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$$L_{3,\infty}$$-solutions of Navier-Stokes equations and backward uniqueness. (English. Russian original) Zbl 1064.35134
Russ. Math. Surv. 58, No. 2, 211-250 (2003); translation from Usp. Mat. Nauk 58, No. 2, 3-44 (2003).
Let $$f(x,t) \in L_s(\mathbb{R}^3)$$ with respect to the variable $$x \in \mathbb{R}^3$$. Let $$L_{s,\infty}$$ denote the space with the norm $$\| f\| _{s,\infty}= \operatorname{ess\,sup}_{t\in (0,T)}\| f(\cdot,t)\| _s$$. The authors prove that any weak Leray-Hopf solution $$v(x,t)$$ of the Cauchy problem for the Navier-Stokes equations satisfying the additional condition $$v \in L_{3,\infty}(Q_T)$$ belongs to $$L_5(Q_T)$$. It is smooth and unique on $$Q_T=\mathbb{R}^3 \times (0,T)$$.
The authors also study the conditions when $$v$$ is Hölder continuous in a ball from $$\mathbb{R}^3$$. The heat operator $$\partial_t+\Delta$$ is considered on $$Q_+=\mathbb{R}_+^n \times (0,1)$$ in a class of generalized functions. It is proved that if $$| \partial_t u+\Delta u| \leq c(| \nabla u| +| u| )$$ on $$Q_+$$, $$u(\cdot,0)=0$$ on $$\mathbb{R}_+^n$$ and $$u(x,t) \leq \exp(M| x| ^2)$$ for all $$(x,t) \in Q_+$$, then $$u(x,t) \equiv 0$$ on $$Q_+$$.

##### 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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