Notes on the Global Well-Posedness for the Maxwell-Navier-Stokes System

Abstract

Masmoudi (2010) obtained global well-posedness for 2D Maxwell-Navier-Stokes system. In this paper, we reprove global existence of regular solutions to the 2D system by using energy estimates and Brezis-Gallouet inequality. Also we obtain a blow-up criterion for solutions to 3D Maxwell-Navier-Stokes system.

1. Introduction

In this paper, we consider Maxwell-Navier-Stokes equations in ℝ^𝑑 (𝑑=2,3) as follows: 𝜕𝑣+()𝜕𝑡𝑣⋅∇𝑣−Δ𝑣+∇𝑝=𝑗×𝐵inℝ^𝑑×(0,𝑇),𝜕𝐸𝜕𝑡−∇×𝐵=−𝑗inℝ^𝑑×(0,𝑇),𝜕𝐵𝜕𝑡+∇×𝐸=0inℝ^𝑑×(0,𝑇),∇⋅𝑣=∇⋅𝐵=0inℝ^𝑑×(0,𝑇),𝑗=𝐸+𝑣×𝐵, (1) subject to the initial data 𝑣(𝑥,0)=𝑣₀(𝑥),𝐸(𝑥,0)=𝐸₀(𝑥),𝐵(𝑥,0)=𝐵₀(𝑥). (2) Here 𝑣, 𝐸, and 𝐵∶ℝ^𝑑×(0,𝑇)→ℝ³ are vector fields defined on ℝ^𝑑 (𝑑=2 or 3). Vector fields 𝑣, 𝐸, and 𝐵 denote fluid velocity, electric fields and magnetic fields, respectively. 𝑝 denotes the scalar pressure and 𝑗 is the electric current given by Ohm’s law. 𝑗×𝐵 represents the Lorentz force. Here we put the viscosity and the electric resistivity to be 1 for the simplification. Note that in 2D case, vector fields 𝑣, 𝐸, and 𝐵 can be understood as 𝑣(𝑥,𝑡)=(𝑣₁(𝑥₁,𝑥₂,𝑡),𝑣₂(𝑥₁,𝑥₂,𝑡),0), and so forth.

For the compatibility of the initial data, we assume that ∇⋅𝑣₀=∇⋅𝐵₀=0. (3)

Since the divergence-free condition of the magnetic field is conserved, ∇⋅𝐵=0 in (1) is not necessary in general if we assume the divergence-free condition for the initial data of the magnetic field in ℝ^𝑑. In many physical situations, current displacement term 𝜕_𝑡𝐸 is neglected because the physical coefficient for this term is very small (∼1/𝑐², where 𝑐 denotes the speed of light). But mathematically, the presence of the term 𝜕_𝑡𝐸 in the second equation (Ampere-Maxwell equation) preserves the hyperbolic nature of the Maxwell equation in the Maxwell-Navier-Stokes equations (see [1, 2] and references therein). Also we remark that full Maxwell-Navier-Stokes equations have been used for the accurate computation of electromagnetic hypersonics in aerothermodynamics (see [3, 4] and references therein). For further physical motivations, see [5].

Neglecting the current displacement term, Maxwell-Navier-Stokes system is reduced to the usual MHD system. There have been many extensive mathematical studies for the existence, blow-up criterion, and regularity criterion of MHD and related models (see [6–12] and references therein). Recently, Maxwell-Navier-Stokes system has been receiving much mathematical attention after pioneering work of Masmoudi [2]. In [2], global existence of regular solutions to (1) in ℝ² is proved by using the Besov-type 𝐿 space technique developed by Chemin and Lerner [13]. In [1, 14], the local existence of mild solution and the global existence of (1) with small data have been studied. Duan [15] studied large time behaviour of solutions to (1). In [16], Ibrahim and Yoneda obtained local-in-time existence for nondecaying initial data in torus. Also Germain and Masmoudi [17] studied global existence of solutions to Euler-Maxwell equations with small data and Jang and Masmoudi [18] mathematically derived Ohm’s law from the kinetic equation.

The aim of this paper is to study the global well-posedness for (1) using the standard energy estimates. We obtain the local-in-time existence of 𝐻² solution by using the standard mollifier technique (see Proposition 4) and re-prove the global existence of 𝐻² solution for 2D Maxwell-Navier-Stokes system (see Theorem 1) by using standard energy estimates and Brezis-Gallouet inequality, which was used to prove global existence of regular solution for the partial viscous Boussinesq equations by Chae [19]. Also we provide blow-up criterion of regular solutions to 3D Maxwell-Navier-Stokes equations (see Theorem 2).

We state our main results in the following.

Theorem 1. Assume that (𝑣₀,𝐸₀,𝐵₀)∈𝐻²(ℝ²) and ∇⋅𝑣₀=∇⋅𝐵₀=0. Then, for any 𝑇>0, there exists a solution to 2D Maxwell-Navier-Stokes system (1) such that (𝑣,𝐸,𝐵)∈𝐶((0,𝑇];𝐻²) and (∇𝑣,𝑗)∈𝐿²(0,𝑇;𝐻²).

Theorem 2. Suppose that (𝑣₀,𝐸₀,𝐵₀)∈𝐻²(ℝ³) and ∇⋅𝑣₀=∇⋅𝐵₀=0. If 𝑇^∗, the maximal existence time of the local existence of regular solution to 3D Maxwell-Navier-Stokes system (1), is finite, then ^𝑇∗₀‖𝑣(𝑡)‖²_𝐿^∞+‖𝐵(𝑡)‖_𝐿^8/3_∞𝑑𝑡=∞. (4)

Remark 3. (1) As logarithmic inequality has been used in [2], Brezis-Gallouet inequality gives logarithmic-type estimates. But it provides double exponential bound compared with exponential bound in [2].
(2) The presence of the current displacement term 𝜕_𝑡𝐸 makes Maxwell-Navier-Stokes system do not enjoy the scaling invariance property of the usual Navier-Stokes system, 𝑣_𝜆(𝑥,𝑡)=𝜆𝑣(𝜆𝑥,𝜆²𝑡). In Theorem 2, ∫^𝑇₀‖𝑣(𝑡)‖²_𝐿^∞𝑑𝑡 is concurrent with the usual scaling invariant norm of solutions to 3D Navier-Stokes equations.

The rest of this paper is organized as follows. In Section 2, we provide the local-in-time existence of regular solution to 2D and 3D Maxwell-Navier-Stokes systems and global existence of 2D Maxwell-Navier-Stokes system with large data. In Section 3, we provide the blow-up criterion for 𝐻² solution to 3D Maxwell-Navier-Stokes system.

2. Local Existence and Global Well-Posedness

At first, we note that one can have the energy identity in two or three dimensions: 12𝑑‖𝑣‖𝑑𝑡^2𝐿2+‖𝐵‖²_𝐿2+‖𝐸‖²_𝐿2+𝑗²_𝐿2+‖∇𝑣‖²_𝐿2=0. (5)

The previously energy inequality can be justified for local in time regular solution in the following proposition. In the following, 𝐶 denotes a harmless constant which may change from one line to the other. We prove local-in-time existence of 𝐻² solution using the standard energy estimates.

Proposition 4. Let (𝑢₀,𝐸₀,𝐵₀)∈𝐻²(ℝ^𝑑) (𝑑=2 or 3) with ∇⋅𝑢₀=∇⋅𝐵₀=0. Then there exists 𝑇=𝑇(‖𝑢₀‖_𝐻2,‖𝐸₀‖_𝐻2,‖𝐵₀‖_𝐻2) such that there exists a unique solution (𝑢,𝐸,𝐵)∈𝐿^∞(0,𝑇;𝐻²(ℝ^𝑑))∩𝐿𝑖𝑝(0,𝑇;𝐿²).

Proof. We use the mollifier method as described in [20]. Although the details are similar to [20], we provide some a priori estimates for the reader’s sake. We consider the standard mollifier operator 𝒥_𝜖𝑓=𝜌_𝜖∗𝑓,𝜌_𝜖(⋅)=1𝜖^𝑑𝜌⋅𝜖^𝑑, (6)
where 𝜌∈𝐶^∞₀(ℝ^𝑑), and 𝜌≥0, ∫_ℝ𝑑𝜌𝑑𝑥=1.
We introduce the following regularized system of (1): 𝜕_𝑡𝑣^𝜖+𝒥_𝜖𝒥_𝜖𝑣^𝜖⋅∇𝒥_𝜖𝑣^𝜖−Δ𝒥²_𝜖𝑣^𝜖+∇𝑝^𝜖=𝒥_𝜖𝒥²_𝜖𝑗^𝜖×𝒥_𝜖𝐵^𝜖inℝ^𝑑×(),𝜕0,𝑇_𝑡𝐸^𝜖−∇×𝒥²_𝜖𝐵^𝜖=−𝒥²_𝜖𝑗^𝜖inℝ^𝑑×(),𝜕0,𝑇_𝑡𝐵^𝜖+∇×𝒥²_𝜖𝐸^𝜖=0inℝ^𝑑×(0,𝑇),∇⋅𝑣^𝜖=∇⋅𝐵^𝜖=0inℝ^𝑑×(),𝑗0,𝑇_𝜖=𝐸^𝜖+𝒥_𝜖𝑣^𝜖×𝒥_𝜖𝐵^𝜖, (7)
with initial data (𝑣^𝜖₀,𝐸^𝜖₀,𝐵^𝜖₀)=(𝒥_𝜖𝑣₀,𝒥_𝜖𝐸₀,𝒥_𝜖𝐵₀).
Taking the 𝐿² inner product of (7)₁, (7)₂, and (7)₃ with 𝑣^𝜖, 𝐸^𝜖, 𝐵^𝜖, respectively, we obtain 12𝑑𝑣𝑑𝑡^𝜖2𝐿2+𝐸^𝜖2_𝐿2+𝐵^𝜖2_𝐿2+∇𝒥_𝜖𝑣^𝜖2_𝐿2+𝒥_𝜖𝑗^𝜖2_𝐿21=−2_ℝ^𝑑𝒥_𝜖𝑣^𝜖⋅∇𝒥_𝜖𝑣^𝜖2+𝑑𝑥_ℝ𝑑∇×𝒥_𝜖𝐵^𝜖⋅𝒥_𝜖𝐸^𝜖−𝑑𝑥_ℝ𝑑∇×𝒥_𝜖𝐸^𝜖⋅𝒥_𝜖𝐵^𝜖+𝑑𝑥_ℝ𝑑𝒥²_𝜖𝑗^𝜖×𝒥_𝜖𝐵^𝜖⋅𝒥_𝜖𝑣^𝜖+𝑑𝑥_ℝ𝑑𝒥²_𝜖𝑗^𝜖⋅𝒥_𝜖𝑣^𝜖×𝒥_𝜖𝐵^𝜖𝑑𝑥=0. (8)
We compute the derivative 𝐷^𝛼, 𝛼 is a multi-index such that |𝛼|≤2, of (7), multiply them by 𝐷^𝛼𝑣^𝜖, 𝐷^𝛼𝐸^𝜖, and 𝐷^𝛼𝐵^𝜖, respectively, and integrate them over ℝ^𝑑 to obtain 12𝑑𝑣𝑑𝑡^𝜖2𝐻2+𝐸^𝜖2_𝐻2+𝐵^𝜖2_𝐻2+∇𝒥_𝜖𝑣^𝜖2_𝐻2+𝒥_𝜖𝑗^𝜖2_𝐻2≤𝐶𝒥_𝜖𝑣^𝜖⊗𝒥_𝜖𝑣^𝜖_𝐻2∇𝒥_𝜖𝑣^𝜖_𝐻2+𝐶𝒥²_𝜖𝑗^𝜖×𝒥_𝜖𝐵^𝜖_𝐻2𝒥_𝜖𝑣^𝜖_𝐻2+𝐶𝒥²_𝜖𝑗^𝜖_𝐻2𝒥_𝜖𝑣^𝜖×𝒥_𝜖𝐵^𝜖_𝐻2≤𝐶𝒥_𝜖𝑣^𝜖4_𝐻2+𝒥_𝜖𝐵^𝜖4_𝐻2+12𝒥_𝜖∇𝑣^𝜖2_𝐻2+𝒥_𝜖𝑗^𝜖2_𝐻2. (9)
In the previously mentioned, 𝒥_𝜖𝑣^𝜖⊗𝒥_𝜖𝑣^𝜖 denotes a tensor (𝒥_𝜖𝑣^𝜖_𝑖𝒥_𝜖𝑣^𝜖_𝑗)_{1≤𝑗≤𝑑}.
Using Picard’s theorem, these estimates imply local existence of solution.

The main ingredient of the proof of Theorem 1 is the following Brezis-Gallouet inequality (logarithmic Sobolev inequality): 𝑓_𝐿∞≤𝐶1+𝑓_𝐿2+∇𝑓_𝐿2log⁺Δ𝑓_𝐿2^1/2,𝑓∈𝐻²ℝ². (10) Here log⁺𝑎 denotes log(𝑒+𝑎).

Proof of Theorem 1. We provide a priori estimates on the regular solutions. Let 𝑇 be a finite maximal time of existence in Proposition 4. By obtaining 𝐻² bound on (0,𝑇] of solution, we can continue solution beyond 𝑇 by using Proposition 4.
Taking curl operator on (1)₁ and 𝜕_𝑖=𝜕/𝜕𝑥_𝑖 (𝑖=1,2) operator on (1)₂ and (1)₃, we have 𝜕𝜔+()𝜕𝑡𝑣⋅∇𝜔−Δ𝜔=∇×𝑗×𝐵,inℝ²×(),𝜕𝜕0,𝑇_𝑖𝐸𝜕𝑡−∇×𝜕_𝑖𝐵=−𝜕_𝑖𝑗,inℝ²×(),𝜕𝜕0,𝑇_𝑖𝐵𝜕𝑡+∇×𝜕_𝑖𝐸=0,inℝ²×(0,𝑇). (11)
(i)  𝐻¹  Estimates. Taking scalar product (11) with 𝜔, 𝜕_𝑖𝐸, and 𝜕_𝑖𝐵, respectively, and summing over 𝑖=1,2, we have 12𝑑‖𝜔‖𝑑𝑡^2𝐿2+‖∇𝜔‖²_𝐿2=_ℝ21∇×𝑗×𝐵⋅𝜔𝑑𝑥,2𝑑‖‖𝑑𝑡∇𝐸²_𝐿2=_𝑖ℝ2∇×𝜕_𝑖𝐵⋅𝜕_𝑖𝐸𝑑𝑥−_ℝ21∇𝑗⋅∇𝐸𝑑𝑥,2𝑑‖‖𝑑𝑡∇𝐵²_𝐿2=−_𝑖ℝ2∇×𝜕_𝑖𝐸⋅𝜕_𝑖𝐵𝑑𝑥. (12)(13)
Using the identity _ℝ2∇×𝜕_𝑖𝐵⋅𝜕_𝑖𝐸𝑑𝑥=_ℝ2∇×𝜕_𝑖𝐸⋅𝜕_𝑖𝐵𝑑𝑥 (14)
and 𝐸=𝑗−𝑣×𝐵, we obtain 12𝑑‖‖𝑑𝑡∇𝐸²_𝐿2+‖∇𝐵‖²_𝐿2+∇𝑗²_𝐿2=_ℝ2∇𝑗⋅∇(𝑣×𝐵)𝑑𝑥. (15)
In the following, 𝜖 denotes a sufficiently small positive number. Since it holds that ∇×(𝑗×𝐵)=(𝐵⋅∇)𝑗, we estimate the right-hand side of (12) using Young’s inequality and interpolation inequality: ^ℝ2≤‖𝐵‖∇×𝑗×𝐵⋅𝜔𝑑𝑥_𝐿4∇𝑗_𝐿2‖𝜔‖_𝐿⁴≤𝐶‖𝐵‖_𝐿^1/2₂‖∇𝐵‖_𝐿^1/2₂‖𝜔‖_𝐿^1/2₂‖∇𝜔‖_𝐿^1/2₂∇𝑗_𝐿2≤𝐶‖𝜔‖²_𝐿2‖∇𝐵‖²_𝐿2+𝜖‖∇𝜔‖²_𝐿2+𝜖∇𝑗²_𝐿2, (16)
where 𝜖 is a small positive number. Also we have _ℝ2()≤∇𝑗⋅∇𝑣×𝐵𝑑𝑥_ℝ2𝑣|+∇𝑗|||∇𝐵𝑑𝑥_ℝ2∇𝑗|𝐵||∇𝑣|𝑑𝑥=𝐼+𝐼𝐼. (17)
We estimate 𝐼≤𝐶‖𝑣‖²_𝐿^∞‖∇𝐵‖²_𝐿2+𝜖∇𝑗²_𝐿2,𝐼𝐼≤‖𝐵‖_𝐿⁴‖∇𝑣‖_𝐿⁴∇𝑗_𝐿2≤𝐶‖𝐵‖²_𝐿2‖∇𝑣‖²_𝐿2‖∇𝐵‖²_𝐿2+𝜖‖Δ𝑣‖²_𝐿2+𝜖∇𝑗²_𝐿2. (18)
Collecting previous estimates, we have 𝑑‖𝜔‖𝑑𝑡^2𝐿2+‖∇𝐸‖²_𝐿2+‖∇𝐵‖²_𝐿2+‖∇𝜔‖²_𝐿2+∇𝑗²_𝐿2≤𝐶‖𝜔‖²_𝐿2‖∇𝐵‖²_𝐿2+𝐶‖𝑣‖²_𝐿^∞‖∇𝐵‖²_𝐿2+𝐶‖∇𝐵‖²_𝐿2‖∇𝑣‖²_𝐿2‖∇𝐵‖²_𝐿2. (19)
(ii)  𝐻² Estimates. Taking Δ operator on (1)₁, (1)₂, and (1)₃ and 𝐿² scalar product with Δ𝑣, Δ𝐸, and Δ𝐵, respectively, we have 12𝑑‖‖𝑑𝑡Δ𝑣²_𝐿2+‖∇Δ𝑣‖²_𝐿2≤𝐶_ℝ2|∇𝑣|𝐷²𝑣+𝑑𝑥_ℝ2Δ𝑗×𝐵|Δ𝑣|𝑑𝑥∶=𝐼₁+𝐼₂,12𝑑‖‖𝑑𝑡Δ𝐸²_𝐿2+‖Δ𝐵‖²_𝐿2+Δ𝑗²_𝐿2≤_ℝ2Δ𝑗|Δ(𝑣×𝐵)|𝑑𝑥∶=𝐼₃. (20)
We estimate 𝐼₁, 𝐼₂, and 𝐼₃ using interpolation inequality, Young’s inequality, and Hölder’s inequality: 𝐼₁≤𝐶‖∇𝑣‖_𝐿⁴‖Δ𝑣‖_𝐿⁴‖Δ𝑣‖_𝐿²≤𝐶‖∇𝑣‖_𝐿^1/2₂‖Δ𝑣‖_𝐿^3/2₂‖∇Δ𝑣‖_𝐿^1/2₂≤𝐶‖∇𝑣‖_𝐿^2/3₂‖Δ𝑣‖²_𝐿2+𝜖‖∇Δ𝑣‖²_𝐿2,𝐼₂≤𝐶_ℝ2∇𝑗|∇𝐵||Δ𝑣|𝑑𝑥+𝐶_ℝ2Δ𝑗|𝐵||Δ𝑣|𝑑𝑥+𝐶_ℝ2𝑗|Δ𝐵||Δ𝑣|𝑑𝑥∶=𝐼₂₁+𝐼₂₂+𝐼₂₃. (21)(22)
Each term can be estimated by the standard interpolation inequality and Young’s inequality as follows: 𝐼₂₁≤𝐶∇𝑗_𝐿4‖∇𝐵‖_𝐿⁴‖Δ𝑣‖_𝐿²≤𝐶𝑗_𝐿^1/4₂Δ𝑗_𝐿^3/4₂‖𝐵‖_𝐿^1/4₂‖Δ𝐵‖_𝐿^3/4₂‖∇𝑣‖_𝐿^1/2₂‖∇Δ𝑣‖_𝐿^1/2₂≤𝐶𝑗_𝐿^2/3₂‖𝐵‖_𝐿^2/3₂‖∇𝑣‖_𝐿^4/3₂‖Δ𝐵‖²_𝐿2+𝜖Δ𝑗²_𝐿2≤𝐶𝑗²_𝐿2+‖∇𝑣‖²_𝐿2‖Δ𝐵‖²_𝐿2+𝜖Δ𝑗²_𝐿2,𝐼₂₂≤𝐶Δ𝑗_𝐿2‖𝐵‖_𝐿⁴‖Δ𝑣‖_𝐿⁴≤𝐶Δ𝑗_𝐿2‖𝐵‖_𝐿^3/4₂‖Δ𝐵‖_𝐿^1/4₂‖Δ𝑣‖_𝐿^1/2₂‖∇Δ𝑣‖_𝐿^1/2₂≤𝜖Δ𝑗²_𝐿2+𝐶‖Δ𝐵‖_𝐿^1/2₂‖Δ𝑣‖_𝐿^1/2₂‖∇Δ𝑣‖_𝐿²≤𝜖Δ𝑗²_𝐿2+𝜖‖∇Δ𝑣‖²_𝐿2+𝐶𝑗_𝐿2‖∇𝑣‖_𝐿²‖Δ𝐵‖²_𝐿2,𝐼₂₃≤𝐶𝑗_𝐿∞‖Δ𝐵‖_𝐿²‖Δ𝑣‖_𝐿²≤𝐶𝑗_𝐿2Δ𝑗_𝐿^1/2₂‖Δ𝐵‖_𝐿²‖∇𝑣‖_𝐿^1/2₂‖∇Δ𝑣‖_𝐿^1/2₂≤𝜖Δ𝑗²_𝐿2+𝐶𝑗_𝐿^2/3₂‖Δ𝐵‖_𝐿^4/3₂‖∇𝑣‖_𝐿^2/3₂‖∇Δ𝑣‖_𝐿^2/3₂≤𝜖Δ𝑗²_𝐿2+𝜖‖∇Δ𝑣‖²_𝐿2+𝐶𝑗_𝐿2‖∇𝑣‖_𝐿²‖Δ𝐵‖²_𝐿2. (23)
𝐼₃ can be written as 𝐼₃≤𝐶_ℝ2Δ𝑗|∇𝑣||∇𝐵|𝑑𝑥+𝐶_ℝ2Δ𝑗|Δ𝑣||𝐵|𝑑𝑥+𝐶_ℝ2Δ𝑗|𝑣||Δ𝐵|𝑑𝑥∶=𝐼₃₁+𝐼₃₂+𝐼₃₃,𝐼₃₁≤𝐶Δ𝑗_𝐿2‖∇𝑣‖_𝐿⁴‖∇𝐵‖_𝐿⁴≤𝐶Δ𝑗_𝐿2‖∇𝑣‖_𝐿^3/4₂‖∇Δ𝑣‖_𝐿^1/4₂‖𝐵‖_𝐿^1/4₂‖Δ𝐵‖_𝐿^3/4₂≤𝜖Δ𝑗²_𝐿2+𝜖‖∇Δ𝑣‖²_𝐿2+𝐶‖∇𝑣‖²_𝐿2‖Δ𝐵‖²_𝐿2. (24)
The same as the estimate of 𝐼₂₂, we obtain 𝐼₃₂≤𝜖Δ𝑗²_𝐿2+𝜖‖∇Δ𝑣‖²_𝐿2+𝐶𝑗_𝐿2‖∇𝑣‖_𝐿²‖Δ𝐵‖²_𝐿2. (25)
Also we have 𝐼₃₃≤𝐶Δ𝑗_𝐿2‖𝑣‖_𝐿^∞‖Δ𝐵‖_𝐿²≤𝜖Δ𝑗²_𝐿2+𝐶‖𝑣‖²_𝐿^∞‖Δ𝐵‖²_𝐿2. (26)
Therefore, we have 𝑑‖‖𝑑𝑡Δ𝑣²_𝐿2+‖Δ𝐸‖²_𝐿2+‖Δ𝐵‖²_𝐿2+‖∇Δ𝑣‖²_𝐿2+Δ𝑗²_𝐿2≤𝐶1+‖∇𝑣‖²_𝐿2+𝑗²_𝐿2+‖𝑣‖²_𝐿^∞×‖Δ𝑣‖²_𝐿2+‖Δ𝐵‖²_𝐿2. (27)
(iii)  Use of Brezis-Gallouet Inequality. Using Brezis-Gallouet inequality, we obtain 𝑑‖‖𝑑𝑡Δ𝑣²_𝐿2+‖Δ𝐸‖²_𝐿2+‖Δ𝐵‖²_𝐿2+‖∇Δ𝑣‖²_𝐿2+Δ𝑗²_𝐿2≤𝐶1+‖∇𝑣‖²_𝐿2+𝑗²_𝐿2+‖𝑣‖²_𝐿2+‖∇𝑣‖²_𝐿2×‖Δ𝑣‖²_𝐿2+‖Δ𝐸‖²_𝐿2+‖Δ𝐵‖²_𝐿2×log⁺‖Δ𝑣‖²_𝐿2+‖Δ𝐸‖²_𝐿2+‖Δ𝐵‖²_𝐿2. (28)
Let 𝑦(𝑡)=‖Δ𝑣‖²_𝐿2+‖Δ𝐸‖²_𝐿2+‖Δ𝐵‖²_𝐿2, and let 𝑧(𝑡)=1+‖Δ𝑣‖²_𝐿2+‖𝑗‖²_𝐿2+‖𝑣‖²_𝐿2+‖∇𝑣‖²_𝐿2. Hence one has 𝑑𝑦(𝑡)(𝑡)𝑦(𝑡)𝑑𝑡≤𝐶𝑧log⁺𝑦(𝑡). (29)
Since ^𝑇₀𝑧(𝑡)𝑑𝑡≤𝐶(1+𝑇), (30)
the bound of 𝑦(𝑡) is immediate as follows: sup_{0≤𝑡≤𝑇}‖Δ𝑣‖²_𝐿2+‖Δ𝐸‖²_𝐿2+‖Δ𝐵‖²_𝐿2≤Δ𝑣₀²_𝐿2+Δ𝐸₀²_𝐿2+Δ𝐵₀²_𝐿2×expexp(𝐶(𝑇+1)). (31)This completes the proof of Theorem 1.

3. Blow-Up Criterion for 3D Maxwell-Navier-Stokes System

In this section, we provide a blow-up criterion for 𝐻² solution in Proposition 4 to 3D Maxwell-Navier-Stokes system.

Proof of Theorem 2. Assume that ^𝑇∗₀‖𝑣(𝑡)‖²_𝐿^∞+‖𝐵(𝑡)‖_𝐿^8/3_∞𝑑𝑡<∞, (32)
where 𝑇^∗ is the finite maximal existence time of a classical solution.
Similar to the computation in Section 2, one has 𝐻¹ estimates of 𝐸 and 𝐵 as follows: 12𝑑‖‖𝑑𝑡∇𝐸²_𝐿2+‖∇𝐵‖²_𝐿2+∇𝑗²_𝐿2=_ℝ3()‖‖∇𝑗⋅∇𝑣×𝐵𝑑𝑥≤𝐶∇(𝑣×𝐵)²_𝐿2+𝜖∇𝑗²_𝐿2≤𝐶‖𝐵‖²_𝐿^∞‖∇𝑣‖²_𝐿2+𝐶‖𝑣‖²_𝐿^∞‖∇𝐵‖²_𝐿2+𝜖∇𝑗²_𝐿2. (33)
𝐻¹ estimates of 𝑣 are as follows: 12𝑑‖‖𝑑𝑡∇𝑣²_𝐿2+‖Δ𝑣‖²_𝐿2≤_ℝ3|𝑣|+||∇𝑣||Δ𝑣𝑑𝑥_ℝ3|‖𝑣‖𝑗×𝐵|Δ𝑣𝑑𝑥≤𝐶²_𝐿^∞‖∇𝑣‖²_𝐿2+𝐶𝑗×𝐵²_𝐿2+𝜖‖Δ𝑣‖²_𝐿2. (34)
The estimate of ‖𝑗×𝐵‖²_𝐿2 is provided in the following: 𝑗×𝐵²_𝐿2≤𝐶‖𝐸×𝐵‖²_𝐿2+𝐶‖(𝑣×𝐵)×𝐵‖²_𝐿2≤𝐶‖𝐸‖²_𝐿2‖𝐵‖²_𝐿^∞+𝐶‖𝑣‖²_𝐿6‖𝐵‖⁴_𝐿6≤𝐶‖𝐸‖²_𝐿2‖𝐵‖²_𝐿^∞+𝐶‖∇𝑣‖²_𝐿2‖𝐵‖_𝐿^4/3₂‖𝐵‖_𝐿^8/3_∞. (35)
Thus we have 𝑑‖‖𝑑𝑡∇𝑣²_𝐿2+‖∇𝐸‖²_𝐿2+‖∇𝐵‖²_𝐿2+‖Δ𝑣‖²_𝐿2+∇𝑗²_𝐿2≤𝐶1+‖𝑣‖²_𝐿^∞+‖𝐵‖_𝐿^8/3_∞×‖∇𝑣‖²_𝐿2+‖∇𝐸‖²_𝐿2+‖∇𝐵‖²_𝐿2+𝐶‖𝐵‖²_𝐿^∞. (36)
Gronwall’s inequality gives us that ‖(∇𝑣,∇𝐸,∇𝐵)‖²_{𝐿∞(0,𝑇∗;𝐿2)}+(Δ𝑣,∇𝑗)²_{𝐿2(0,𝑇∗;𝐿2)}≤𝐶<∞. (37)
Next, we consider 𝐻² estimates.
Integrating by parts and using Young’s inequality, it follows that 12𝑑‖‖𝑑𝑡Δ𝐸²_𝐿2+‖Δ𝐵‖²_𝐿2+Δ𝑗²_𝐿2≤𝐶‖Δ(𝑣×𝐵)‖²_𝐿2+𝜖Δ𝑗²_𝐿2≤𝐶‖Δ𝑣‖²_𝐿2‖𝐵‖²_𝐿^∞+𝐶‖𝑣‖²_𝐿^∞‖Δ𝐵‖²_𝐿2+𝐶‖∇𝑣‖²_𝐿4‖∇𝐵‖²_𝐿4+𝜖Δ𝑗²_𝐿2≤𝐶‖Δ𝑣‖²_𝐿2‖𝐵‖²_𝐿^∞+𝐶‖𝑣‖²_𝐿^∞‖Δ𝐵‖²_𝐿2+𝐶‖𝑣‖_𝐿^∞‖𝐵‖_𝐿^∞‖Δ𝑣‖_𝐿²‖Δ𝐵‖_𝐿²+𝜖Δ𝑗²_𝐿2. (38)
Similarly, it follows that 12𝑑‖‖𝑑𝑡Δ𝑣²_𝐿2≤𝐶‖∇(𝑣⋅∇𝑣)‖²_𝐿2+𝐶∇𝑗×𝐵²_𝐿2+𝜖‖∇Δ𝑣‖²_𝐿2≤𝐶‖∇𝑣‖⁴_𝐿4+𝐶‖𝑣‖²_𝐿^∞‖Δ𝑣‖²_𝐿2+𝐶‖∇𝐸‖²_𝐿6‖𝐵‖²_𝐿3+𝐶‖𝐸‖²_𝐿6‖∇𝐵‖²_𝐿3+𝐶‖∇(𝑣×𝐵)‖²_𝐿2‖𝐵‖²_𝐿^∞+𝐶‖𝑣×𝐵‖²_𝐿3‖∇𝐵‖²_𝐿6. (39)
Using the interpolation inequality, one has ‖∇𝑣‖⁴_𝐿4≤𝐶‖𝑣‖²_𝐿^∞‖Δ𝑣‖²_𝐿2. (40)
Interpolation inequality and Young’s inequality produce that ‖𝐸‖²_𝐿6‖∇𝐵‖²_𝐿3≤𝐶‖∇𝐸‖²_𝐿2‖∇𝐵‖_𝐿²‖Δ𝐵‖_𝐿²≤𝐶‖∇𝐸‖²_𝐿2‖∇𝐵‖²_𝐿2+‖Δ𝐵‖²_𝐿2. (41)
Similarly, we estimate that ‖∇(𝑣×𝐵)‖²_𝐿2‖𝐵‖²_𝐿^∞≤𝐶‖∇𝑣‖²_𝐿3‖∇𝐵‖²_𝐿2+‖∇𝑣‖²_𝐿2‖∇𝐵‖²_𝐿3,‖𝑣×𝐵‖²_𝐿3‖∇𝐵‖²_𝐿6≤𝐶‖𝑣‖²_𝐿^∞‖𝐵‖²_𝐿3‖Δ𝐵‖²_𝐿2. (42)
We already know that ‖(∇𝑣,∇𝐸,∇𝐵)‖²_{𝐿∞(0,𝑇∗;𝐿2)}<𝐶. (43)
Gathering all the estimates, we achieve 𝑑‖‖𝑑𝑡Δ𝑣²_𝐿2+‖Δ𝐸‖²_𝐿2+‖Δ𝐵‖²_𝐿2+‖∇Δ𝑣‖²_𝐿2+Δ𝑗²_𝐿2≤𝐶1+‖𝑣‖²_𝐿^∞+‖𝐵‖²_𝐿^∞×1+‖Δ𝑣‖²_𝐿2+‖Δ𝐸‖²_𝐿2+‖Δ𝐵‖²_𝐿2. (44)
Using Gronwall’s inequality, we conclude that ‖(Δ𝑣,Δ𝐸,Δ𝐵)‖²_{𝐿∞(0,𝑇∗;𝐿2)}+∇Δ𝑣,Δ𝑗²_{𝐿2(0,𝑇∗;𝐿2)}≤𝐶<∞. (45)This completes the proof of Theorem 2.

Acknowledgments

Ensil Kang’s work was supported by research fund from Chosun University, 2009, and Jihoom Lee’s work was partly supported by the National Research Foundation of Korea (NRF-2011-0006697) and Chung-Ang University Research Grants in 2013.