On the construction of suitable weak solutions to the 3D Navier–Stokes equations in a bounded domain by an artificial compressibility method

Abstract

In this paper we will prove that suitable weak solutions of three dimensional Navier-Stokes equations in bounded domain can be constructed by a particular type of artificial compressibility approximation.

Suitable weak solutions to the 3D Navier–Stokes equations are constructed with the Voigt approximation

“…Recent results in an exterior domain with the Yosida approximation are also those by Farwig, Kozono, and Sohr [18]. Finally, we mention also that the existence of suitable weak solution has been proved by using some artificial compressibility method, see [11,17].…”

A New Proof of the Existence of Suitable Weak Solutions and Other Remarks for the Navier-Stokes Equations

“…Our results can be compared to other previous proofs of existence: the one in the Appendix in [2] (based on the construction of a family of time delayed linear approximations), the main result in Da Veiga [15] (relying on regularization with vanishing fourth-order terms), the main result in Guermond [3] (where Faedo-Galerkin techniques are employed) and, also, the results by Berselli [17], where the Voigt approximations and the artificial compressibility method are shown to converge.…”

From quasi-incompressible to semi-compressible fluids

“…It actually combines the simple timeregularization (4b) with a mere elliptic regularization div v = 1 H ∆π as used e.g. in [11,18,49].…”

“…8]) and used also e.g. in [11,21,25,26]. A physical meaning in the energy balance of this term is the variation of the kinetic energy 2 |v| 2 within the volume changes div v and is needed to gain the correct energetics, cf.…”