Backward Uniqueness of Phase-Lock Equations of Superconductivity

In this article we will show that the phase-pock equations of Superconductivity have backward uniqueness. Backward uniqueness guarantees that the semi group generated by the solutions of phase-lock equations is injective, which plays an important role in understanding the long time dynamic properties of the solutions. INTRODUCTION In the previous article [1-3], we proposed the following phase-lock equations to model the superconductivity phenomena: 2 2 2 ( 1) 0, κ + − − ∆ + = t f f f f q f ( ) 2 2 0, 1.1 η + + − = t q f q curl q curlh 0, = divq with the following boundary conditions, · 0, , · 0, , = × = × = ∂ grad f n curl q n h n q n on Ω Where f is a real-valued function and q is vector-valued real function. In [3,4] we proved the existence and uniqueness of both strong solutions and weak solutions. These results show that the phase-lock equations are well-post. In this short article, we will show that the phaselock equations have backward uniqueness, which guarantees that the semi group generated by the solutions of phase-lock equations is injective. This result shall enable us to understand the long term dynamics of the solutions to the phase-lock equations. The Main Results Let 2 { ( ) | 0, · | 0,} ∞ ∂ = ∈ = = V q C divq q n Ω Ω , we introduce two Sobolev spaces 1 2 1 2 , = × = × H H H V V V Where ( ) ( ) { } 2 1 1 1 2 2 2 1 2 2 , , , { }. = = = − = − H L V H H the closure of V under the L norm V the closure of V under the H norm Ω Ω Notice that 1 2 2 }   =     ∫ Q curlq d Ω Ω is an equivalent norm of V2 And from [5], we have 2 2 1 2 { | 0, · | 0}, { | 0, · | 0}. ∂ ∂ = ∈ = = = ∈ = = H q L div q q n V q H div q q n Ω