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The Navier-Stokes equation for an incompressible fluid in $$\mathbb{R}^ 2$$ with a measure as the initial vorticity. (English) Zbl 0826.35094
The author studies global in time solutions of the Navier-Stokes equation for an incompressible fluid in $$\mathbb{R}^2$$ $\partial_t u - \Delta u + \Pi \partial (u \otimes u) = 0, \quad u=\Pi u,$ where $$\Pi$$ is the projection onto solenoidal vectors along gradients, $$u \otimes u$$ is a tensor with $$jk$$-component $$u_k u_j$$ and $$\partial (u \otimes u)$$ is a vector with $$j$$-th component $$\partial_k (u_ku_j) = u_k \partial_k u_j$$. Besides, the equation for the associated vorticity $$\zeta = \partial_1 u_2 - \partial_2 u_1$$ is studied: $\partial_t \zeta - \Delta \zeta + \partial \cdot (\zeta S* \zeta) = 0, \quad S(x) = (2 \pi)^{-1} |x |^{-2} (x_2, -x_1).$ The following estimates for $$u$$ and $$\zeta$$ in terms of the total variation $$|\omega |$$ of the initial vorticity $$\omega$$ are obtained: \begin{aligned} t^{1 - 1/q} |\zeta (t) |_q \leq k^{-(1- 1/q)} |\omega |, \quad & q \in [1, \infty) \\ t^{1/2 - 1/p} |u(t) |_p \leq \sigma_{2p/(p + 2)} k^{- (1/2 - 1/p)} |\omega |,\quad & p \in (2, \infty). \end{aligned} Global estimates for derivatives of $$\zeta$$ and $$u$$ are deduced \begin{aligned} t^{n + k/2 + 1 - 1/q} \biggl |\partial^n_t (- \Delta)^{k/2} \zeta (t) \biggr |_q \leq K, \quad & q \in (1, \infty), \\ t^{n + k/2 + 1/2 - 1/p} \biggl |\partial^n_t (-\Delta)^{k/2} u(t) \biggr |_p \leq K, \quad & p \in (1, \infty). \end{aligned} It is shown that $$|\zeta (t) |_q$$ is monotone nonincreasing in $$t > 0$$ for each fixed $$q \in [1, \infty)$$. Uniqueness is proved under a certain condition on the atomic part of $$\omega$$.

##### MSC:
 35Q30 Navier-Stokes equations 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 35K05 Heat equation 76D05 Navier-Stokes equations for incompressible viscous fluids