## Interior regularity of weak solutions to the perturbed Navier-Stokes equations.(English)Zbl 1265.35246

The author studies the perturbed Navier-Stokes system $\partial _tv-\Delta v+(v\cdot \nabla )v+\nabla \pi _1=-\partial _tv_0-(\partial _t\omega _0)\times x-\omega _0\times (\omega _0\times x)-2\omega _0\times v,\; \text{ div }v=0, \tag{1}$ which describes the motion of a viscous incompressible fluid relative to the coordinate system, moving with a translational velocity $$v_0(t)$$ and angular velocity $$\omega _0(t)$$. The special form of the Navier-Stokes equation from (1) comes from [G. K. Batchelor, An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967; Zbl 0152.44402), page 140]. Setting $$u = v + v_0 +\omega _0\times x$$, $$\pi =\pi _1-v_0\cdot (\omega _0\times x)$$, system (1) is transformed to $\partial _tu-\Delta u+(u\cdot \nabla )u=(v_0\cdot \nabla )u+(\omega _0\times x)\cdot \nabla u-\omega _0\times u,\; \text{ div }u=0 \text{ in } \Omega \times (0,T), \tag{2}$ $$\Omega$$ is supposed to be a domain in $$\mathbb {R}^2$$ or $$\mathbb {R}^3$$ with a smooth boundary. The author defines the notions of a weak and suitable weak solution of the system (2) and proves a series of theorems about the interior regularity of a weak or a suitable weak solution $$u$$. The theorems are analogous to the known theorems on regularity of weak or suitable weak solutions, coming from [J. Serrin, Arch. Ration. Mech. Anal. 9, 187–195 (1962; Zbl 0106.18302); L. Caffarelli, R. Kohn and L. Nirenberg, Commun. Pure Appl. Math. 35, 771–831 (1982; Zbl 0509.35067); S. Takahashi, Manuscr. Math. 69, No. 3, 237–254 (1990; Zbl 0718.35022)]. Generally, the author shows that the perturbing terms have only a little influence on regularity of the solution. It is not much surprising, especially in those situations, when the author treats the regularity in a bounded region or in the neighbourhood of a fixed point, because $$v_0$$ and $$\omega _0$$ are supposed to depend “smoothly” on $$t$$. On the other hand, since one of the perturbing terms on the right hand side of the first equation of (2) contains factor $$x$$, the theorems estimating the solution (or informating about its regularity) in an exterior domain, are of a greater interest. Comparing Theorem 1.4 of Han’s paper, which brings a statement analogous to Caffarelli-Kohn-Nirenberg’s well known criterion for regularity of a suitable weak solution to the non-perturbed Navier-Stokes equation, deducing concretely the regularity at the point $$(x_0,t_0)$$ from the limit superior (for $$r\rightarrow 0$$) of $$r^{-1} \int _{Q_r(x_0,t_0)} | \nabla u| ^2 \text{d}x\;\text{d}t$$, one can observe that the factor $$r^{-1}$$ is missing in Han’s Theorem 1.4.

### MSC:

 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 35B65 Smoothness and regularity of solutions to PDEs

### Citations:

Zbl 0152.44402; Zbl 0106.18302; Zbl 0509.35067; Zbl 0718.35022
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### References:

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