Introduction to Navier-Stokes Equation

January 07, 2025

In this blog, we will see some basic information about Navier-Stokes Equations, Euler Equations, and related problems.

Navier-Stokes Equation

Navier-Stokes equations (NS eqn for abbreviation) describe the viscous incompressible flow. It is a PDE system taking the following form:

\begin{equation} \begin{aligned} \partial_t u - \nu \Delta u + u\cdot \nabla u + \nabla p = f \\ \mbox{div } u = 0 \end{aligned} \tag{1} \label{NS} \end{equation}

where \( u(t,x):\ \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n \) is a vector field, in physical model it denotes the velocity field; \( p:\mathbb{R} \times \mathbb{R}^n\to \mathbb{R} \) is a single-valued function, which denotes the pressure; and \( f \) is also a vector field, who denotes the force in physical model. When we say \( \nabla u \), div \( u \), we always mean the derivative to the space variable \( x \). \( \nu \) is a constant, called 'kinetic viscosity', or 'viscosity' for simplicity.

Given \( f \), our goal is to find a pair of \( (u,p) \) to satisfy the equations. However, in most contexts, we omit \( p \) since it is not essential in the integral equation, only say to find a solution \( u \).

The nonlinear term of NS eqn is \( u\cdot \nabla u \).

Main Problems

• Initial Value Problem. In \( \mathbb{R}^n \), given \( u_0 \) satisfying compatible condition div \( u_0 = 0 \), find a solution \( u(t,x) \) such that it solves PDE \(\eqref{NS}\) and \( u(0,x) = u_0 \).

Stationary Navier-Stokes Equation

Also called Steady Navier-Stokes Equation in some literature. We look for a time-independent solution of Navier-Stokes Equations, i.e., we assume \( u(x), p(x), f(x) \) all independent of time variable \( t \). Then we got the following Stationary Navier-Stokes Equation:

\begin{equation} \begin{aligned} -\nu \Delta u + u \cdot \nabla u + \nabla p = f \\ \mbox{div } u = 0 \end{aligned} \tag{2} \label{SNS} \end{equation}

Intuitively, if a global solution to NS eqn \(\eqref{NS} u(t,x)\) has a limit in some sense when \( t \to \infty \), i.e. \( u(t,x) \to u_{\infty}(x) \) for some \( u_{\infty} \), then \( u_{\infty} \) shall solve the Stationary NS eqn. One would notice that the Stationary NS eqn is an elliptic eqn.

Problems

• Boundary Value Problem (BVP). Especially in dimension 2 and 3. Given \( \Omega \) bounded \( C^1 \) open set, \( u = a \) on \( \partial \Omega \). By divergence free condition div \( u = 0 \), we have compatible condition for BVP:
  \[ \int_{\partial\Omega} a \cdot n\ dS = \int_{\Omega} \mbox{div } u\ dx = 0 \]
  We call \( \mathcal{F} = \int_{\partial \Omega} a \cdot n \ dS \) total flux. With total flux zero condition, the well-posedness problem of Stationary NS.
• Exterior Domain Problem. Let \( \Omega \) be the complement of some bounded domain, consider Stationary NS eqn in \( \Omega \). The well-posedness problem, or well-posedness property under further assumption like small data on \( \partial\Omega \) or \( \lim_{\|x\|\to \infty} \|u(x)\| \).

For example, the Liouville-type problem is to prove the triviality of \( u \) when \( \Omega = \mathbb{R}^n \), \( f = 0 \), and \( \lim_{\|x\|\to\infty} \|u(x)\| = 0 \).

Euler Equation

Euler Equation describe the inviscid incompressible flow. 'Inviscid' means the viscosity constant \( \nu = 0 \). Thus the equation takes the form of:

\begin{equation} \begin{aligned} \partial_t u + u\cdot \nabla u + \nabla p = f \\ \mbox{div } u = 0 \end{aligned} \tag{3} \label{Euler} \end{equation}

Stokes Problem

Stokes Problem is the linear part of Stationary NS eqn:

\begin{equation} \begin{aligned} -\nu \Delta u + \nabla u = f \\ \mbox{div } u = 0 \end{aligned} \tag{4} \label{Stokes} \end{equation}

Reference for Stationary Navier-Stokes Equations:

• Ladyzhenskaya, O. The Mathematical Theory of Viscous Incompressible Flow

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