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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
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