Flow of an autonomous ordinary differential equation

Question

We had the following definition in the real analysis class:

Let $f \in C(\mathbb{R}^n,\mathbb{R}^n)$ be Lipschitz continuous. The flow $\varphi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ of the ordinary differential equation $y'(x) = f(y(x))$ evaluated at $x$ and $y_0$ is defined to be the solution of the initial value problem $y(0) = y_0$ evaluated at $x$.

We did some proofs of properties of the flow, i.e. that $(\varphi_x)_{x \in \mathbb{R}}$ is a $1$-parameter group of diffeomorphisms. Since we did no exercises about it and I will have an exam in a month, I am a bit unsure, what I should learn about this concept. I have already taken a course on numerical solutions to ODEs where we also encountered a little bit the notion of the flow, i.e. one step methods which can be reversed. So my questions are the following two:

• Why is the concept of the flow of an ODE important? For what purpose it is used?
• Are there any standard exercises which one can do using the concept of the flow?

Thanks.

Answer 1

The flow is an important idea in the sense that it allows us to examine the behavior of solutions to an ODE at a "larger scale." What I mean by this is that instead of tracking a single solution, emanating from a point, we can track an ensemble of solutions, emanating from a set of points, all at the same time, and we can use this to study how the ODE distorts these points.

This is particularly important in applications of ODEs in PDE, and especially important in continuum mechanics. For instance, let's consider the following transport equation: $$ \partial_t u(t,x) + a(x) \cdot \nabla u(t,x) =0 $$ where $a: \mathbb{R}^n \to \mathbb{R}^n$ is Lipschitz. If we let $\varphi : \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ denote the flow map associated to $a$ then we have that $$ \frac{d}{dt} u(t,\varphi(t,x)) = \partial_t u(t,\varphi(t,x)) + \partial_t \varphi(x,t) \cdot \nabla u(t,\varphi(t,x)) \\ = \partial_t u(t,\varphi(t,x)) + a(\varphi(t,x)) \cdot \nabla u(t,\varphi(t,x)) =0 $$ which tells us that $u$ is "constant along the flow." In particular, if we specify the initial condition $u(0,x) = g(x)$, then $$ u(t,\varphi(t,x)) = u(0,\varphi(0,x)) = u(0,x) = g(x) $$ and hence we can solve for $u$ via $$ u(t,x) = g(\varphi^{-1}(t,x)) = g(\varphi(-t,x)). $$ This establishes a key relationship between the flow of a vector field and the "transport" operator / "convective derivative" induced by the vector field $a$. This plays an essential role in continuum mechanics in going back and forth between the material (Lagrangian) and laboratory (Eulerian) coordinate frames.

Since you already know that the flow map induces a $1-$parameter family of diffeomorphisms, I would suggest studying flows by trying to prove another one of the main theorems about them, Liouville's theorem. It says that $$ \det D \varphi(t,x) = \exp\left( \int_0^t \text{div}f(\varphi(s,x)) ds \right) $$ whenever $\varphi$ is the flow associated to $f$. This is another extremely important result in continuum mechanics, as it builds a link between incompressibility (the divergence free condition) and the local preservation of volume. Indeed, if $\text{div} f=0$, then Liouville shows that $$ \det D \varphi(t,x) =1 \text{ for all }t,x, $$ and so if $U \subseteq \mathbb{R}^n$ is a measurable set, then $$ |\varphi(t,U)| = \int_{U} |\det D \varphi(t,x)| dx = \int_U dx = |U|, $$
which tells us that the volume, or Lebesgue measure, of $U$ is preserved along the flow for all measurable sets.