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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 402793, 6 pages
Notes on the Global Well-Posedness for the Maxwell-Navier-Stokes System
1Department of Mathematics, Chosun University, Gwangju 501-759, Republic of Korea
2Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea
Received 28 March 2013; Accepted 26 June 2013
Academic Editor: Changxing Miao
Copyright © 2013 Ensil Kang and Jihoon Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
In this paper, we consider Maxwell-Navier-Stokes equations in () as follows: subject to the initial data Here , , and are vector fields defined on ( or 3). Vector fields , , and denote fluid velocity, electric fields and magnetic fields, respectively. denotes the scalar pressure and is the electric current given by Ohm’s law. 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 , , and can be understood as , and so forth.
For the compatibility of the initial data, we assume that
Since the divergence-free condition of the magnetic field is conserved, in (1) is not necessary in general if we assume the divergence-free condition for the initial data of the magnetic field in . In many physical situations, current displacement term is neglected because the physical coefficient for this term is very small (, 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 .
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 . In , global existence of regular solutions to (1) in is proved by using the Besov-type space technique developed by Chemin and Lerner . In [1, 14], the local existence of mild solution and the global existence of (1) with small data have been studied. Duan  studied large time behaviour of solutions to (1). In , Ibrahim and Yoneda obtained local-in-time existence for nondecaying initial data in torus. Also Germain and Masmoudi  studied global existence of solutions to Euler-Maxwell equations with small data and Jang and Masmoudi  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 solution by using the standard mollifier technique (see Proposition 4) and re-prove the global existence of 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 . 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 and . Then, for any , there exists a solution to 2D Maxwell-Navier-Stokes system (1) such that and .
Theorem 2. Suppose that and . If , the maximal existence time of the local existence of regular solution to 3D Maxwell-Navier-Stokes system (1), is finite, then
Remark 3. (1) As logarithmic inequality has been used in , Brezis-Gallouet inequality gives logarithmic-type estimates. But it provides double exponential bound compared with exponential bound in .
(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, . In Theorem 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 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:
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 solution using the standard energy estimates.
Proposition 4. Let ( or 3) with . Then there exists such that there exists a unique solution .
Proof. We use the mollifier method as described in . Although the details are similar to , we provide some a priori estimates for the reader’s sake. We consider the standard mollifier operator
where , and , .
We introduce the following regularized system of (1):
with initial data .
Taking the inner product of (7)1, (7)2, and (7)3 with , , , respectively, we obtain
We compute the derivative , is a multi-index such that , of (7), multiply them by , , and , respectively, and integrate them over to obtain
In the previously mentioned, denotes a tensor .
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 denotes .
Proof of Theorem 1. We provide a priori estimates on the regular solutions. Let be a finite maximal time of existence in Proposition 4. By obtaining bound on of solution, we can continue solution beyond by using Proposition 4.
Taking curl operator on (1)1 and () operator on (1)2 and (1)3, we have
(i) Estimates. Taking scalar product (11) with , , and , respectively, and summing over , we have
Using the identity
and , we obtain
In the following, denotes a sufficiently small positive number. Since it holds that , we estimate the right-hand side of (12) using Young’s inequality and interpolation inequality:
where is a small positive number. Also we have
Collecting previous estimates, we have
(ii) Estimates. Taking operator on (1)1, (1)2, and (1)3 and scalar product with , , and , respectively, we have
We estimate , , and using interpolation inequality, Young’s inequality, and Hölder’s inequality:
Each term can be estimated by the standard interpolation inequality and Young’s inequality as follows:
can be written as
The same as the estimate of , we obtain
Also we have
Therefore, we have
(iii) Use of Brezis-Gallouet Inequality. Using Brezis-Gallouet inequality, we obtain
Let , and let . Hence one has
the bound of is immediate as follows: 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 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.
Similar to the computation in Section 2, one has estimates of and as follows:
estimates of are as follows:
The estimate of is provided in the following:
Thus we have
Gronwall’s inequality gives us that
Next, we consider estimates.
Integrating by parts and using Young’s inequality, it follows that
Similarly, it follows that
Using the interpolation inequality, one has
Interpolation inequality and Young’s inequality produce that
Similarly, we estimate that
We already know that
Gathering all the estimates, we achieve
Using Gronwall’s inequality, we conclude that This completes the proof of Theorem 2.
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.
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