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A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier-Stokes equations. (English) Zbl 1283.35069

Summary: The partial regularity of the suitable weak solutions to the Navier-Stokes equations in \(\mathbb R^n\) with \(n=2,3,4\) and the stationary Navier-Stokes equations in \(\mathbb R^n\) for \(n=2,3,4,5,6\) are investigated in this paper. Using some elementary observation of these equations together with De Giorgi iteration method, we present a unified proof on the results of L. Caffarelli et al. [Commun. Pure Appl. Math. 35, 771–831 (1982; Zbl 0509.35067)], M. Struwe [Commun. Pure Appl. Math. 41, No. 4, 437–458 (1988; Zbl 0632.76034)], H. Dong and D. Du [Commun. Math. Phys. 273, No. 3, 785–801 (2007; Zbl 1156.35442)] and H. Dong and R. M. Strain [Indiana Univ. Math. J. 61, No. 6, 2211–2229 (2012; Zbl 1286.35193)]. Particularly, we obtain the partial regularity of the suitable weak solutions to the 4d non-stationary Navier-Stokes equations, which improves the previous result of [Dong et al., 2007, loc. cit.], where Dong and Du studied the partial regularity of smooth solutions of the 4d Navier-Stokes equations at the first blow-up time.

MSC:

35Q30 Navier-Stokes equations
35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35D30 Weak solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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References:

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