## Uniqueness for mild solutions of Navier-Stokes equations in $$L^3(\mathbb{R}^3)$$ and other limit functional spaces. (Unicité dans $$L^3(\mathbb{R}^3)$$ et d’autres espaces fonctionnels limites pour Navier-Stokes.)(French)Zbl 0970.35101

Consider $\begin{cases} \overrightarrow{\nabla}\vec u=0 \cr \partial_t\vec u=\triangle\vec u-(\vec u \overrightarrow{\nabla})\vec u-\overrightarrow{\nabla}p \end{cases}\tag{1}$ the Navier-Stokes (N-S) - equations for an incompressible and homogeneous viscous fluid filling all the space and in the absence of exterior forces (the density and the viscosity constants being taken $$1$$) where $$\vec u(t,x):\mathbb{R}^+\times \mathbb{R}^3 \to \mathbb{R}^3$$ is the speed vector and $$p(t,x):\mathbb{R}^+\times \mathbb{R}^3 \to \mathbb{R}$$ is the pressure.
Let $$T \in ]0,\infty]$$. A weak solution on $$]0,T[$$ of (1) is a field of vectors $$\vec u(t,x)\in (L^2_{\text{loc}}(]0,T[\times \mathbb{R}^3))^3$$ which verifies $\begin{cases} \overrightarrow{\nabla}\vec u=0\cr \text{there exists} \;p\in D'(]0,T[\times \mathbb{R}^3) \;\text{such that } \partial_t\vec u=\triangle\vec u-\overrightarrow{\nabla}\cdot \vec u \otimes \vec{u}-\overrightarrow{\nabla}p. \end{cases}$
If, in addition $$\vec{u}(t,x) \in C([0,T[,(L^p)^3)$$, one says that $$\vec u$$ is a mild solution in $$L^p$$. The main result of the paper is the proof of uniqueness for mild solutions of the (N-S) equations in $$L^3(\mathbb{R}^3)$$ given by the
Theorem 1. Let $$\vec u \in C([0,T[,(L^3)^3), \text \;\overrightarrow{v} \in C([0,T'[,(L^3)^3)$$, such that
i) $$\vec u \;\text{is a weak solution of the (N-S) - equations on} \;]0,T[$$,
ii) $$\overrightarrow{v} \;\text{is a weak solution of the (N-S) - equations on} \;]0,T'[$$,
iii) $$\vec u |_{t=0}=\overrightarrow{v} |_{t=0}$$.
Then $$\vec u=\overrightarrow{v} \;\text{on} \;[0,\text{inf}\{T,T'\}[$$.

### MSC:

 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids

### Keywords:

uniqueness; Navier- Stokes equations; weak solution
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