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