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

Han, Pigong

Appl. Math., Praha 57, No. 5, 427-444 (2012).

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.

Reviewer: Jiří Neústupa (Praha)

MSC:

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

Keywords:

perturbed Navier-Stokes equation; interior regularity; partial regularity

Citations:

Zbl 0152.44402; Zbl 0106.18302; Zbl 0509.35067; Zbl 0718.35022

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