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

Authors: Ensil Kang; Jihoon Lee
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction

In this paper, we consider Maxwell-Navier-Stokes equations inℝd (d=2,3) as follows:
(1)∂v∂t+(v·∇)v-Δv+∇p=j×Binℝd×(0,T),∂E∂t-∇×B=-jinℝd×(0,T),∂B∂t+∇×E=0inℝd×(0,T),∇·v=∇·B=0inℝd×(0,T),j=E+v×B,
subject to the initial data
(2)v(x,0)=v0(x),E(x,0)=E0(x),B(x,0)=B0(x).
Here v, E, and B:ℝd×(0,T)→ℝ3 are vector fields defined on ℝd (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(3)∇·v0=∇·B0=0.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 ℝd. 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
(4)∫0T*‍∥v(t)∥L∞2+∥B(t)∥L∞8/3dt=∞.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, vλ(x,t)=λv(λx,λ2t). In Theorem 2, ∫0T‍∥v(t)∥L∞2dt 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 Section2, 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:(5)12ddt(∥v∥L22+∥B∥L22+∥E∥L22)+∥j∥L22+∥∇v∥L22=0.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(ℝd) (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(ℝd))∩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
(6)𝒥ϵf=ρϵ*f,ρϵ(·)=1ϵdρ(·ϵd),
whereρ∈C0∞(ℝd), and ρ≥0, ∫ℝd‍ρdx=1.
We introduce the following regularized system of (1):
(7)∂tvϵ+𝒥ϵ(𝒥ϵvϵ·∇)𝒥ϵvϵ-Δ𝒥ϵ2vϵ+∇pϵ=𝒥ϵ(𝒥ϵ2jϵ×𝒥ϵBϵ)inℝd×(0,T),∂tEϵ-∇×𝒥ϵ2Bϵ=-𝒥ϵ2jϵinℝd×(0,T),∂tBϵ+∇×𝒥ϵ2Eϵ=0inℝd×(0,T),∇·vϵ=∇·Bϵ=0inℝd×(0,T),jϵ=Eϵ+𝒥ϵvϵ×𝒥ϵBϵ,
with initial data(v0ϵ,E0ϵ,B0ϵ)=(𝒥ϵv0,𝒥ϵE0,𝒥ϵB0).
Taking theL2 inner product of (7)1, (7)2, and (7)3 with vϵ, Eϵ, Bϵ, respectively, we obtain
(8)12ddt(∥vϵ∥L22+∥Eϵ∥L22+∥Bϵ∥L22)+∥∇𝒥ϵvϵ∥L22+∥𝒥ϵjϵ∥L22=-12∫ℝd‍(𝒥ϵvϵ)·∇(𝒥ϵvϵ)2dx+∫ℝd‍(∇×𝒥ϵBϵ)·𝒥ϵEϵdx-∫ℝd‍(∇×𝒥ϵEϵ)·𝒥ϵBϵdx+∫ℝd‍(𝒥ϵ2jϵ×𝒥ϵBϵ)·𝒥ϵvϵdx+∫ℝd‍𝒥ϵ2jϵ·(𝒥ϵvϵ×𝒥ϵBϵ)dx=0.
We compute the derivativeDα, α is a multi-index such that |α|≤2, of (7), multiply them by Dαvϵ, DαEϵ, and DαBϵ, respectively, and integrate them over ℝd to obtain
(9)12ddt(∥vϵ∥H22+∥Eϵ∥H22+∥Bϵ∥H22)+∥∇𝒥ϵvϵ∥H22+∥𝒥ϵjϵ∥H22≤C∥𝒥ϵvϵ⊗𝒥ϵvϵ∥H2∥∇𝒥ϵvϵ∥H2+C∥𝒥ϵ2jϵ×𝒥ϵBϵ∥H2∥𝒥ϵvϵ∥H2+C∥𝒥ϵ2jϵ∥H2∥𝒥ϵvϵ×𝒥ϵBϵ∥H2≤C(∥𝒥ϵvϵ∥H24+∥𝒥ϵBϵ∥H24)+12(∥𝒥ϵ∇vϵ∥H22+∥𝒥ϵjϵ∥H22).
In the previously mentioned,𝒥ϵvϵ⊗𝒥ϵvϵ denotes a tensor (𝒥ϵviϵ𝒥ϵvjϵ)1≤j≤d.
Using Picard’s theorem, these estimates imply local existence of solution.The main ingredient of the proof of Theorem1 is the following Brezis-Gallouet inequality (logarithmic Sobolev inequality):
(10)∥f∥L∞≤C(1+∥f∥L2+∥∇f∥L2(log+∥Δf∥L2)1/2),f∈H2(ℝ2).
Here log+a denotes log(e+a).Proof of Theorem1.
We provide a priori estimates on the regular solutions. LetT 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=∂/∂xi (i=1,2) operator on (1)2 and (1)3, we have
(11)∂ω∂t+(v·∇)ω-Δω=∇×(j×B),inℝ2×(0,T),∂(∂iE)∂t-∇×∂iB=-∂ij,inℝ2×(0,T),∂(∂iB)∂t+∇×∂iE=0,inℝ2×(0,T).
(i)  H1  Estimates. Taking scalar product (11) with ω, ∂iE, and ∂iB, respectively, and summing over i=1,2, we have
(12)12ddt∥ω∥L22+∥∇ω∥L22=∫ℝ2‍∇×(j×B)·ωdx,(13)12ddt∥∇E∥L22=∑i‍∫ℝ2‍∇×∂iB·∂iEdx-∫ℝ2‍∇j·∇Edx,12ddt∥∇B∥L22=-∑i‍∫ℝ2‍∇×∂iE·∂iBdx.
Using the identity(14)∫ℝ2‍∇×∂iB·∂iEdx=∫ℝ2‍∇×∂iE·∂iBdx
andE=j-v×B, we obtain
(15)12ddt(∥∇E∥L22+∥∇B∥L22)+∥∇j∥L22=∫ℝ2‍∇j·∇(v×B)dx.
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:
(16)|∫ℝ2‍∇×(j×B)·ωdx|≤∥B∥L4∥∇j∥L2∥ω∥L4≤C∥B∥L21/2∥∇B∥L21/2∥ω∥L21/2∥∇ω∥L21/2∥∇j∥L2≤C∥ω∥L22∥∇B∥L22+ϵ∥∇ω∥L22+ϵ∥∇j∥L22,
whereϵ is a small positive number. Also we have
(17)|∫ℝ2‍∇j·∇(v×B)dx|≤∫ℝ2‍|∇j||v||∇B|dx+∫ℝ2‍|∇j||B||∇v|dx=I+II.
We estimate(18)I≤C∥v∥L∞2∥∇B∥L22+ϵ∥∇j∥L22,II≤∥B∥L4∥∇v∥L4∥∇j∥L2≤C∥B∥L22∥∇v∥L22∥∇B∥L22+ϵ∥Δv∥L22+ϵ∥∇j∥L22.
Collecting previous estimates, we have(19)ddt(∥ω∥L22+∥∇E∥L22+∥∇B∥L22)+∥∇ω∥L22+∥∇j∥L22≤C∥ω∥L22∥∇B∥L22+C∥v∥L∞2∥∇B∥L22+C∥∇B∥L22∥∇v∥L22∥∇B∥L22.
(ii)  H2Estimates. Taking Δ operator on (1)1, (1)2, and (1)3 and L2 scalar product with Δv, ΔE, and ΔB, respectively, we have
(20)12ddt∥Δv∥L22+∥∇Δv∥L22≤C∫ℝ2‍|∇v||D2v|dx+∫ℝ2‍|Δ(j×B)||Δv|dx:=I1+I2,12ddt(∥ΔE∥L22+∥ΔB∥L22)+∥Δj∥L22≤∫ℝ2‍|Δj||Δ(v×B)|dx:=I3.
We estimateI1, I2, and I3 using interpolation inequality, Young’s inequality, and Hölder’s inequality:
(21)I1≤C∥∇v∥L4∥Δv∥L4∥Δv∥L2≤C∥∇v∥L21/2∥Δv∥L23/2∥∇Δv∥L21/2≤C∥∇v∥L22/3∥Δv∥L22+ϵ∥∇Δv∥L22,(22)I2≤C∫ℝ2‍|∇j||∇B||Δv|dx+C∫ℝ2‍|Δj||B||Δv|dx+C∫ℝ2‍|j||ΔB||Δv|dx:=I21+I22+I23.
Each term can be estimated by the standard interpolation inequality and Young’s inequality as follows:(23)I21≤C∥∇j∥L4∥∇B∥L4∥Δv∥L2≤C∥j∥L21/4∥Δj∥L23/4∥B∥L21/4∥ΔB∥L23/4∥∇v∥L21/2∥∇Δv∥L21/2≤C∥j∥L22/3∥B∥L22/3∥∇v∥L24/3∥ΔB∥L22+ϵ∥Δj∥L22≤C(∥j∥L22+∥∇v∥L22)∥ΔB∥L22+ϵ∥Δj∥L22,I22≤C∥Δj∥L2∥B∥L4∥Δv∥L4≤C∥Δj∥L2∥B∥L23/4∥ΔB∥L21/4∥Δv∥L21/2∥∇Δv∥L21/2≤ϵ∥Δj∥L22+C∥ΔB∥L21/2∥Δv∥L21/2∥∇Δv∥L2≤ϵ∥Δj∥L22+ϵ∥∇Δv∥L22+C∥j∥L2∥∇v∥L2∥ΔB∥L22,I23≤C∥j∥L∞∥ΔB∥L2∥Δv∥L2≤C∥j∥L2∥Δj∥L21/2∥ΔB∥L2∥∇v∥L21/2∥∇Δv∥L21/2≤ϵ∥Δj∥L22+C∥j∥L22/3∥ΔB∥L24/3∥∇v∥L22/3∥∇Δv∥L22/3≤ϵ∥Δj∥L22+ϵ∥∇Δv∥L22+C∥j∥L2∥∇v∥L2∥ΔB∥L22.
I
3 can be written as
(24)I3≤C∫ℝ2‍|Δj||∇v||∇B|dx+C∫ℝ2‍|Δj||Δv||B|dx+C∫ℝ2‍|Δj||v||ΔB|dx:=I31+I32+I33,I31≤C∥Δj∥L2∥∇v∥L4∥∇B∥L4≤C∥Δj∥L2∥∇v∥L23/4∥∇Δv∥L21/4∥B∥L21/4∥ΔB∥L23/4≤ϵ∥Δj∥L22+ϵ∥∇Δv∥L22+C∥∇v∥L22∥ΔB∥L22.
The same as the estimate ofI22, we obtain
(25)I32≤ϵ∥Δj∥L22+ϵ∥∇Δv∥L22+C∥j∥L2∥∇v∥L2∥ΔB∥L22.
Also we have(26)I33≤C∥Δj∥L2∥v∥L∞∥ΔB∥L2≤ϵ∥Δj∥L22+C∥v∥L∞2∥ΔB∥L22.
Therefore, we have(27)ddt(∥Δv∥L22+∥ΔE∥L22+∥ΔB∥L22)+∥∇Δv∥L22+∥Δj∥L22≤C(1+∥∇v∥L22+∥j∥L22+∥v∥L∞2)×(∥Δv∥L22+∥ΔB∥L22).
(iii)  Use of Brezis-Gallouet Inequality. Using Brezis-Gallouet inequality, we obtain
(28)ddt(∥Δv∥L22+∥ΔE∥L22+∥ΔB∥L22)+∥∇Δv∥L22+∥Δj∥L22≤C(1+∥∇v∥L22+∥j∥L22+∥v∥L22+∥∇v∥L22)×(∥Δv∥L22+∥ΔE∥L22+∥ΔB∥L22)×log+(∥Δv∥L22+∥ΔE∥L22+∥ΔB∥L22).
Lety(t)=∥Δv∥L22+∥ΔE∥L22+∥ΔB∥L22, and let z(t)=1+∥Δv∥L22+∥j∥L22+∥v∥L22+∥∇v∥L22. Hence one has
(29)ddty(t)≤Cz(t)y(t)log+y(t).
Since(30)∫0T‍z(t)dt≤C(1+T),
the bound ofy(t) is immediate as follows:
(31)sup0≤t≤T(∥Δv∥L22+∥ΔE∥L22+∥ΔB∥L22)≤(∥Δv0∥L22+∥ΔE0∥L22+∥ΔB0∥L22)×exp(exp(C(T+1))).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 forH2 solution in Proposition 4 to 3D Maxwell-Navier-Stokes system.Proof of Theorem2.
Assume that
(32)∫0T*‍∥v(t)∥L∞2+∥B(t)∥L∞8/3dt<∞,
whereT* is the finite maximal existence time of a classical solution.
Similar to the computation in Section2, one has H1 estimates of E and B as follows:
(33)12ddt(∥∇E∥L22+∥∇B∥L22)+∥∇j∥L22=∫ℝ3‍∇j·∇(v×B)dx≤C∥∇(v×B)∥L22+ϵ∥∇j∥L22≤C∥B∥L∞2∥∇v∥L22+C∥v∥L∞2∥∇B∥L22+ϵ∥∇j∥L22.
H
1 estimates of v are as follows:
(34)12ddt∥∇v∥L22+∥Δv∥L22≤∫ℝ3‍|v||∇v||Δv|dx+∫ℝ3‍|j×B||Δv|dx≤C∥v∥L∞2∥∇v∥L22+C∥j×B∥L22+ϵ∥Δv∥L22.
The estimate of∥j×B∥L22 is provided in the following:
(35)∥j×B∥L22≤C∥E×B∥L22+C∥(v×B)×B∥L22≤C∥E∥L22∥B∥L∞2+C∥v∥L62∥B∥L64≤C∥E∥L22∥B∥L∞2+C∥∇v∥L22∥B∥L24/3∥B∥L∞8/3.
Thus we have(36)ddt(∥∇v∥L22+∥∇E∥L22+∥∇B∥L22)+∥Δv∥L22+∥∇j∥L22≤C(1+∥v∥L∞2+∥B∥L∞8/3)×(∥∇v∥L22+∥∇E∥L22+∥∇B∥L22)+C∥B∥L∞2.
Gronwall’s inequality gives us that(37)∥(∇v,∇E,∇B)∥L∞(0,T*;L2)2+∥(Δv,∇j)∥L2(0,T*;L2)2≤C<∞.
Next, we considerH2 estimates.
Integrating by parts and using Young’s inequality, it follows that(38)12ddt(∥ΔE∥L22+∥ΔB∥L22)+∥Δj∥L22≤C∥Δ(v×B)∥L22+ϵ∥Δj∥L22≤C∥Δv∥L22∥B∥L∞2+C∥v∥L∞2∥ΔB∥L22+C∥∇v∥L42∥∇B∥L42+ϵ∥Δj∥L22≤C∥Δv∥L22∥B∥L∞2+C∥v∥L∞2∥ΔB∥L22+C∥v∥L∞∥B∥L∞∥Δv∥L2∥ΔB∥L2+ϵ∥Δj∥L22.
Similarly, it follows that(39)12ddt∥Δv∥L22≤C∥∇(v·∇v)∥L22+C∥∇(j×B)∥L22+ϵ∥∇Δv∥L22≤C∥∇v∥L44+C∥v∥L∞2∥Δv∥L22+C∥∇E∥L62∥B∥L32+C∥E∥L62∥∇B∥L32+C∥∇(v×B)∥L22∥B∥L∞2+C∥v×B∥L32∥∇B∥L62.
Using the interpolation inequality, one has(40)∥∇v∥L44≤C∥v∥L∞2∥Δv∥L22.
Interpolation inequality and Young’s inequality produce that(41)∥E∥L62∥∇B∥L32≤C∥∇E∥L22∥∇B∥L2∥ΔB∥L2≤C∥∇E∥L22(∥∇B∥L22+∥ΔB∥L22).
Similarly, we estimate that(42)∥∇(v×B)∥L22∥B∥L∞2≤C(∥∇v∥L32∥∇B∥L22+∥∇v∥L22∥∇B∥L32),∥v×B∥L32∥∇B∥L62≤C∥v∥L∞2∥B∥L32∥ΔB∥L22.
We already know that(43)∥(∇v,∇E,∇B)∥L∞(0,T*;L2)2<C.
Gathering all the estimates, we achieve(44)ddt(∥Δv∥L22+∥ΔE∥L22+∥ΔB∥L22)+∥∇Δv∥L22+∥Δj∥L22≤C(1+∥v∥L∞2+∥B∥L∞2)×(1+∥Δv∥L22+∥ΔE∥L22+∥ΔB∥L22).
Using Gronwall’s inequality, we conclude that(45)∥(Δv,ΔE,ΔB)∥L∞(0,T*;L2)2+∥(∇Δv,Δj)∥L2(0,T*;L2)2≤C<∞.This completes the proof of Theorem 2.



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