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

and Applied Analysis 3 We compute the derivative D, α is a multi-index such that |α| ≤ 2, of (7), multiply them by DV, DE, and DB, respectively, and integrate them over R to obtain 1 2 d dt ( 󵄩󵄩󵄩󵄩V ε󵄩󵄩󵄩󵄩 2 H2 + 󵄩󵄩󵄩󵄩E ε󵄩󵄩󵄩󵄩 2 H2 + 󵄩󵄩󵄩󵄩B ε󵄩󵄩󵄩󵄩 2 H2 ) + 󵄩󵄩󵄩󵄩∇JεV ε󵄩󵄩󵄩󵄩 2 H2 + 󵄩󵄩󵄩󵄩Jεj ε󵄩󵄩󵄩󵄩 2 H2 ≤ C 󵄩󵄩󵄩󵄩JεV ε ⊗ JεV ε󵄩󵄩󵄩󵄩H2 󵄩󵄩󵄩󵄩∇JεV ε󵄩󵄩󵄩󵄩H2 + C 󵄩󵄩󵄩󵄩 J 2 ε j ε × JεB 󵄩󵄩󵄩󵄩H2 󵄩󵄩󵄩󵄩JεV ε󵄩󵄩󵄩󵄩H2 + C 󵄩󵄩󵄩󵄩 J 2 ε j 󵄩󵄩󵄩󵄩H2 󵄩󵄩󵄩󵄩JεV ε × JεB ε󵄩󵄩󵄩󵄩H2 ≤ C ( 󵄩󵄩󵄩󵄩JεV ε󵄩󵄩󵄩󵄩 4 H2 + 󵄩󵄩󵄩󵄩JεB ε󵄩󵄩󵄩󵄩 4 H2 ) + 1 2 ( 󵄩󵄩󵄩󵄩Jε∇V ε󵄩󵄩󵄩󵄩 2 H2 + 󵄩󵄩󵄩󵄩Jεj ε󵄩󵄩󵄩󵄩 2 H2 ) . (9) In the previouslymentioned,JεV ε ⊗JεV ε denotes a tensor (JεV ε i JεV 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∞ ≤ C (1 + 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩L2 + 󵄩󵄩󵄩󵄩∇f 󵄩󵄩󵄩󵄩L2(log +󵄩󵄩󵄩󵄩Δf 󵄩󵄩󵄩󵄩L2) 1/2 ) , f ∈ H 2 (R 2 ) . (10) Here loga 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 obtainingH boundon (0, T] of solution, we can continue solution beyond T by using Proposition 4. Taking curl operator on (1)1 and ∂i = ∂/∂xi (i = 1, 2) operator on (1)2 and (1)3, we have ∂ω ∂t + (V ⋅ ∇) ω − Δω = ∇ × (j × B) , in R × (0, T) , ∂ (∂iE) ∂t − ∇ × ∂iB = −∂ij, in R 2 × (0, T) , ∂ (∂iB) ∂t + ∇ × ∂iE = 0, in R 2 × (0, T) . (11) (i) H Estimates. Taking scalar product (11) with ω, ∂iE, and ∂iB, respectively, and summing over i = 1, 2, we have 1 2 d dt ‖ω‖ 2 L2 + ‖∇ω‖ 2 L2 = ∫ R2 ∇ × (j × B) ⋅ ω dx, (12) 1 2 d dt ‖∇E‖ 2 L2 = ∑ i ∫ R2 ∇ × ∂iB ⋅ ∂iE dx − ∫ R2 ∇j ⋅ ∇E dx, 1 2 d dt ‖∇B‖ 2 L2 = −∑ i ∫ R2 ∇ × ∂iE ⋅ ∂iB dx. (13) Using the identity ∫ R2 ∇ × ∂iB ⋅ ∂iE dx = ∫ R2 ∇ × ∂iE ⋅ ∂iB dx (14) and E = j − V × B, we obtain 1 2 d dt (‖∇E‖ 2 L2 + ‖∇B‖ 2 L2 ) + 󵄩󵄩󵄩󵄩∇j 󵄩󵄩󵄩󵄩 2 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 ∇ × (j × B) ⋅ ω dx 󵄨󵄨󵄨󵄨󵄨󵄨 ≤ ‖B‖L4 󵄩󵄩󵄩󵄩∇j 󵄩󵄩󵄩󵄩L2‖ω‖L4 ≤ C‖B‖ 1/2 L2 ‖∇B‖ 1/2 L2 ‖ω‖ 1/2 L2 ‖∇ω‖ 1/2 L2 󵄩󵄩󵄩󵄩∇j 󵄩󵄩󵄩󵄩L2 ≤ C‖ω‖ 2 L2 ‖∇B‖ 2 L2 + ε‖∇ω‖ 2 L2 + ε 󵄩󵄩󵄩󵄩∇j 󵄩󵄩󵄩󵄩 2 L2 , (16) where ε is a small positive number. Also we have 󵄨󵄨󵄨󵄨󵄨󵄨 ∫ R2 ∇j ⋅ ∇ (V × B) dx 󵄨󵄨󵄨󵄨󵄨󵄨 ≤ ∫ R2 󵄨󵄨󵄨󵄨∇j 󵄨󵄨󵄨󵄨 |V| |∇B| dx + ∫ R2 󵄨󵄨󵄨󵄨∇j 󵄨󵄨󵄨󵄨 |B| |∇V| dx = I + II. (17)

For the compatibility of the initial data, we assume that Since the divergence-free condition of the magnetic field is conserved, ∇ ⋅  = 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  .In many physical situations, current displacement term    is neglected because the physical coefficient for this term is very small (∼1/ 2 , where  denotes the speed of light).But mathematically, the presence of the term    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][7][8][9][10][11][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 R 2 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  2 solution by using the standard mollifier technique (see Proposition 4) and re-prove the global existence of  2 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.
But it provides double exponential bound compared with exponential bound in [2].
(2) The presence of the current displacement term    makes Maxwell-Navier-Stokes system do not enjoy the scaling invariance property of the usual Navier-Stokes system, V  (, ) = V(,  2 ).In Theorem 2, ∫  0 ‖V()‖ 2  ∞  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  2 solution to 3D Maxwell-Navier-Stokes system.

Local Existence and Global Well-Posedness
At first, we note that one can have the energy identity in two or three dimensions: The previously energy inequality can be justified for local in time regular solution in the following proposition.In the following,  denotes a harmless constant which may change from one line to the other.We prove local-in-time existence of  2 solution using the standard energy estimates.
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 where  ∈  ∞ 0 (R  ), and  ≥ 0, ∫ R   = 1.We introduce the following regularized system of (1): with initial data (V  0 ,   0 ,   0 ) = (J  V 0 , J   0 , J   0 ).Taking the  2 inner product of (7) 1 , (7) 2 , and (7) 3 with V  ,   ,   , respectively, we obtain We compute the derivative   ,  is a multi-index such that || ≤ 2, of ( 7), multiply them by   V  ,     , and     , respectively, and integrate them over R  to obtain In the previously mentioned, J  V  ⊗J  V  denotes a tensor (J  V   J  V   ) 1≤≤ .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): Here log +  denotes log( + ).

Blow-Up Criterion for 3D Maxwell-Navier-Stokes System
In this section, we provide a blow-up criterion for  2 solution in Proposition 4 to 3D Maxwell-Navier-Stokes system.
Proof of Theorem 2. Assume that where  * is the finite maximal existence time of a classical solution.