On prouve l’unicité des solutions du système de Navier-Stokes incompressible dans $𝒞([0,T);L^d{\left(\Omega \right)}^d)$, où $\Omega$ est un domaine lipschitzien borné de ${ℝ}^d$ ($d\ge 3$).

The main result of this paper is the proof of uniqueness for mild solutions of the Navier-Stokes equations in L3(R3). This result is extended as well to some Morrey-Campanato spaces.

Consider the Navier-Stokes equation with the initial data $a\in L_{\sigma }^2\left({ℝ}^d\right)$. Let $u$ and $v$ be two weak solutions with the same initial value $a$. If $u$ satisfies the usual energy inequality and if $\nabla v\in L^2((0,T);{\dot{X}}_1{\left({ℝ}^d\right)}^d)$ where ${\dot{X}}_1\left({ℝ}^d\right)$ is the multiplier space, then we have $u=v$.

Existence of solutions to many kinds of PDEs can be proved by using a fixed point argument or an iterative argument in some Banach space. This usually yields uniqueness in the same Banach space where the fixed point is performed. We give here two methods to prove uniqueness in a more natural class. The first one is based on proving some estimates in a less regular space. The second one is based on a duality argument. In this paper, we present some results obtained in collaboration with Pierre-Louis...