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Navier–Stokes Equations : A Mathematical Analysis

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Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition of the Subject

Introduction

Derivation of the Navier–Stokes Equations and Preliminary Considerations

Mathematical Analysis of the Boundary Value Problem

Mathematical Analysis of the Initial‐Boundary Value Problem

Future Directions

Acknowledgment

Bibliography

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Abbreviations

Steady‐State flow:

Flow where both velocity and pressure fields are time‐independent.

Three‐dimensional (or 3D) flow:

Flow where velocity and pressure fields depend on all three spatial variables.

Two‐dimensional (or planar, or 2D) flow:

Flow where velocity and pressure fields depend only on two spatial variables belonging to a portion of a plane, and the component of the velocity orthogonal to that plane is identically zero.

Local solution:

Solution where velocity and pressure fields are known to exist only for a finite interval of time.

Global solution:

Solution where velocity and pressure fields exist for all positive times.

Regular solution:

Solution where velocity and pressure fields satisfy the Navier–Stokes equations and the corresponding initial and boundary conditions in the ordinary sense of differentiation and continuity.

At times, we may interchangeably use the words “flow” and “solution”.

Basic notation:

\({\mathbb{N}}\) is the set of positive integers. \({\mathbb{R}}\) is the field of real numbers and \({\mathbb{R}^N}\), \({N\in\mathbb{N}}\), is the set of all N‑tuple \({\boldsymbol{x}=(x_1,\dots,x_N)}\). The canonical base in \({\mathbb{R}^N}\) is denoted by \({\{\boldsymbol{e}_1,\boldsymbol{e}_2,\boldsymbol{e}_3,\ldots, \boldsymbol{e}_N\}\equiv\{\boldsymbol{e}_i\}}\). For \({a,b\in\mathbb{R}}\), \({b > a}\), we set \({(a,b)=\{x\in\mathbb{R}: a < x < b\}}\), \({[a,b]=\{x\in\mathbb{R}: a\le x\le b\}}\), \( [a,b)=\{x\in\mathbb{R}: a\le x < b\}\) and \({(a,b]=\{x\in\mathbb{R}: a < x\le b\}}\). By \({\bar{\mathcal{A}}}\) we indicate the closure of the subset \({\mathcal{A}}\) of \({\mathbb{R}^N}\). A domain is an open connected subset of \({\mathbb{R}^N}\). Given a second‐order tensor \({\boldsymbol{A}}\) and a vector \({\boldsymbol{a}}\), of components \({\{A_{ij}\}}\) and \({\{a_{i}\}}\), respectively, in the basis \({\{\boldsymbol{e}_{i}\}}\), by \({\boldsymbol{a}\cdot\boldsymbol{A}}\) [respectively, \({\boldsymbol{A}\cdot\boldsymbol{a}}\)] we mean the vector with components \({A_{ij}a_i}\) [respectively, \({A_{ij}a_j}\)]. (We use the Einstein summation convention over repeated indices, namely, if an index occurs twice in the same expression, the expression is implicitly summed over all possible values for that index.) Moreover, we set \({|\boldsymbol{A}|=\sqrt{A_{ij}A_{ij}}}\). If \({\boldsymbol{h}(\boldsymbol{z})\equiv\{h_i(\boldsymbol{z})\}}\) is a vector field, by \({\nabla \boldsymbol{h}}\) we denote the second‐order tensor field whose components \({\{\nabla \boldsymbol{h}\}_{ij}}\) in the given basis are given by \({\{\partial h_j/\partial z_i\}}\).

Function spaces notation:

If \({\mathcal{A}\subseteq\mathbb{R}^N}\) and \( k\in \mathbb{N}\cup\{0\}\), by \({C^k(\mathcal{A})}\) [respectively, \({C^k(\bar{\mathcal{A}})}\)] we denote the class of functions which are continuous in \({\mathcal{A}}\) up to their kth derivatives included [respectively, are bounded and uniformly continuous in \({\mathcal{A}}\) up to their kth derivatives included]. The subset of \({C^k(\mathcal{A})}\) of functions vanishing outside a compact subset of \({\mathcal{A}}\) is indicated by \({C^k_0(\mathcal{A})}\). If \({u\in C^k(\mathcal{A})}\) for all \({k\in \mathbb{N}\cup\{0\}}\), we shall write \({u\in C^\infty(\mathcal{A})}\). In an analogous way we define \({C^\infty(\bar{\mathcal{A}})}\) and \({C_0^\infty(\mathcal{A})}\). The symbols \({L^q(\mathcal{A})}\), \({W^{m,q}(\mathcal{A})}\), \({m\ge 0}\), \({1\le q \le \infty}\), denote the usual Lebesgue and Sobolev spaces, respectively (\({W^{0,q}(\mathcal{A})= L^q(\mathcal{A})}\)). Norms in \({L^q(\mathcal{A})}\) and \({W^{m,q}(\mathcal{A})}\) are denoted by \( \smash{ \|\,\cdot\,\|_{q,\mathcal{A}}}\), \( \smash{ \|\,\cdot\,\|_{m,q,\mathcal{A}}}\). The trace space on the boundary, \({\partial\mathcal{A}}\), of \({\mathcal{A}}\) for functions from \({W^{m,q}(\mathcal{A})}\) will be denoted by \({W^{m-1/q,q}(\partial\mathcal{A})}\) and its norm by \( \smash{ \|\,\cdot\|_{m-1/q,q,\partial\mathcal{A}}}\).

By \({D^{k,q}(\mathcal{A})}\), \({k\ge1}\), \({1 < q < \infty}\), we indicate the homogeneous Sobolev space of order \({(m,q)}\) on \({\mathcal{A}}\), that is, the class of functions u that are (Lebesgue) locally integrable in \({\mathcal{A}}\) and with \({D^\beta u\in L^q(\mathcal{A})}\), \({|\beta|=k}\), where \( D^{\beta} ={\partial ^{|\beta|}}/{\partial x_1 ^{\beta_1}\partial x_2^{\beta_2}\dots\partial x_N^{\beta_N}}\, \), \( |\beta|=\beta_1+\beta_2+\dots+\beta_N \). For \({u\in D^{k,q}(\mathcal{A})}\), we put

$$ |u|_{k,q,\mathcal{A}}= \left(\sum_{|\beta|=k} \int_\mathcal{A}\mid D^{\beta} u\mid^{q}\right)^{1/q}\:. $$

Notice that, whenever confusion does not arise, in the integrals we omit the infinitesimal volume or surface elements. Let

$$ \mathcal{D}(\mathcal{A})=\{\boldsymbol{\varphi}\in C^{\infty}_0(\mathcal{A}):\mathrm{div}\,\boldsymbol{\varphi} = 0\}\:. $$

By \({L^q_\sigma(\mathcal{A})}\) we denote the completion of \({\mathcal{D}(\mathcal{A})}\) in the norm \({\|\cdot\|_q}\). If \({\mathcal{A}}\) is any domain in \({\mathbb{R}^N}\) we have \({L^2(\mathcal{A})=L^2_\sigma(\mathcal{A})\oplus G(\mathcal{A})}\), where \( G(\mathcal{A})=\{\boldsymbol{h}\in L^2(\mathcal{A}): \boldsymbol{h}=\nabla p\,,\text{ for some } p\in D^{1,2}(\mathcal{A})\}\); (see Sect. III.1 in [31]). We denote by P the orthogonal projection operator from \({L^2(\mathcal{A})}\) onto \({L^2_\sigma(\mathcal{A})}\). By \({\mathcal{D}_0^{1,2}(\mathcal{A})}\) we mean the completion of \({\mathcal{D}(\mathcal{A})}\) in the norm \({|\,\cdot\,|_{1,2,\mathcal{A}}}\). \({\mathcal{D}_0^{1,2}(\mathcal{A})}\) is a Hilbert space with scalar product \({[\boldsymbol{v}_1,\boldsymbol{v}_2]:=\int_\mathcal{A}(\partial\boldsymbol{v}_1/\partial{x_i})\cdot(\partial\boldsymbol{v}_2/\partial{x_i})}\). Furthermore, \({\mathcal{D}_0^{-1,2}(\mathcal{A})}\) is the dual space of \({\mathcal{D}_0^{1,2}(\mathcal{A})}\) and \({\langle\cdot,\cdot\rangle_{\mathcal{A}}}\) is the associated duality pairing.

If \({\boldsymbol{g}\equiv\{g_i\}}\) and \({\boldsymbol{h}\equiv\{h_i\}}\) are vector fields on \({\mathcal{A}}\), we set

$$ (\boldsymbol{g},\boldsymbol{h})_\mathcal{A}=\int_\mathcal{A} g_i h_i\,, $$

whenever the integrals make sense.

In all the above notation, if confusion will not arise, we shall omit the subscript \({\mathcal{A}}\).

Given a Banach space X, and an open interval \({(a,b)}\), we denote by \({L^q(a,b;X)}\) the linear space of (equivalence classes of) functions \({f:(a,b)\to X}\) whose X‑norm is in \({L^q(a,b)}\). Likewise, for r a non‐negative integer and I a real interval, we denote by \({C^r(I;X)}\) the class of continuous functions from I to X, which are differentiable in I up to the order r included.

If X denotes any space of real functions, we shall use, as a rule, the same symbol X to denote the corresponding space of vector and tensor‐valued functions.

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Acknowledgment

This work was partially supported by the National Science Foundation, Grants DMS-0404834 and DMS-0707281.

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  1. University of Pittsburgh, Pittsburgh, USA

    Giovanni P. Galdi

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  1. Giovanni P. Galdi
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2012 Springer-Verlag

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Galdi, G.P. (2012). Navier–Stokes Equations : A Mathematical Analysis. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_60

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