Research Article Uniqueness of Weak Solutions to an Electrohydrodynamics Model

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1 Abstract and Applied Analysis Volume 2012, Article ID , 14 pages doi: /2012/ Research Article Uniqueness of Weak Solutions to an Electrohydrodynamics Model Yong Zhou 1 and Jishan Fan 2 1 Department of Mathematics, Zhejiang Normal University, Jinhua , China 2 Department of Applied Mathematics, Nanjing Forestry University, Nanjing , China Correspondence should be addressed to Yong Zhou, yzhoumath@zjnu.edu.cn Received 11 October 2011; Accepted 3 March 2012 Academic Editor: Narcisa C. Apreutesei Copyright q 2012 Y. Zhou and J. Fan. This is an open access article distributed under the Creative Commons Attribution icense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper studies uniqueness of weak solutions to an electrohydrodynamics model in R d d 2, 3. Whend 2, we prove a uniqueness without any condition on the velocity. For d 3, 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 R d 0, 1, 2 : t u u u π Δu Δφ φ, div u 0, t n u n ( n n φ ), t p u p ( p p φ ), Δφ n p, ( u, n, p ) x, 0 ( u0,n 0,p 0 ) x, x R d d 2, The unknowns u, π, φ, n, and p denote the velocity, pressure, electric potential, anion concentration, and cation concentration, respectively.

2 2 Abstract and Applied Analysis Equations are known as the electrochemical equations 3 or semiconductor equations 4, 5, and electro-rheological systems 2, 6 when formally setting u and 1.2 are Navier-Stokes equations with the orentz force Δφ φ. The uniqueness of weak solutions to the Navier-Stokes equations is still open. In 1962, Serrin 7 gave the first uniqueness condition: u r( 0,T; s( R 3)) with 2 r 3 s, 3 <s. 1.7 Kozono and Taniuchi 8 proved the following uniqueness criterion: u 2( ( 0,T;BMO R 3)). 1.8 Here BMO denotes the functions of bounded mean oscillation. Ogawa and Taniuchi 9 obtained the uniqueness criterion: ( ( u og 0,T; Ḃ, 0 R 3)) 1.9 with { ( ) T og 0,T; Ḃ, 0 : f; fḃ0, log (e } ) fḃ0, dt < Here it should be noted that Kozono et al. 10 proved that u is smooth if u 1( 0,T; Ḃ 0, ( R 3)) Here Ḃ, 0 is the homogeneous Besov space. Kurokiba and Ogawa 4 considered the semiconductor equations when u 0 and proved that the existence and uniqueness of weak solutions with p initial data n 0,p 0 when p d/2 d 3 and 1 <p<2 d 2. Note that the system holds its form under the scaling u, π, φ, n, p u λ,π λ,φ λ,n λ,p λ : λu, λ 2 π, φ, λ 2 n, λ 2 p λ 2 t, λx. Under this scaling, the space r 0,T; s is invariant for u when 2/r d/s and the space r 0,T; s is invariant for n, p when 2/r d/s 2. Furthermore, d for u 0 and d/2 for n 0,p 0 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 in a bounded domain Ω. When Ω R d, 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.

3 Abstract and Applied Analysis 3 Theorem 1.1. et n 0,p 0 1 R 2 log R 2, n 0,p 0 0 in R 2, n 0 dx p 0 dx, φ 0 2, and u 0 2. Then there exists a unique weak solution u, n, p, φ to the problem satisfying ( n, p ) ( 0,T; 1 log ) 2( 0,T; 2) 4/3( 0,T; W 1,4/3), n,p 0 in R 2 0,T ( t n, t p ) 4/3( 0,T; W 1,4/3), φ ( 0,T; 2) 2( 0,T; H 1) 4( 0,T; 4), u ( 0,T; 2) 2( 0,T; H 1) 4( 0,T; 4), t u 4/3( 0,T; H 1) for any T> Remark 1.2. We can assume n 0 p 0 H 1 Hardy space and Δφ 0 n 0 p 0 gives φ 0 2 R 2. Theorem 1.3 d 3. et n 0,p 0 3/2, n 0,p 0 0 in R 3, n0 dx p 0 dx, and u 0 2. Suppose that 1.9 holds true, then there exists a unique weak solution u, n, p, φ to the problem satisfying ( n 3/4,p 3/4) ( 0,T; 2) 2( 0,T; H 1), n,p 0 in R 3 0,T, ( n, p ) ( 0,T; 3/2) 5/2( 0,T; 5/2) 5/3( 0,T; W 1,5/3) 4( 0,T; 2), ( t n, t p ) 5/3( 0,T; W 1,3/2), φ ( 0,T; W 1,3/2) 5/2( 0,T; W 1,5/2), φ ( 0,T; 3) 5/2( 0,T; 15), u ( 0,T; 2) 2( 0,T; H 1), t u 2( 0,T; W 1,3/2) 1.13 for any T>0. et η j,j 0, ±1, ±2, ±3,..., be the ittlewood-paley dyadic decomposition of unity that satisfies η C 0 B 2 \ B 1/2, η j ξ η 2 j ξ, and j η j ξ 1 except ξ 0. To fill the origin, we put a smooth cut off ψ S R 3 with ψ ξ C 0 B 1 such that ψ η j ξ. j The homogeneous Besov space Ḃ s p,q : {f S : f Ḃ s p,q fḃs p,q : 2 js η j f q p j < } is introduced by the norm 1/q, 1.15 for s R, 1 p, q. 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.

4 4 Abstract and Applied Analysis 2. Proof of Theorem 1.1 First, by the maximum principle, it is easy to prove that n, p 0 in R d 0,. 2.1 Testing 1.3 by 1 log n and testing 1.4 by 1 log p, respectively, using 1.2, summing up the resulting equality, we obtain T n log n p log pdx 4 n 2 T p 2 dx dt 0 0 n 0 log n 0 p 0 log p 0 dx. Δφ 2 dx dt 2.2 Substracting 1.4 from 1.3, weseethat t ( n p ) u ( n p ) ( ( n p ) ( n p ) φ ). 2.3 Testing the above equation by φ,using 1.5 and 2.1,weseethat φ 2 dx u Δφ φdx Δφ (n 2 ) dx p φ 2 dx Whence φ 2 dx uδφ φdx Δφ (n 2 ) dx p φ 2 dx Testing 1.1 by u,using 1.2, wefindthat u 2 dx u 2 dx uδφ φdx. 2.6 Summing up 2.5 and 2.6, weget u 2 φ 2 dx u 2 Δφ 2 ( ) n p φ 2 dx 0, 2.7 whence 1 u 2 T φ 2 dx u 2 Δφ 2 ( ) n p φ 2 x dt u φ0 2 dx. 2.8

5 Abstract and Applied Analysis 5 Integrating 1.3 and 1.4, we have ndx n 0 dx p 0 dx pdx. 2.9 Using the Gagliardo-Nirenberg inequality, u 2 4 C u 2 u 2, 2.10 we deduce that u 4 0,T; 4 C, φ 4 0,T; 4 C, n, p 2 0,T; 2 C Since n 2 n n, n 2 0,T; 2, n 4 0,T; 4, we easily infer that n 4/3( 0,T; 4/3), 2.14 by the Hölder inequality. Similarly, we have p 4/3( 0,T; 4/3) It is easy to show that ( t n, t p ) 4/3( 0,T; W 1,4/3), t u 4/3( 0,T; H 1) Now we are in a position to prove the uniqueness. et u i,π i,n i,p i,φ i i, 2 be two weak solutions to the problem Also let us denote u : u 1 u 2, π : π 1 π 2, n : n 1 n 2, p : p 1 p 2, φ : φ 1 φ We define N and P satisfying the following equations: ΔN N n in R d 0,, ΔP P p in R d 0,

6 6 Abstract and Applied Analysis It is easy to verify that t n u 1 n un 2 Δn (n φ 1 n 2 φ ), t p (u 1 p up 2 ) Δp ( p φ1 p 2 φ ), t ( n p ) ( u1 ( n p ) u ( n2 p 2 )) Δ ( n p ) (( n p ) φ1 ( n 2 p 2 ) φ ) Testing 2.20 by N, we derive N 2 N 2 dx N 2 ΔN 2 dx n φ 1 N n 2 φ N u 1 n N un 2 Ndx : I 1 I 2 I 3 I Using 2.10, 2.18 and 2.19, each term I i i, 2, 3, 4 can be bounded as follows: I 1 ΔN 2 φ1 4 N 4 N 4 φ1 4 N 2 C ΔN 2 φ1 4 N 1/2 ΔN 1/2 C φ1 2 4 N 2 2 H 1 18 ΔN 2 C 2 φ1 4 N 2 4 C 2 φ1 4 N 2 H, 1 I 2 n 2 2 φ 4 N 4 n 2 2 φ 2 n N 2 4 C n 2 2 φ 2 Δφ 2 C n 2 2 N 2 ΔN 2 Δφ 2 ( ΔN 2 C n φ ) 2 2 N 2 2, 2 I 3 u 1 4 ΔN 2 N C u 1 4 ΔN 2 N 1/2 ΔN 1/ ΔN 2 C u N 2 4, 2 I 4 n 2 2 u 4 N 4 n 2 2 u 2 4 n 2 2 N 2 4 C n 2 2 u 2 u 2 C n 2 2 N 2 ΔN 2 18 u ΔN 2 2 C n ( u 2 2 N 2 2 ).

7 Abstract and Applied Analysis 7 Substituting these estimates into 2.23, weobtain N 2 N 2 dx N 2 ΔN 2 dx 2 ( )( C φ1 4 n u N 2 4 N 2 2 u 2 2 ) φ Δφ u Similarly for the p-equation, we get p 2 p 2 1 p dx 2 Δp 2 dx 2 ( C φ 1 4 p )( u p 2 p u 2 2 φ ) Δφ u Testing 2.22 by φ, using 1.5, we deduce that φ Δφ 2 dx 2 dx (n p ) φ1 φ ( n 2 p 2 )( φ ) 2 u1 Δφ φ u ( n 2 p 2 ) φdx 2.27 : J 1 J 2 J 3 J 4. Using 2.10, 2.18,and 2.19, each term J i i, 2, 3, 4 can be bounded as follows: J 1 n p 2 φ 1 4 φ 4 C ΔN 2 ΔP 2 N 2 P 2 φ 1 4 φ 1/2 Δφ 1/ ΔN ΔP 2 C N 2 2 C P Δφ 2 C φ1 4 φ 2, J 2 0, J 3 u 1 4Δφ 2 φ 4

8 8 Abstract and Applied Analysis C u 1 4Δφ 2 φ 1/2 Δφ 1/2 2 2 Δφ 2 C u φ 2, 4 2 J 4 u 4 n2 p 2 2 φ 4 n2 p 2 2 u 2 4 n2 p 2 2 φ 2 4 C n2 p 2 2 u 2 u 2 C n2 p 2 2 φ 2 Δφ 2 18 u 2 Δφ 2 C 2 n2 p ( u 2 φ 2 ) Substituting these estimates into 2.27, we have φ 2 1 Δφ dx 2 dx 2 ( C φ1 4 u )( 4 n2 p 2 2 u 2 2 ) φ 2 2 N 2 2 P ΔN ΔP u It is easy to find that u satisfies t u u 2 u u u 1 π Δu Δφ φ 1 Δφ 2 φ Testing this equation by u,using 1.2, we have u 2 dx u 2 dx Δφ φ 1 u Δφ 2 φ u u u 1 udx : l 1 l 2 l Using 2.10, each term l i i, 2, 3 can be bounded as follows: l 1 Δφ 2 φ1 4 u 4 C Δφ 2 φ1 4 u 1/2 u 1/2 2 2 Δφ u 2 C 2 φ1 4 u 2 4, 2 l 2 Δφ 2 2 φ 4 u 4

9 Abstract and Applied Analysis 9 C Δφ 2 2 φ 1/2 Δφ 1/2 2 u 1/2 u 1/ Δφ u 2 C ( 2 Δφ2 2 u 2 2 φ 2 ), 2 2 l 3 u u u x u 4 u 2 u 1 4 C u 1/2 2 u 3/2 2 u u 2 2 C u u Substituting these estimates into 2.31, we have u 2 dx u 2 dx Δφ 2 ( C φ1 4 )( Δφ u u 2 4 φ 2 ) Combining 2.25, 2.26, 2.29, and 2.33, using 2.8, 2.11, 2.12, 2.13, andthe Gronwall inequality, we conclude that N P 0, u 0, φ 0, 2.34 and thus n p 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 But each term I i i, 2, 3, 4 can be bounded as follows: I 1 ΔN 2 N 30/13 φ1 15 N 6 φ1 3 N 2 C ΔN 2 N 4/5 2 ΔN 1/5 2 φ1 15 C N ΔN 2 2 C φ1 5/2 15 N 2 2 C N 2 2, 3.1

10 10 Abstract and Applied Analysis by the Gagliardo-Nirenberg inequality, N 30/13 C N 4/5 ΔN 1/5, 2 2 I 2 n 2 2 φ 6 N 3 C n 2 2 Δφ 2 N 1/2 ΔN 1/2 2 2 C n 2 2 ΔN N ΔP P 2 N 1/2 ΔN 1/ ΔN ΔP 2 2 C N 2 2 C P 2 2 C n N by the Gagliardo-Nirenberg inequality, N 2 C N 3 2 ΔN 2, I 3 u 1 n Ndx u 1 ΔN Ndx. 3.3 Now we decompose u 1 into three parts in the phase variable: u 1 M η j u 1 η j u 1 η j u 1 j< M j M j>m 3.4 : u l 1 um 1 uh 1. Thus I 3 u l 1 ΔN Ndx i : I 31 I 32 I 33. i u m 1 in Ndx u h 1 ΔN Ndx 3.5 Recalling the Bernstein inequality, η j u q C2 3j 1/p 1/q η j u p, 1 p q, 3.6

11 Abstract and Applied Analysis 11 the low-frequency part is estimated as I 31 N 6 ΔN 2u l 1 3 C ΔN 2 2 j/2 2 ηj u 1 2 j< M C ΔN 2 2 j< M 2 j 1/2 j ηj u /2 3.7 C2 M/2 ΔN 2 2 u 1 2 C2 M/2 ΔN 2 2. The second term can be bounded as follows: I 32 i i N 2 N 2 i u m 1 C N 2 u m 2 C N 2 2 M j M 1 ηj u CM N 2 2 u 1 Ḃ0,. as On the other hand, the last term is simply bounded by the Hausdorff-Young inequality I 33 ΔN 2 N 2 u h 1 { } ΔN 2 N 2 Δ 1/2 η j 1 η j η j 1 η j Δ/2 u 1 j>m C ΔN 2 N 2 2 j ηj Δ/2 u 1 j>m C ΔN 2 N 2 2 j u 1 Ḃ0, j>m 3.9 C2 M ΔN 2 N 2 u 1 Ḃ0, C2 M N 2 2 u 1 2 Ḃ 0, C2 M ΔN 2 2.

12 12 Abstract and Applied Analysis Choosing M properly large so that C2 M/2 /36 and C2 M u 1 Ḃ0,, we reach I 3 )) 8 ΔN 2 C N 2 2 u 2 1 Ḃ0, (1 log (e u 1 Ḃ0,, I 4 u 6 n 2 2 N 3 C u 2 n 2 2 N 1/2 ΔN 1/ ΔN u 2 2 C n 4 2 N Substituting the above estimates into 2.23, weobtain N 2 N 2 dx N 2 ΔN 2 dx 2 ( )( ) C φ1 5/2 n N 2 2 P 2 2 N )) C N 2 u 2 1 Ḃ0, (1 log (e u 1 Ḃ0, ΔP u 2 2. Similarly for the p-equation, we have P 2 P 2 dx P 2 ΔP 2 dx ( C φ1 5/2 )( ) p N 2 2 P 2 2 P )) (e u 1 Ḃ0, C P 2 2 u 1 Ḃ0, (1 log ΔN u 2 2. As in Section 2, we still have But each term l i i, 2, 3 can be bounded as follows: l 1 φ1 15 Δφ 2 u 30/13 C φ1 15 ΔN 2 ΔP 2 N 2 P 2 u 4/5 u 1/ ΔN ΔP 2 2 C N 2 2 C P u 2 2 C φ1 5/2 15 u 2 2, 3.13

13 Abstract and Applied Analysis 13 by the Gagliardo-Nirenberg inequality, u 30/13 C u 4/5 u 1/5, 2 2 l 2 Δφ 2 2 φ 6 u 3 C Δφ 2 2 Δφ 2 u 1/2 u 1/2 2 2 C Δφ2 2 ΔN 2 ΔP 2 N 2 P 2 u 1/2 u 1/ ΔN ΔP 2 2 C N 2 2 C P u 2 2 C Δφ2 4 2 u 2 2, 3.14 by the Gagliardo-Nirenberg inequality u 2 3 C u 2 u By the similar calculations as that of I 3, l 3 can be bounded as follows: l 3 )) 18 u 2 C u 2 2 u 2 1 Ḃ0, (1 log (e u 1 Ḃ0, Substituting the above estimates into 2.31, we have u 2 dx u 2 dx 2 9 ΔN ΔP 2 C N 2 2 C P ( φ1 5/2 Δφ ) u )) C u 2 u 2 1 Ḃ0, (1 log (e u 1 Ḃ0, Combining 3.11, 3.12, and 3.17, using 1.13 and the Gronwall inequality, we arrive at N P 0, u 0, 3.18 as thus n p 0, φ This completes the proof.

14 14 Abstract and Applied Analysis 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. R , andnsfc Grant no References 1 M. Bazant and T. Squires, Induced-charge electro-kinetic phenomena, theory and microfluidic applications, Physical Review etters, vol. 92, no. 6, Article ID , 4 pages, R. E. Problestein, Physicochemical Hydrodynamics, An introduction, John Wiley & Sons, K. T. Chu and M. Z. Bazant, Electrochemical thin films at and above the classical limiting current, SIAM Journal on Applied Mathematics, vol. 65, no. 5, pp , M. Kurokiba and T. Ogawa, Well-posedness for the drift-diffusion system in p arising from the semiconductor device simulation, Mathematical Analysis and Applications, vol. 342, no. 2, pp , P. Biler, W. Hebisch, and T. Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Analysis, vol. 23, no. 9, pp , S. Thamida and H. C. Chang, Nonlinear electro-kinetic ejection and entrainment due to polarization at nearly insulated wedges, Physics of Fluids, vol. 14, pp , J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Archive for Rational Mechanics and Analysis, vol. 9, pp , H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Mathematische Zeitschrift, vol. 235, no. 1, pp , T. Ogawa and Y. Taniuchi, The limiting uniqueness criterion by vorticity for Navier-Stokes equations in Besov spaces, The Tohoku Mathematical Journal, vol. 56, no. 1, pp , H. Kozono, T. Ogawa, and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Mathematische Zeitschrift, vol. 242, no. 2, pp , J. Fan and H. Gao, Uniqueness of weak solutions to a model of electro-kinetic fluid, Communications in Mathematical Sciences, vol. 7, no. 2, pp , R. J. Ryham, Existence, uniqueness, regularity and long-term behavior for dissipative systems modeling electrohydrodynamics, submitted to Analysis of PDEs, 13 M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Mathematical Models and Methods in Applied Sciences, vol. 19, no. 6, pp , J. W. Jerome, Analytical approaches to charge transport in a moving medium, Transport Theory and Statistical Physics, vol. 31, no. 4 6, pp , J. Zhao, C. Deng, and S. Cui, Global well-posedness of a dissipative system arising in electrohydrodynamics in negative-order Besov spaces, Mathematical Physics, vol. 51, no. 9, Article ID , 17 pages, 2010.

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