Regularity up to the boundary of solutions to boundary-value problems of the theory of generalized Newtonian liquids
The author studies the boundary value problem that describes a~slow stationary flow of a~generalized Newtonian liquid \[ \begin{gathered} -{\text{div}}{\partial D\over \partial \tau}\bigl(\varepsilon(u)\bigr) =f-\nabla p\quad \text{in } \Omega\subset \mathbb R^n, \\ \text{div }u=0\quad\text{in } \Omega, \\ u=0\quad\text{on } \partial\Omega. \end{gathered} \] Here \(u\) is the unknown velocity field, \(p\) is the unknown pressure field, \(\varepsilon(u)={1\over 2}\bigl(\nabla u+(\nabla u)^T\bigr)\) is the~strain rate tensor, \(D\) is a~dissipative potential on the space of real symmetric matrices of order~\(n\). For a~generalized solution to this problem, the author proves that \(u\in W^2_2(\Omega)\) and \(p\in W^1_2(\Omega)\) and the Hölder continuity of the strain rate tensor \(\varepsilon(u)\) on some set ~ \(\Omega_0\subset\overline{\Omega}\). In two dimensions, the Hölder continuity of \(\varepsilon(u)\) is proven everywhere on the set ~\(\overline{\Omega}\). Finally, \(\varepsilon(u)\) belongs to \(L^\infty\) for a~special class of potentials (\(u(x)\) stands for a~generalized solution).
stationary flow of a generalized Newtonian liquid
regularity of the strain velocity tensor
scientific article; zbMATH DE number 1495305