REMARK ON UNIQUENESS OF WEAK SOLUTIONS TO THE NAVIER-STOKES EQUATIONS
(N-S) du — Au + u • Vu + Vp = / in χ e Ω, 0 < t < Τ, at div u = 0 in χ e Ω, 0 < t < Τ, u = 0 on 3Ω, w|t=o a, where u = u(x,t) = («'(χ, i), · · •, u(x, t)) and ρ = p(x,t) denote the unknown velocity vector and pressure of the fluid at the point (x,t) € Ω χ (0,T), while α = a(x) = (a(x), • ••, a(x)) is the given initial velocity vector field; / = f(x, t) = ( f l { x , t), • • •, f(x, t)) is a given external force. The purpose of the present paper is to improve the criterion on uniqueness of weak solutions to (N-S) in the class ¿°°(0,Γ; £(Ω)). We know that for every a G Ζ£(Ω) and every / e ¿'(Ο,T; L(ù)), there exists at least one weak solution u of (N-S) satisfying the energy inequality. Here we mean by the weak solution a function u in L°°(0, Τ; Ζ^(Ω)) Π L 2 ( 0 , T ; H ^ m satisfying (N-S) in the sense of distributions (Definition of Section 1). For the energy inequality, see (1.1) below.