Uniqueness of Weak Solutions to an Electrohydrodynamics Model

and Applied Analysis 3 Theorem 1.1. Let n0, p0 ∈ L1 R2 ∩ L logL R2 , n0, p0 ≥ 0 in R2, ∫ n0 dx ∫ p0 dx,∇φ0 ∈ L2, and u0 ∈ L2. Then there exists a unique weak solution u, n, p, φ to the problem 1.1 – 1.6 satisfying ( n, p ) ∈ L∞0, T ;L1 ∩ L logL ∩ L20, T ;L2 ∩ L4/30, T ;W1,4/3, n, p ≥ 0 in R2 × 0, T ( ∂tn, ∂tp ) ∈ L4/30, T ;W−1,4/3, ∇φ ∈ L∞0, T ;L2 ∩ L20, T ;H1 ∩ L40, T ;L4, u ∈ L∞0, T ;L2 ∩ L20, T ;H1 ∩ L40, T ;L4, ∂tu ∈ L4/3 ( 0, T ;H−1 ) for any T > 0. 1.12 Remark 1.2. We can assume n0−p0 ∈ H1 Hardy space andΔφ0 n0−p0 gives∇φ0 ∈ L2 R2 . Theorem 1.3 d 3 . Let n0, p0 ∈ L3/2, n0, p0 ≥ 0 in R3, ∫ n0 dx ∫ p0 dx, and u0 ∈ L2. Suppose that 1.9 holds true, then there exists a unique weak solution u, n, p, φ to the problem 1.1 – 1.6 satisfying ( n3/4, p3/4 ) ∈ L∞0, T ;L2 ∩ L20, T ;H1, n, p ≥ 0 in R3 × 0, T , ( n, p ) ∈ L∞0, T ;L3/2 ∩ L5/20, T ;L5/2 ∩ L5/30, T ;W1,5/3 ∩ L40, T ;L2, ( ∂tn, ∂tp ) ∈ L5/30, T ;W−1,3/2, ∇φ ∈ L∞0, T ;W1,3/2 ∩ L5/20, T ;W1,5/2, ∇φ ∈ L∞0, T ;L3 ∩ L5/20, T ;L15, u ∈ L∞0, T ;L2 ∩ L20, T ;H1, ∂tu ∈ L2 ( 0, T ;W−1,3/2 ) 1.13 for any T > 0. Let ηj , j 0,±1,±2,±3, . . ., be the Littlewood-Paley dyadic decomposition of unity that satisfies η̂ ∈ C∞ 0 B2 \ B1/2 , η̂j ξ η̂ 2−j ξ , and ∑∞ j −∞ η̂j ξ 1 except ξ 0. To fill the origin, we put a smooth cut off ψ ∈ S R3 with ψ̂ ξ ∈ C∞ 0 B1 such that


1.11
HereḂ 0 ∞,∞ is the homogeneous Besov space. Kurokiba and Ogawa 4 considered the semiconductor equations 1.3 -1.5 when u 0 and proved that the existence and uniqueness of weak solutions with L p initial data n 0 , p 0 when p d/2 d ≥ 3 and 1 < p < 2 d 2 . Note that the system 1.1 -1.5 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 L r 0, T; L s is invariant for u when 2/r d/s 1 and the space L r 0, T; L s is invariant for n, p when 2/r d/s 2. Furthermore, L d for u 0 and L 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 1. The aim of this paper is to generalize the results of 4, 9 . We will prove the following results.
Let η j , j 0, ±1, ±2, ±3, . . ., be the Littlewood-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 The homogeneous Besov spaceḂ s p,q : {f ∈ S : f Ḃ s p,q < ∞} is introduced by the norm 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.

Proof of Theorem 1.1
First, by the maximum principle, it is easy to prove that 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 Testing the above equation by −φ, using 1.5 and 2.1 , we see that Testing 1.1 by u, using 1.2 , we find that Summing up 2.5 and 2.6 , we get Using the Gagliardo-Nirenberg inequality, we deduce that 14 by the Hölder inequality. Similarly, we have ∇p ∈ L 4/3 0, T; L 4/3 .

2.16
Now we are in a position to prove the uniqueness. Let u i , π i , n i , p i , φ i i 1, 2 be two weak solutions to the problem 1.1 -1.6 . Also let us denote We define N and P satisfying the following equations: It is easy to verify that Testing 2.20 by N, we derive

2.24
Abstract and Applied Analysis 7 Substituting these estimates into 2.23 , we obtain

2.25
Similarly for the p-equation, we get

2.27
Using 2.10 , 2.18 , and 2.19 , each term J i i 1, 2, 3, 4 can be bounded as follows: Abstract and Applied Analysis

2.30
Testing this equation by u, using 1.2 , we have

2.32
Substituting these estimates into 2.31 , we have This completes the proof.

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 I i i 1, 2, 3, 4 can be bounded as follows: Abstract and Applied Analysis by the Gagliardo-Nirenberg inequality, by the Gagliardo-Nirenberg inequality,

3.3
Now we decompose u 1 into three parts in the phase variable:

3.5
Recalling the Bernstein inequality, the low-frequency part is estimated as

3.7
The second term can be bounded as follows:

3.8
On the other hand, the last term is simply bounded by the Hausdorff-Young inequality as

3.10
Substituting the above estimates into 2.23 , we obtain

3.11
Similarly for the p-equation, we have

3.12
As in Section 2, we still have 2.31 . But each term i i 1, 2, 3 can be bounded as follows:

3.13
Abstract and Applied Analysis 13 by the Gagliardo-Nirenberg inequality, u L 30/13 ≤ C u 4/5 L 2 ∇u 1/5 L 2 , 3.14 by the Gagliardo-Nirenberg inequality By the similar calculations as that of I 3 , 3 can be bounded as follows: This completes the proof.