Besov regularity for the stationary Navier–Stokes equation on bounded Lipschitz domains

Abstract

We use the scale , , , to study the regularity of the stationary Stokes equation on bounded Lipschitz domains , , with connected boundary. The regularity in these Besov spaces determines the order of convergence of nonlinear approximation schemes. Our proofs rely on a combination of weighted Sobolev estimates and wavelet characterizations of Besov spaces. Using Banach’s fixed point theorem, we extend this analysis to the stationary Navier–Stokes equation with suitable Reynolds number and data, respectively.

Keywords:

• Stokes equation
• Navier–Stokes equation
• Besov space
• weighted Sobolev estimate
• wavelet
• nonlinear approximation
• fixed point theorem

AMS Subject Classifications:

• Primary: 35B65
• 35Q30
• 76D05
• 76D07
• Secondary: 42C40
• 46E35
• 65T60

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author has been supported by the Deutsche Forschungsgemeinschaft (DFG) [grant number DA 360/19-1]. The work of the last two authors has been supported by the Deutsche Forschungsgemeinschaft (DFG) [grant number DA 360/20-1]. The second author has been also partially supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand.