## Differential and Integral Equations

- Differential Integral Equations
- Volume 17, Number 11-12 (2004), 1359-1394.

### Optimal results for the Brezzi-Pitkäranta approximation of viscous flow problems

S. A. Nazarov and M. Specovius-Neugebauer

#### Abstract

By introducing the term $-{\varepsilon} ^2 \Delta p$ into the equation of continuity and an additional Neumann boundary condition for the pressure $p$, a strongly elliptic system is obtained which is a singular perturbation of the Stokes system. We use parameter-dependent Sobolev norms to derive asymptotically precise estimates for solutions to the perturbed problem as ${\varepsilon} \searrow 0$. This results in optimal estimates for the difference between solutions to both problems; such estimates are not available by the usually applied energy methods. Under additional regularity assumptions for the data, for the energy estimates, the order of convergence with respect to ${\varepsilon}$ is improved, and convergence in $H^{s+1}$ and $H^s$ norms is obtained for the velocity and pressure with $s\in [0,3/2)$. We verify the asymptotic precision of the estimates by constructing the boundary layers.

#### Article information

**Source**

Differential Integral Equations Volume 17, Number 11-12 (2004), 1359-1394.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060251

**Mathematical Reviews number (MathSciNet)**

MR2100032

**Zentralblatt MATH identifier**

1150.35508

**Subjects**

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Secondary: 35B25: Singular perturbations 76D07: Stokes and related (Oseen, etc.) flows 76N20: Boundary-layer theory

#### Citation

Nazarov, S. A.; Specovius-Neugebauer, M. Optimal results for the Brezzi-Pitkäranta approximation of viscous flow problems. Differential Integral Equations 17 (2004), no. 11-12, 1359--1394.https://projecteuclid.org/euclid.die/1356060251