Geodesic
Uniqueness of weak solutions of the Navier-Stokes equations
Applications of Mathematics, Tome 53 (2008) no. 6, pp. 561-582.

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Consider the Navier-Stokes equation with the initial data $a\in L_{\sigma }^2( \Bbb R^d) $. Let $u$ and $v$ be two weak solutions with the same initial value $a$. If $u$ satisfies the usual energy inequality and if $\nabla v\in L^2(( 0,T) ;\dot X _1(\Bbb R^d)^d)$ where $\dot X_1(\Bbb R^d)$ is the multiplier space, then we have $u=v$.
DOI : 10.1007/s10492-008-0042-9
Classification : 35D30, 35Q30, 76D03, 76D05
Keywords: Navier-Stokes equations; solution uniqueness; weak Leray-Hopf solution; multiplier space 
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Gala, Sadek. Uniqueness of weak solutions of the Navier-Stokes equations. Applications of Mathematics, Tome 53 (2008) no. 6, pp. 561-582. doi : 10.1007/s10492-008-0042-9. http://geodesic.mathdoc.fr/articles/10.1007/s10492-008-0042-9/

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