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The Oseen and Navier–Stokes equations in a non‐solenoidal framework
Author(s) -
Amrouche Chérif,
RodríguezBellido María Ángeles
Publication year - 2014
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3337
Subject(s) - solenoidal vector field , mathematics , navier–stokes equations , bounded function , domain (mathematical analysis) , mathematical analysis , divergence (linguistics) , norm (philosophy) , weak solution , vector field , geometry , physics , compressibility , law , linguistics , philosophy , political science , thermodynamics
The very weak solution for the Stokes, Oseen and Navier–Stokes equations has been studied by several authors in the last decades in domains ofR n , n ≥ 2 . The authors studied the Oseen and Navier–Stokes problems assuming a solenoidal convective velocity in a bounded domain Ω ⊂ R 3of classC 1 , 1for v ∈ L s (Ω) for s ≥3 in some previous papers. The results for the Navier–Stokes equations were obtained by using a fixed‐point argument over the Oseen problem. These results improve those of Galdi et al. , Farwig et al. and Kim for the Navier–Stokes equations, because a less regular domain Ω ⊂ R 3and more general hypothesis on the data are considered. In particular, the external forces must not be small. In this work, existence of weak, strong, regularised and very weak solution for the Oseen problem are proved, mainly assuming that v ∈ L 3 (Ω) and its divergence ∇· v are sufficiently small in the W −1,3 (Ω)‐norm. In this sense, one extends the analysis made by the authors for a given solenoidal v in some previous papers. As a consequence, the existence of very weak solution for the Navier–Stokes problem ( u , π ) ∈ L 3 ( Ω ) × W − 1 , 3 ( Ω ) / R for a non‐zero divergence condition is obtained in the 3D case. Copyright © 2014 John Wiley & Sons, Ltd.