## 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

### Citations:

Zbl 0577.35058; Zbl 0573.76029
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