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arXiv · 2404.18333

$H^2$-regularity for stationary and non-stationary Bingham problems with perfect slip boundary condition

Takeshi Fukao·Takahito Kashiwabara
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Abstract

$H^2$-spatial regularity of stationary and non-stationary problems for Bingham fluids formulated with the pseudo-stress tensor is discussed. The problem is mathematically described by an elliptic or parabolic variational inequality of the second kind, to which weak solvability in the Sobolev space $H^1$ is well known. However, higher regularity up to the boundary in a bounded smooth domain seems to remain open. This paper indeed shows such $H^2$-regularity if the problems are supplemented with the so-called perfect slip boundary condition and if the yield stress vanishes on the boundary. For the stationary Bingham--Stokes problem, the key of the proof lies in a priori estimates for a regularized problem avoiding investigation of higher pressure regularity, which seems difficult to get in the presence of a singular diffusion term. The $H^2$-regularity for the stationary case is then directly applied to establish strong solvability of the non-stationary Bingham--Navier--Stokes problem, based on discretization in time and on the truncation of the nonlinear convection term.

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Takeshi Fukao, Takahito Kashiwabara. 2025-03-26. $H^2$-regularity for stationary and non-stationary Bingham problems with perfect slip boundary condition. https://arxiv.org/abs/2404.18333

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