ON THE UNIQUENESS OF BOUNDED WEAK SOLUTIONS TO THE NAVIER-STOKES CAUCHY PROBLEM

Abstract. In this note we give a uniqueness theorem for solutions (u, pi) to the Navier-Stokes Cauchy problem, assuming that u belongs to L∞((0, T) × Rn) and (1 + |x|)−n−1pi ∈ L1(0, T;L1(Rn)), n ≥ 2. The interest to our theorem is motivated by the fact that a possible pressure field p̃i, belonging to L1(0, T; BMO), satisfies in a suitable sense our assumption on the pressure, and by the fact that the proof is very simple. 1