u0,b0)≤C∥P(ρ)−P(ρ¯)∥L2+∥b−b¯∥L42+∥b−b¯∥L2∥ux∥L2+C≤C∥ρ−ρ¯∥L2+∥b−b¯∥L42+1∥ux∥L2+C≤εμ∥ux∥L22+C(1+∥b−b¯∥L44),

which together with (3.58) gives (choosing ε > 0 small enough)

∫μux2+νbx2+(b−b¯)4+∫0T∫ρu˙2+(b−b¯)6+μν(b−b¯)2bx2+ν2bxx2≤C(1+∥b−b¯∥L44)+C∫0T1+∥ux∥L22+ν∥bx∥L22∥ux∥L22+ν∥bx∥L22. (3.59)

Now, integrating (3.50) over [0, T] with respect to t, and then adding the resulting inequality multiplied by 4C into (3.59) show that

∫μux2+νbx2+(b−b¯)4+∫0T∫ρu˙2+(b−b¯)6+μν(b−b¯)2bx2+ν2bxx2≤C+C∫0T1+∥ux∥L22+ν∥bx∥L22∥ux∥L22+ν∥bx∥L22,

which together with Gronwall's inequality and Lemma 3.1 yields

∫μux2+νbx2+(b−b¯)4dx+∫0T∫(b−b¯)6+μν(b−b¯)2bx2dxdt+∫0T∫ν2bxx2dxdt+∫0T∫ρu˙2dxdt≤C(T).

Thus, it follows from (3.17) and (3.18) that

∥u∥L22≤C(1+∥ux∥L22)≤C,

and

∥u∥L∞≤C∥u∥L212∥ux∥L212≤C(1+∥ux∥L2)≤C.

Then, we complete the proof of Lemma 3.4. □

To get the uniform upper bound of the magnetic field b, we need to re-estimate the ∥b_x∥_L² independent of ν as follows.

Lemma 3.5

Let (ρ, u, b) be a smooth solution of (1.1)-(1.2). Then for any T > 0, it holds that

sup0≤t≤T∥b∥L∞(R)+∥ρx∥L2(R)2+∥bx∥L2(R)2+∫0Tμ∥uxx∥L2(R)2+ν∥bxx∥L2(R)2dt≤C.

Proof

Differentiating the equality (1.1)₃ with respect to x, then multiplying the resulting equation by b_x and integrating by parts over ℝ, we have

12ddt∫bx2+ν∫bxx2=−2∫bx2ux−∫bbxuxx−∫ubxbxx=−32∫bx2ux−∫bbxuxx. (3.60)

To control the second term on the right-hand side of (3.60), multiplying the momentum equation (1.1)₂ by u_xx gives

μ∫uxx2=∫ρu˙uxx+γ∫ργ−1ρxuxx+∫bbxuxx. (3.61)

Combining (3.60) and (3.61) yields

12ddt∫bx2+ν∫bxx2+μ∫uxx2=−32∫bx2ux+∫ρu˙uxx+γ∫ργ−1ρxuxx=−32∫bx2F+P(ρ)−P(ρ¯)+b2−b¯22μ+∫ρu˙uxx+γ∫ργ−1ρxuxx≤−32∫bx2F−P(ρ¯)+−b¯22μ+∫ρu˙uxx+γ∫ργ−1ρxuxx≤C∥F∥L∞∥bx∥L22+∥bx∥L22+∥ρu˙∥L2∥uxx∥L2+∥ρx∥L2∥uxx∥L2≤ε∥uxx∥L22+C∥F∥L∞+1∥bx∥L22+C∥ρx∥L22+∥ρu˙∥L22≤ε∥uxx∥L22+C∥ρu˙∥L22+∥b−b¯∥L66+1∥bx∥L22+C∥ρx∥L22+∥ρu˙∥L22, (3.62)

where we have used (3.53) and Lemma 3.4.

Next, differentiating the density equation (1.1)₁ with respect to x, multiplying the resulting equation by ρ_x, and integrating it over ℝ implies that

12ddt∫ρx2=−2∫ρx2ux−∫uρxρxx−∫ρρxuxx=−32∫ρx2ux−∫ρρxuxx=−32∫ρx2F+P(ρ)−P(ρ¯)+b2−b¯22μ−∫ρρxuxx≤−32μ∫ρx2F+32μ∫P(ρ¯)+b¯22ρx2+C∥ρx∥L2∥uxx∥L2≤ε∥uxx∥L22+C∥F∥L∞+1∥ρx∥L22≤ε∥uxx∥L22+C∥ρu˙∥L22+∥b−b¯∥L66+1∥ρx∥L22, (3.63)

where we have used (3.53) and Lemma 3.4. Then adding the (3.63) into (3.62), we have

12ddt∫bx2+ρx2+ν∫bxx2+μ∫uxx2≤C∥ρu˙∥L22+∥b−b¯∥L66+11+∥ρx∥L22+∥bx∥L22,

which together with Gronwall's inequality and Lemma 3.4 gives

∫ρx2+bx2+∫0T∫νbxx2+μuxx2≤C(T).

Thus, it follows from Lemma 3.1 and the Gagliardo-Nirenberg inequality (2.1) that

∥b∥L∞≤∥b−b¯∥L∞+b¯≤C∥b−b¯∥L212∥bx∥L212+b¯≤C(T).

Thus, the proof of Lemma 3.5 is finished. □

With the help of the Lemma 3.5, we can get the estimates of the first order derivative with respect to t of the density and the magnetic field, respectively.

Lemma 3.6

Let (ρ, u, b) be a smooth solution to (1.1)-(1.2). Then for any T > 0, it holds that

sup0≤t≤T∥ρt∥L2(R)+∥bt∥L2(0,T;L2(R))≤C.

Proof

By direct calculation, we have

∥ρt∥L2≤C∥ρux∥L2+∥uρx∥L2≤C∥ρ∥L∞∥ux∥L2+∥u∥L∞∥ρx∥L2∈L∞(0,T),

and

∥bt∥L2≤C∥bux∥L2+∥ubx∥L2+∥νbxx∥L2≤C∥b∥L∞∥ux∥L2+∥u∥L∞∥bx∥L2+∥νbxx∥L2≤C1+∥νbxx∥L2∈L2(0,T),

where we have used Lemma 3.4 and Lemma 3.5. □

The following estimates of the second order derivative of the velocity play an important role in the analysis of the non-resistive limit.

Lemma 3.7

Let (ρ, u, b) be a smooth solution to (1.1)-(1.2). Then for any T > 0, it holds that

sup0≤t≤T∥ρu˙∥L2(R)+∫0Tμ∥uxt∥L2(R)2≤C,

and

sup0≤t≤T∥uxx∥L2(R)+∥ρut∥L2(R)≤C.

Proof

Differentiating the momentum equation (1.1)₂ with respect to t, we have

ρtu˙+ρu˙t+P(ρ)+b22xt=μuxxt.

Multiplying the above equation by u̇ and integrating the resulting equation over ℝ yields

12ddt∫ρu˙2+μ∫uxt2=−12∫ρtu˙2−∫P(ρ)+b22xtu˙−μ∫uxtux2+uuxx=:L1+L2+L3. (3.64)

Now, we estimate the terms L₁-L₃ as

L1=12∫(ρu)xu˙2=∫−ρuu˙u˙x≤C∥ρu˙∥L2∥u∥L∞∥u˙x∥L2≤C∥ρu˙∥L2∥uxt∥L2+∥ux∥L42+∥u∥L∞∥uxx∥L2≤C∥ρu˙∥L2∥uxt∥L2+∥ux∥L232∥uxx∥L212+∥uxx∥L2≤ε∥uxt∥L22+C∥ρu˙∥L22+∥uxx∥L22+1, (3.65)

and similarly, we also have

L2=∫P(ρ)+b22tu˙x≤C∥ργ−1ρt+bbt∥L2∥u˙x∥L2≤C∥ρt∥L2+∥bt∥L2∥uxt+ux2+uuxx∥L2≤C1+∥bt∥L2∥uxt∥L2+∥ux∥L232∥uxx∥L212+∥u∥L∞∥uxx∥L2≤ε∥uxt∥L22+C∥uxx∥L22+∥bt∥L22+1, (3.66)

and

L3≤C∥uxt∥L2∥ux∥L42+∥u∥L∞∥uxx∥L2≤ε∥uxt∥L22+C1+∥uxx∥L22, (3.67)

where we have used the Gagliardo-Nirenberg inequality, Lemma 3.4 and Lemma 3.5. Thus, substituting (3.65)-(3.67) into (3.64) and choosing ε > 0 small enough, we have

12ddt∫ρu˙2+μ∫uxt2≤C1+∥ρu˙∥L22+∥uxx∥L22+∥bt∥L22,

which combining Gronwall's inequality, Lemma 3.4 and Lemma 3.5 gives

12∫ρu˙2+∫0T∫μuxt2≤C(T).

This together with the momentum equation yields that

μ∥uxx∥L2≤C∥ρu˙∥L2+∥P(ρ)+b22x∥L2≤C∥ρu˙∥L2+∥ρx∥L2+∥bx∥L2≤C,

and

∥ρut∥L2≤C∥ρu˙−ρuux∥L2≤C∥ρu˙∥L2+C∥ρ∥L∞∥u∥L∞∥ux∥L2≤C.

We complete the proof of Lemma 3.7. □

In conclusion, we have established a global ν-independent priori estimates for the smooth solution (ρ^δ, u^δ, b^δ) to (1.1)-(1.2) with the initial density away from vacuum (ρ0δ≥δ>0), which is uniform of δ.

4 Proof of the Theorem 1.1

Before giving the proof of the Theorem 1.1, we first give the global existence of strong solutions to 1D non-resistive MHD equations (1.3)-(1.4) when the initial density and initial magnetic approach non-zero constants at infinity.

Proposition 4.1

Suppose that the initial data (ρ̃₀, ũ₀, b̃₀)(x) satisfies

ρ~0−ρ¯∈H1(R),b~0−b¯∈H1(R),u~0∈H2(R),12ρ~0u~02+Φ(ρ~0)+(b~0−b¯)22|x|α∈L1(R) (4.1)

for some 1 < α ≤ 2, and the compatibility condition

μu~0x−P(ρ~0)−12b~02x=ρ~0g~(x),x∈R (4.2)

with some g̃ ∈ L²(ℝ). Then for any T > 0, there exists a unique global strong solution (ρ̃, ũ, b̃) to the Cauchy problem (1.3) and (1.4) such that

0≤ρ~(x,t)≤C,∀(x,t)∈R×[0,T], (4.3)

sup0≤t≤T12ρ~u~2+Φ(ρ~)+(b~−b¯)22(1+|x|α)L1+∫0Tμu~x(1+|x|α2)L22dt≤C, (4.4)

and

sup0≤t≤T∥u~∥H2+∥b~x∥L2+∥ρ~x∥L2+∥ρ~u~˙∥L2+∫0Tμ∥u~xt∥L22≤C. (4.5)

Proof

For the details, the readers can refer to [1]. □

The existence and uniqueness of global strong solution in Theorem 1.1:

Before proving Theorem 1.1, by the standard fixed point Theorem and the global uniform of δ priori estimates derived in section 3, we have the following auxiliary theorem.

Theorem 4.2

Under the same assumption in Theorem 1.1, then for any δ ∈ (0, 1), there exists a unique global solution (ρ^δ, u^δ, b^δ) to (1.1)-(1.2) with the initial data replaced by (ρ0δ,u0δ,b0) such that for any T > 0,

0<δe−CT≤ρδ(x,t)≤C,∀(x,t)∈R×[0,T],

sup0≤t≤T12(ρδuδ)2+Φ(ρδ)+(bδ−b¯)22(1+|x|α)L1(t)+∫0Tμuxδ(1+|x|α2)L22+νbxδ(1+|x|α2)L22dt≤C,

and

sup0≤t≤T∥uδ∥L2+∥uxxδ∥L2+∥uxδ∥L2+∥bxδ∥L2+∥ρxδ∥L2+∥ρδu˙δ∥L2(t)+∫0T∥uxtδ∥L22+ν∥bxxδ∥L22dt≤C,

where C is independent of ν and δ.

From Theorem 4.2, we know that there exists a unique global strong solution (ρ^δ, u^δ, b^δ), such that Lemma 3.1-Lemma 3.7 are valid when we replace (ρ, u, b) by (ρ^δ, u^δ, b^δ). With the uniform estimates for δ, we let δ → 0⁺ (take subsequence if necessary) to get a solution to (1.1)-(1.2) still denoted by (ρ, u, b) which satisfies Lemma 3.1-Lemma 3.7 by the lower semicontinuity of the norms. By a standard continuity argument c.f. [27], we can obtain the global existence of the solution as in Theorem 1.1. The uniqueness of the solution can be proved by the standard L² energy method, here we omit the details for brevity. Hence, the first part of the proof of Theorem 1.1 is completed.

The non-resistive limit in Theorem 1.1: To justify the non-resistive limit as ν → 0, we consider the the difference of these two solutions (ρ-ρ̃, u-ũ, b-b̃) which satisfy the following equations:

(ρ−ρ~)t+(ρ−ρ~)ux+ρ~(u−u~)x+(ρ−ρ~)xu+ρ~x(u−u~)=0,ρ(u−u~)t+ρu(u−u~)x−μ(u−u~)xx=−(ρ−ρ~)(u~t+u~u~x)−ρ(u−u~)u~x−P(ρ)−P(ρ~)x−b2−b~22x,(b−b~)t+ux(b−b~)+b~(u−u~)x+u(b−b~)x+(u−u~)b~x=νbxx. (4.6)

Firstly, multiplying the equation (4.6)₁ by 2(ρ − ρ̃), integrating the resultant over ℝ, and integrating by parts, it follows from Hölder's inequality, the Gagliardo-Nirenberg inequality, the Cauchy inequality, Lemma 3.4 and Proposition 4.1 that

ddt∫(ρ−ρ~)2=−∫(ρ−ρ~)2ux−2∫ρ~(ρ−ρ~)(u−u~)x−2∫ρ~x(ρ−ρ~)(u−u~)≤C(∥ux∥L∞∥ρ−ρ~∥L22+∥ρ~∥L∞∥ρ−ρ~∥L2∥(u−u~)x∥L2+∥ρ~x∥L2∥ρ−ρ~∥L2∥u−u~∥L∞)≤C∥ux∥L212∥uxx∥L212∥ρ−ρ~∥L22+∥(u−u~)x∥L2+∥u−u~∥L212∥(u−u~)x∥L212∥ρ−ρ~∥L2≤ε∥(u−u~)x∥L22+C1+∥uxx∥L2∥ρ−ρ~∥L22+1. (4.7)

Secondly, we multiply (4.6)₂ by 2(u − ũ), integrate the resultant over ℝ, integrate by parts, and use Hölder's inequality, the Gagliardo-Nirenberg inequality, Young's inequality, the fact that ∥u−u~∥L22 ≤ C1+∥(u−u~)x∥L22 (see (3.17)) and Proposition 4.1 to get

ddt∫ρ(u−u~)2+2μ∫(u−u~)x2=−∫2(ρ−ρ~)(u−u~)(u~t+u~u~x)−∫2ρ(u−u~)2u~x−∫2P(ρ)−P(ρ~)x(u−u~)−∫(b2−b~2)x(u−u~)≤C(∥u~˙∥L2∥ρ−ρ~∥L2∥u−u~∥L∞+∥u~x∥L∞∥ρ(u−u~)∥L22+(∥ρ−ρ~∥L2+∥b−b~∥L2)∥(u−u~)x∥L2)≤C[∥u~˙∥L2∥u−u~∥L212∥(u−u~)x∥L212∥ρ−ρ~∥L2+∥u~x∥L212∥u~xx∥L212∥ρ(u−u~)∥L22+(∥ρ−ρ~∥L2+∥b−b~∥L2)∥(u−u~)x∥L2]≤ε∥(u−u~)x∥L22+C1+∥ρ−ρ~∥L22+∥b−b~∥L22+∥ρ(u−u~)∥L221+∥u~˙∥L22+∥u~xx∥L2. (4.8)

Thirdly, multiplying the equation (4.6)₃ by 2(b − b̃), integrating the resultant over ℝ, integrating by parts and using Proposition 4.1, the third equation of (1.3), the fact that ∥b~∥L∞≤C∥b~∥L212∥b~x∥L212, we deduce

ddt∫(b−b~)2=−∫(b−b~)2ux−2∫b~(b−b~)(u−u~)x−2∫(b−b~)(u−u~)b~x+2ν∫(b−b~)bxx≤C(∥ux∥L∞∥b−b~∥L22+∥b~∥L∞∥b−b~∥L2∥(u−u~)x∥L2+∥b−b~∥L2∥b~x∥L2∥u−u~∥L∞+ν∥b−b~∥L2∥bxx∥L2)≤ε∥(u−u~)x∥L22+C1+∥uxx∥L2∥b−b~∥L22+ν2∥bxx∥L22. (4.9)

Then combining (4.7)-(4.9) and choosing ε > 0 sufficiently small, we have

ddt∫(ρ−ρ~)2+ρ(u−u~)2+(b−b~)2+2μ∫(u−u~)x2≤C1+∥uxx∥L2+∥u~xx∥L2+∥u~˙∥L22∥ρ−ρ~∥L22+∥b−b~∥L22+∥ρ(u−u~)∥L22+1+ν2∥bxx∥L22,

which together with Gronwall's inequality, Lemma 3.5 and Proposition 4.1 yields

∫(ρ−ρ~)2+ρ(u−u~)2+(b−b~)2+2μ∫(u−u~)x2≤Cν2∫0T∥bxx∥L22dtexpC∫0T1+∥uxx∥L2+∥u~xx∥L2+∥u~˙∥L22dt≤Cν, (4.10)

where we have used Lemmas 3.5 and 3.7, Proposition 4.1 and the following fact:

∥u~˙∥L2≤C∥ρ¯u~˙∥L2≤C∥(ρ¯−ρ~)u~˙∥L2+∥ρ~u~˙∥L2≤C∥ρ~−ρ¯∥L2∥u~˙∥L∞+1≤ε∥u~˙∥L2+C(∥u~˙x∥L2+1),

which gives

∥u~˙∥L2≤C(∥u~t+u~u~xx∥L2+1)≤C∥u~xt∥L2+∥u~x∥L42+∥u~u~xx∥L2+1≤C∥u~xt∥L2+∥u~xx∥L2+1∈L2(0,T).

Moreover, we have

∫ρ¯(u−u~)2=∫(ρ¯−ρ)(u−u~)2+∫ρ(u−u~)2≤C∥ρ−ρ¯∥L2∥u−u~∥L42+Cν≤C∥ρ−ρ¯∥L2∥u−u~∥L232∥(u−u~)x∥L212+Cν≤ε∥u−u~∥L22+C∥ρ−ρ¯∥L24∥(u−u~)x∥L22+Cν≤ε∥u−u~∥L22+Cν2+Cν,

which together with (4.10) yields that

sup0≤t≤T∥u−u~∥L22≤Cν. (4.11)

Then, we obtain the desired results of Theorem 1.1. □

Acknowledgments

Z. Li is supported by the National Natural Science Foundation of China (No.11671319, No.11931013), Fund of HPU (No.B2016-57). H. Wang is supported by the National Natural Science Foundation of China (Grant No.11901066), the Natural Science Foundation of Chongqing (Grant No.cstc2019jcyj-msxmX0167) and Project No.2019CDXYST0015 and No. 2020CDJQY-A040 supported by the Fundamental Research Funds for the Central Universities. Y. Ye is supported by the National Natural Science Foundation of China (No.11701145) and China Postdoctoral Science Foundation (No.2020M672196).

Conflict of interest: The authors declare that there is no conflict of interests regarding the publication of this article.

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Received: 2021-05-26

Accepted: 2021-09-12

Published Online: 2021-11-29

Published in Print: 2021-11-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

On non-resistive limit of 1D MHD equations with no vacuum at infinity

Abstract

1 Introduction and Main Results

Theorem 1.1

Remark 1.1

Remark 1.2

2 Preliminaries

Lemma 2.1

Lemma 2.2

Proof

Remark 2.1

Lemma 2.3

3 Global ν-independent estimates for (1.1) and (1.2)

Lemma 3.1

Proof

Lemma 3.2

Proof

Lemma 3.3

Proof

Lemma 3.4

Proof

Lemma 3.5

Proof

Lemma 3.6

Proof

Lemma 3.7

Proof

4 Proof of the Theorem 1.1

Proposition 4.1

Proof

Theorem 4.2

Acknowledgments

References

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