The Pressure Stabilization Method for Viscous Flow Problems

Let $\Omega\subset \mathbb{R}^3$ be a domain with smooth boundary $\partial\Omega$. We investigate a mixed boundary value problem for the following strongly elliptic system of second order differential equations \[ \begin{aligned} & S_{\epsilon} u = \left(-\Delta v+ \nabla p, - {\epsilon}^2 \Delta p + \operatorname{div} v \right)=(f',f_4) \quad \text{ in }\Omega, \\ & v= g',\quad \partial_n p=g_4 \quad \text{ on }\partial \Omega, \end{aligned} \qquad (\text{ S$_\epsilon$ }) \] where we focus our interest on asymptotically precise estimates for the solutions describing their behavior as $\epsilon \to 0$. This system ought be considered as a singular perturbation of the Stokes system (S$_0$) which appears if we set $\epsilon=0$ and cancel the Neumann boundary condition for $p$, in particular the type of ellipticity is changed with $\epsilon =0$. With $f_4=0$ and vanishing boundary values, the above system appears in numerical schemes for the Navier-Stokes equations on bounded domains, namely, in the so-called pressure-stabilization methods. If $\Omega$ is a bounded domain, the energy methods applied there to estimate the error between the solution $(v^{\epsilon},p^{\epsilon})$ of the system (S$_{\epsilon}$) and the solution $(v^0,p^0)$ to the Stokes system are of the form \[ \|v^{\epsilon} -v^0; H^1(\Omega)\| + \|p^{\epsilon} -p^0;L^2(\Omega)_\bot\| \leq C\,\epsilon\, \|f';L^2(\Omega)\|. \] (the index $\bot$ stands for functions with vanishing mean value).

To obtain asymptotically precise estimates we introduce Sobolev norms depending on the small parameter $\epsilon \gt 0$. It turns out that for bounded domains, under the additional smoothness assumption $f' \in H^1(\Omega)$, these estimates can be improved up to convergence of $v^{\epsilon}$ in $H^2(\Omega)$ and $p^\varepsilon$ in $H^1(\Omega)_\bot$.

Apparently up to now the corresponding nonlinear problems as well as the case of unbounded domains were not yet considered. Here the interest is focused on the exterior Dirichlet problem for the linear systems. The appropriate function spaces for the investigations are step weighted, parameter dependent Sobolev spaces. This leads to asymptotically precise estimates as $\epsilon \to 0^{+}$, and enables us to derive the complete asymptotics of the solutions to (S$_\epsilon$) as $|x| \to \infty$. The latter is remarkable insofar as this system itself for $\epsilon \gt 0$ the usual arguments of Kondratiev theory, the differential operator $S_\epsilon$ is not admissible at infinity.

The results presented here are obtained in a joint work with S. A. Nazarov, St. Petersburg.