Boundary partial regularity for the high dimensional Navier–Stokes equations

Abstract

We consider suitable weak solutions of the incompressible Navier-Stokes equations in two cases: the 4D time-dependent case and the 6D stationary case. We prove that up to the boundary, the twodimensional Hausdorff measure of the set of singular points is equal to zero in both cases.2010 Mathematics Subject Classification. 35Q30, 35B65, 76D05.

Regularity criteria for suitable weak solutions to the four dimensional incompressible magneto-hydrodynamic equations near boundary

“…However, the partial regularity to the four dimensional incompressible MHD equations for the boundary case seems still to be open. Meanwhile, similar problems for Navier-Stokes equations were studied in [7,8,9,35]. The main idea in [7,8,9] is to first establish a weak decay estimate of certain scale-invariant quantities, and then successively improve this decay estimate by a bootstrap argument and the elliptic or parabolic regularity theory, thus the proof do not involve any compactness argument.…”

“…Meanwhile, similar problems for Navier-Stokes equations were studied in [7,8,9,35]. The main idea in [7,8,9] is to first establish a weak decay estimate of certain scale-invariant quantities, and then successively improve this decay estimate by a bootstrap argument and the elliptic or parabolic regularity theory, thus the proof do not involve any compactness argument. Motivated by these works, we study the four dimensional incompressible MHD equations' partial regularity for the boundary case in this paper by following the main idea in [7,8,9].…”

“…The main idea in [7,8,9] is to first establish a weak decay estimate of certain scale-invariant quantities, and then successively improve this decay estimate by a bootstrap argument and the elliptic or parabolic regularity theory, thus the proof do not involve any compactness argument. Motivated by these works, we study the four dimensional incompressible MHD equations' partial regularity for the boundary case in this paper by following the main idea in [7,8,9]. Our results extend Kang's boundary regularity result from three dimension to four dimension and extend [12]'s interior result to the boundary case.…”

“…We explain the main steps and the main difficulties to prove the main results slightly below. Our proofs mainly follow the scheme in [8,9] mentioned before. Firstly, we carry out the estimates for some scaled norms.…”

“…And as a result of this step, we can show that if the assumption of Theorem 1.2 or Theorem 1.4 is satisfied, then the assumption of Theorem 1.1 is satisfied. Compared to the Navier-Stokes equations ( [8,9]), the main difference in the MHD equations is to control the pressure Π and the nonlinear term H · ∇H. As we explained before, motivated by [15,34], we can control the terms induced by Π and H · ∇H in the local energy inequality (1.3) with the assumption of Theorem 1.2.…”