Note on quasi-optimal error estimates for the pressure for shear-thickening fluids (Q6652080)

The authors consider the steady \(p\)-Navier-Stokes system: \(-\operatorname{div}S(D\mathbf{v})+[\nabla \mathbf{v}]v+\nabla q=f\), \(\operatorname{div}\mathbf{v}=0\), in \(\Omega \), with the homogeneous Dirichlet boundary condition \(\mathbf{v}=0\) on \(\partial \Omega \), where \(\Omega \subset \mathbb{R}^{d}\), \(d=2,3\), is a bounded, polygonal (if \(d=2\)) or polyhedral (if \(d=3\)) Lipschitz domain, \(\mathbf{v} :\Omega \rightarrow \mathbb{R}^{d}\) the velocity field, \(q:\Omega \rightarrow \mathbb{R}\) the kinematic pressure, \(D\mathbf{v}\) the symmetric part of the velocity gradient, \(S(A):\mathbb{R}^{d\times d}\rightarrow \mathbb{R}_{sym}^{d\times d}\) an extra stress tensor that is given a \((p,\delta)\)-structure: \(S(A)=\mu _{0}(\delta +\left\vert A^{sym}\right\vert)^{p-2}A^{sym}\), with \(p\in (1,\infty)\), \(\delta \geq 0\), and \(\mu _{0}>0\), and \(f:\Omega \rightarrow \mathbb{R}^{d}\) represents the external forces. The purpose of the paper is to prove quasi-optimal error estimates for the pressure, when considering the local discontinuous Galerkin approximation the authors previously developed in the case of shear-thickening, i.e., \(p>2\), and imposing a new Muckenhoupt regularity condition on the viscosity of the fluid. \N\NThe authors recall the notion of \(N\)-function in Orlicz spaces and the associated \(\varepsilon \)-Young inequality. They impose some regularity and boundedness hypotheses on the extra stress tensor \(S\) and they derive properties of the extra stress tensor. With the above problem, they associate the spaces \(\overset{o}{V}=(W_{0}^{1,p}(\Omega))^{d}\), \(\overset{o}{Q}=L_{0}^{p^{\prime}}(\Omega)\), problem (Q): For given \(f\in (L^{p^{\prime}}(\Omega))^{d}\), find \((\mathbf{v},q)^{\intercal}\in \overset{o}{V}\times \overset{o}{Q}\) such that for all \((\mathbf{z}, z)^{\intercal}\in \overset{o}{V}\times \overset{o}{Q}\), it holds that \((S(D \mathbf{v}),D\mathbf{z})_{\Omega}+([\nabla \mathbf{v}]\mathbf{v},\mathbf{z})_{\Omega}-(q,\operatorname{div}\mathbf{z})_{\Omega}=(f,\mathbf{z})_{\Omega}\), \((\operatorname{div} \mathbf{v},z)_{\Omega}=0\), and problem (P): For given \(f\in (L^{p^{\prime}}(\Omega))^{d}\), find \(\mathbf{v}\in \overset{o}{V}(0)\) such that for all \(\mathbf{z}\in \overset{o}{V}(0)\), it holds that \((S(D\mathbf{v}),D\mathbf{z})_{\Omega}+([\nabla \mathbf{v}]\mathbf{v},\mathbf{z})_{\Omega}=(f,\mathbf{z})_{\Omega}\), where \(\overset{o}{V}(0)=\{\mathbf{z}\in \overset{o}{V}\mid \operatorname{div}\mathbf{z}=0\}\). They discuss the well-posedness of these problems and recall a regularity result for a weak solution to problem (P). They recall the Muckenhoupt regularity condition for a solution \(\mathbf{v}\in (W^{1,p}(\Omega))^{d}\) to problem (P) or (Q). They then recall the construction of the discontinuous Galerkin approximation, the problems (P\(_{h}\)) and (Q\(_{h}\)) and a priori estimates on the velocity vector field. \N\NThe main result proves a priori estimates on the pressure, under the Muckenhoupt regularity condition on the velocity. This solves a conjecture posed in the previous paper by the authors in [SIAM J. Numer. Anal. 61, No. 4, 1763--1782 (2023; Zbl 1518.76037)]. For the proof, the authors prove a discrete convex conjugation inequality and stability results for (local) \(L^{2}\)-projection operators, a Bogovski operator, and lifting operators. The paper ends with the presentation of numerical results that prove the quasi-optimality of the proved a priori error estimates.