Boundary partial regularity for the Navier-Stokes equations. (Russian, English) Zbl 1095.35031
J. Math. Sci., New York 132, No. 3, 339-358 (2006); translation from Zap. Nauchn. Semin. POMI 310, 158-190, 228 (2004).
The authors study the local regularity of solutions to the incompressible Navier-Stokes equations near the generally curved boundary. Under the assumption that the boundary is uniformly \(C^2\), they prove the regularity for the suitable weak solution near the boundary in the forms being known either in the interior of the domain or near the flat boundary. As an easy consequence it follows that the one-dimensional parabolic measure of points, where the solution is not Hölder continuous in any neighborhood of the point (i.e. the one-dimensional parabolic measure of the singular set), is zero.
The proof is based on local transformation of the coordinates near the boundary which makes the boundary flat, however, changes slightly the Navier-Stokes system. Thus, a carefull study of the perturbed system is also performed.
The proof is based on local transformation of the coordinates near the boundary which makes the boundary flat, however, changes slightly the Navier-Stokes system. Thus, a carefull study of the perturbed system is also performed.
Reviewer: Milan Pokorný (Praha)
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35D10 | Regularity of generalized solutions of PDE (MSC2000) |
Keywords:
incompressible Navier-Stokes equations; weak solution; singular set; partial regularity; regularity near boundaryReferences:
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