# Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces

## TL;DR

This paper develops a method for approximating functions on smooth bounded domains using eigenspaces of the Laplacian and Stokes operators, ensuring convergence in both Sobolev and Lebesgue norms, with applications to fluid dynamics equations.

## Contribution

It introduces an abstract framework for simultaneous approximation in Lebesgue and Sobolev spaces via fractional power spaces, applied explicitly to Laplacian and Stokes operators.

## Key findings

- Established convergence of approximations in Sobolev and Lebesgue norms.
- Proved energy equality for weak solutions of Brinkman--Forchheimer equations.
- Provided a new approximation technique applicable to PDE analysis.

## Abstract

We approximate functions defined on smooth bounded domains by elements of the eigenspaces of the Laplacian or the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces. We prove an abstract result referred to fractional power spaces of positive, self-adjoint, compact-inverse operators on Hilbert spaces, and then obtain our main result by using the explicit form of these fractional power spaces for the Dirichlet Laplacian and Stokes operators. As a simple application, we prove that all weak solutions of the incompressible convective Brinkman--Forchheimer equations posed on a bounded domain in ${\mathbb R}^3$ satisfy the energy equality.