Abstract
We consider the Navier–Stokes equations with a given eddy viscosity and a wall law as boundary condition. Once the functional background is properly established, we prove the existence of a weak solution to this problem, obtained by approximations based on a singular perturbation of the incompressibility constraint. We investigate two ways of establishing the existence of approximate solutions: the standard Galerkin method and the linearization method by Schauder’s fixed-point theorem. Estimates for the velocity are deduced from energy equalities, whereas estimates for the pressure are derived from appropriate potential vectors. Cases where the solution is unique are also investigated. To achieve our goal, we also develop several theoretical tools that will be used for the analysis of general turbulent models in the following chapters, such as the convergence of families of variational problems or the energy method.
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Notes
- 1.
According to a private communication by L. Tartar, the results of this chapter should work for Lipschitz domains.
- 2.
We implicitly number the equations of a given coupled system in the order they are set out with roman numerals.
- 3.
The Stokes formula may also be referred to as Green’s formula.
- 4.
if \(\boldsymbol{\varphi }\in \mathcal{D}(\overline{\varOmega })\), \(\gamma _{0}\boldsymbol{\varphi } =\boldsymbol{\varphi }\vert _{\varGamma }\), \(\gamma _{n}\boldsymbol{\varphi } =\boldsymbol{\varphi } \cdot \mathbf{n}\vert _{\varGamma }\).
- 5.
Appendix A at the end of the book, also referred to as [TB].
- 6.
Recall that for all measurable sets U whose measure is denoted by λ, ∀ u ∈ L p(U), \(\forall v \in L^{p^{{\prime}} }(U)\), we denote for simplicity \((u,v)_{U} =\int _{U}u(\mathbf{x})v(\mathbf{x})d\lambda (\mathbf{x})\).
- 7.
\(\forall \,j \in \mathbb{N}^{\star }\), \((e_{i},e_{j})_{\varOmega } =\delta _{ i}^{j}\), and \(\forall \,u \in H_{0}^{1}(\varOmega )\), \(u_{n} =\sum _{ j=1}^{n}(e_{ j},u)_{\varOmega }e_{j} \rightarrow u\) in L 2(Ω).
- 8.
Some authors refer to strong solutions rather than mild solutions. In this book, strong solutions are those satisfying (6.16).
- 9.
This result will be used throughout this chapter, which will be not systematically mentioned.
- 10.
\(\vert \vert \nu _{t}\vert \vert _{\infty }\) stands for \(\vert \vert \nu _{t}\vert \vert _{0,\infty,\varOmega }\) for simplicity.
- 11.
In case of a family \((\zeta _{\varepsilon })_{\varepsilon >0}\), subsequential limits are limits of sequences of the form \((\zeta _{\varepsilon _{n}})_{n\in \mathbb{N}}\), where \(\varepsilon _{n} \rightarrow 0\) as n → ∞.
- 12.
- 13.
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Chacón Rebollo, T., Lewandowski, R. (2014). Steady Navier–Stokes Equations with Wall Laws and Fixed Eddy Viscosities. In: Mathematical and Numerical Foundations of Turbulence Models and Applications. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-0455-6_6
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