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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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