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

Takeshi Fukao, Takahito Kashiwabara

TL;DR

The paper proves $H^2$-regularity up to the boundary for stationary and non-stationary Bingham flows with a perfect slip boundary, under the condition that the yield stress vanishes on the boundary. It achieves this by introducing a regularized problem, flattening the boundary to a half-space, and deriving sharp tangential and normal derivative estimates that separate the effects of the singular diffusion and the pressure. The results yield strong solvability for the non-stationary Bingham--Navier--Stokes system via time discretization and a truncation of the convection term. This advances the understanding of boundary regularity for incompressible viscoplastic flows and provides a rigorous route to strong time-dependent solutions with $H^2$-spatial regularity.

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.

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

TL;DR

The paper proves -regularity up to the boundary for stationary and non-stationary Bingham flows with a perfect slip boundary, under the condition that the yield stress vanishes on the boundary. It achieves this by introducing a regularized problem, flattening the boundary to a half-space, and deriving sharp tangential and normal derivative estimates that separate the effects of the singular diffusion and the pressure. The results yield strong solvability for the non-stationary Bingham--Navier--Stokes system via time discretization and a truncation of the convection term. This advances the understanding of boundary regularity for incompressible viscoplastic flows and provides a rigorous route to strong time-dependent solutions with -spatial regularity.

Abstract

-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 is well known. However, higher regularity up to the boundary in a bounded smooth domain seems to remain open. This paper indeed shows such -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 -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.
Paper Structure (20 sections, 24 theorems, 183 equations)

This paper contains 20 sections, 24 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Weak formulations for the stationary problem
  3. $H^2$-regularity for the stationary problem
  4. Coordinate transformation to flatten the boundary
  5. Local coordinates
  6. Reduction to a half-space problem
  7. Regularity in the tangential direction
  8. Estimates for $\nabla\nabla' \tilde{\bm u}$ uniform in $\epsilon$
  9. Estimates for $\nabla' \tilde{p}$ (not uniform in $\epsilon$)
  10. Regularity in the normal direction
  11. Proof of $\partial_d^2 \tilde{\bm u} \in \bm L^2(K_+)$ and $\partial_d \tilde{p} \in L^2(K_+)$
  12. Satisfaction of the slip boundary condition and $H^3$-approximation for $\bm u$
  13. Estimates for $\nabla \partial_d \tilde{\bm u}$ uniform in $\epsilon$
  14. Proof of (\ref{['eq: key estimate for pressure term']})
  15. $H^2$-regularity for the non-stationary problem
  16. ...and 5 more sections

Key Result

Proposition 2.1

(i) Let $(\bm u, p)$ be a sufficiently smooth solution of (eq: strong form)--(eq: slipBC). Then we have (ii) Let $\bm f \in \bm V_\sigma'$ and $g \in L^2(\Omega), \, g \ge 0$. Then there exists a unique solution $\bm u \in \bm V_\sigma$ of (eq: VI).

Theorems & Definitions (55)

  • Proposition 2.1
  • proof
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.2
  • ...and 45 more