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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 \(\omega\).

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