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Abstract and Applied Analysis
Volume 2012, Article ID 864186, 14 pages
http://dx.doi.org/10.1155/2012/864186
Research Article

Uniqueness of Weak Solutions to an Electrohydrodynamics Model

1Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
2Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China

Received 11 October 2011; Accepted 3 March 2012

Academic Editor: Narcisa C. Apreutesei

Copyright © 2012 Yong Zhou and Jishan Fan. 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.

Abstract

This paper studies uniqueness of weak solutions to an electrohydrodynamics model in . When , we prove a uniqueness without any condition on the velocity. For , we prove a weak-strong uniqueness result with a condition on the vorticity in the homogeneous Besov space.

1. Introduction

We consider the following model of electrokinetic fluid in [1, 2]:

The unknowns , , , , and denote the velocity, pressure, electric potential, anion concentration, and cation concentration, respectively.

Equations (1.3)–(1.5) are known as the electrochemical equations [3] or semiconductor equations [4, 5], and electro-rheological systems [2, 6] when formally setting . (1.1) and (1.2) are Navier-Stokes equations with the Lorentz force .

The uniqueness of weak solutions to the Navier-Stokes equations is still open. In 1962, Serrin [7] gave the first uniqueness condition: Kozono and Taniuchi [8] proved the following uniqueness criterion: Here denotes the functions of bounded mean oscillation. Ogawa and Taniuchi [9] obtained the uniqueness criterion: with Here it should be noted that Kozono et al. [10] proved that is smooth if Here is the homogeneous Besov space.

Kurokiba and Ogawa [4] considered the semiconductor equations (1.3)–(1.5) when and proved that the existence and uniqueness of weak solutions with initial data when and .

Note that the system (1.1)–(1.5) holds its form under the scaling . Under this scaling, the space is invariant for when and the space is invariant for when . Furthermore, for and for are invariant spaces under this scaling. Fan and Gao [11], Ryham [12], and Schmuck [13] proved the existence, uniqueness, and regularity of global weak solutions to system (1.1)–(1.6) in a bounded domain . When , Jerome [14] established the first existence result in Kato’s semigroup framework. Zhao et al. [15] obtained global well-posedness for small initial data in Besov spaces with negative index.

The aim of this paper is to generalize the results of [4, 9]. We will prove the following results.

Theorem 1.1. Let , in , and . Then there exists a unique weak solution to the problem (1.1)–(1.6) satisfying

Remark 1.2. We can assume (Hardy space) and gives .

Theorem 1.3 (). Let , in , and . Suppose that (1.9) holds true, then there exists a unique weak solution to the problem (1.1)–(1.6) satisfying for any .

Let , be the Littlewood-Paley dyadic decomposition of unity that satisfies , and except . To fill the origin, we put a smooth cut off with such that The homogeneous Besov space is introduced by the norm for .

It is easy to prove the existence of weak solutions [14] and thus we omit the details here; we only need to derive the estimates (1.12) and (1.13) and prove the uniqueness.

2. Proof of Theorem 1.1

First, by the maximum principle, it is easy to prove that

Testing (1.3) by and testing (1.4) by , respectively, using (1.2), summing up the resulting equality, we obtain

Substracting (1.4) from (1.3), we see that

Testing the above equation by , using (1.5) and (2.1), we see that Whence

Testing (1.1) by , using (1.2), we find that

Summing up (2.5) and (2.6), we get whence

Integrating (1.3) and (1.4), we have

Using the Gagliardo-Nirenberg inequality, we deduce that Since , we easily infer that by the Hölder inequality. Similarly, we have

It is easy to show that Now we are in a position to prove the uniqueness. Let be two weak solutions to the problem (1.1)–(1.6). Also let us denote We define and satisfying the following equations:

It is easy to verify that Testing (2.20) by , we derive

Using (2.10), (2.18) and (2.19), each term can be bounded as follows:

Substituting these estimates into (2.23), we obtain

Similarly for the -equation, we get

Testing (2.22) by , using (1.5), we deduce that

Using (2.10), (2.18), and (2.19), each term can be bounded as follows: Substituting these estimates into (2.27), we have

It is easy to find that satisfies

Testing this equation by , using (1.2), we have

Using (2.10), each term can be bounded as follows:

Substituting these estimates into (2.31), we have

Combining (2.25), (2.26), (2.29), and (2.33), using (2.8), (2.11), (2.12), (2.13), and the Gronwall inequality, we conclude that and thus This completes the proof.

3. Proof of Theorem 1.3

By the same calculations as that in [11], we can prove (1.13) and thus we omit the details here.

Now we are in a position to prove the uniqueness. We still use the same notations as that in Section 2, and similarly we get (2.23). But each term can be bounded as follows: by the Gagliardo-Nirenberg inequality, by the Gagliardo-Nirenberg inequality,

Now we decompose into three parts in the phase variable: Thus

Recalling the Bernstein inequality, the low-frequency part is estimated as

The second term can be bounded as follows:

On the other hand, the last term is simply bounded by the Hausdorff-Young inequality as

Choosing properly large so that and , we reach

Substituting the above estimates into (2.23), we obtain

Similarly for the -equation, we have

As in Section 2, we still have (2.31). But each term can be bounded as follows: by the Gagliardo-Nirenberg inequality, by the Gagliardo-Nirenberg inequality

By the similar calculations as that of , can be bounded as follows:

Substituting the above estimates into (2.31), we have

Combining (3.11), (3.12), and (3.17), using (1.13) and the Gronwall inequality, we arrive at as thus This completes the proof.

Acknowledgments

The authors would like to thank the referee for careful reading and helpful suggestions. This work is partially supported by Zhejiang Innovation Project (Grant no. T200905), ZJNSF (Grant no. R6090109), and NSFC (Grant no. 11171154).

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