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Bilinear estimates in \(BMO\) and the Navier-Stokes equations. (English) Zbl 0970.35099

The authors investigate blow-up phenomena of smooth solutions to the Cauchy problem for the Navier-Stokes system in \({\mathbb R}^n\), \((n\geq 3)\): \[ u_t-\Delta u+u\cdot \nabla u +\nabla p =0,\quad \text{ div} u =0,\quad u(x,0)=a(x). \] It is well known that for every initial datum \(a\in W^{s,2}({\mathbb R}^n)\), \(s>n/2-1\), there exists \(T=T(\|a\|_{W^{s,2}})\) and a unique regular solution \(u(x,t)\) to this Cauchy problem on the interval \([0,T)\). Then there is the important question whether the solution loses its regularity at \(t=T\). Y. Giga [J. Differ. Equations 61, 186-212 (1986; Zbl 0577.35058)] showed that if \(\int_0^T \|u(t)\|^\kappa_{L^r} dt<\infty\) for \(2/\kappa +n/r=1\) with \(n<r< \infty\), then \(u\) can be continued beyond \(t=T\). Moreover, in \({\mathbb R}^3\), J. T. Beale, T. Kato and A. Majda [Commun. Math. Phys. 94, 61-66 (1984; Zbl 0573.76029)] dealt with the vorticity \(\omega=\text{ rot} u\) and proved an analogous result under the condition \(\int_0^T \|\omega (t)\|_{L^\infty} dt<\infty.\)
The purpose of this paper is to extend these results to the marginal space \(BMO\) (which is larger than \(L^\infty\)), and it was proved that solutions can be extended beyond \(t=T\) provided \(\int_0^T \|u (t)\|^2_{BMO} dt <\infty.\) Moreover, the authors characterize the blow-up time by means of the vorticity \(\omega\) in \(BMO\) obtaining the result analogous to the one by Beal, Kato and Majda with \(L^\infty\) replaced by \(BMO\).

MSC:

35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
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