EDO e flussi

Flow §

Given an ODE, we can define a flow in its state space. Each starting conditon moves guided by this flow.

Def (flow) A flow on a set $X$ is a group action of the additive group of real numbers $(\mathbb{R},+)$. More explicitly

$$\Phi :X\times \mathbb{R}\to X$$

such that, for all $x\in X$ and $s,t\in \mathbb{R}$ the flow has the property

$${\Phi }^0(x)=x{\Phi }^s\circ {\Phi }^t(x)={\Phi }^{s+t}(x)$$

From the defnition follows that ${\Phi }^t:X\to X$ is a bijection with inverse ${\Phi }^{-t}:X\to X$.

Usually we require the flow to be compatible with some properties of $X$, for example if $X$ is equipped with a differentiable structure, then $\Phi$ is required to be differentiable. In this case the flow forms a one-parameter group of diffeomorphism (differentiable with differentiable inverse).

Flow of an ODE §

Given the ODE

$$\dot{x}=u(t,x)$$

we can define the flow ${\Phi }^{t,s}(x)$ which solves the ode, given any initial condition $x(s)=x$ (abuse of notation here). Note that we have $2$ real parameters, in the usual definition of flow is assumed $s=0$ (initial time). That is, the flow satisfies

$$\frac{d}{dt}{\Phi }^{t,s}(x)=u(t,{\Phi }^{t,s}(x))$$

with initial condition

$${\Phi }^{s,s}(x)=x$$

We suppose that all the conditions that ensure global existence and uniquness are satisfied Teorema di Picard–Lindelöf.

From the definition of flow of an ODE, follows the property:

$${\Phi }^{t+ϵ,s}(x)=x+u(x,t)ϵ+o({ϵ}^2)$$

The advantage of working with a flow, is that we can ask and answer new question, consider the evolution of a region of initial condition all at once. For example, suppose (as in reality) that the initial conditions are not known exaclty, we can model this uncertainty with a region of inital condition, equipped with a probability distribution. How far apart does the solutions evolve?

Suppose the initial state is uniformly distribuited in a set $B$, then its distributio is

$$f_0(x)=\frac{1}{|B|}{1}_{x\in B}$$

where $|B|$ is the measure of the set $B$. What can be said about the evolution of such system? Clearly, if we define the evolution of the set $B$ as a whole, (with a slight abuse of notation)

$$B_t={\Phi }^{t,0}(B)=\{{\Phi }^{t,0}(x)|x\in B)\}$$

the probability that the system at time $t$ is in $B_t$ is one (solutions don’t disappear!), but there’s no reason to belive that the distribution is still uniform. How does $f$ evolve? To answer this question, we need to develop some properties of the flow.

Prop Let ${\Phi }^{t,s}$ the flow of the field $u(t,x)$ and $J(x,t,s)=\det \frac{\partial {\Phi }^{t,s}}{\partial x}$ the determinant of the jacobian of the flux. Then the jacobian satisfies the following linear equation with non-constant coefficients

$$\frac{d}{dt}{\partial }_x{\Phi }^{t,s}(x)={\partial }_{x}u{|}_{t,{\Phi }^{t,s}(x)}{\partial }_x{\Phi }^{t,s}(s)$$

and $J$

$$\frac{d}{dt}J(x,t,s)=\nabla \cdot u{|}_{t,{\Phi }^{t,s}(x)}J(x,t,s)$$

Proof The first equation is straight forward, just exchange the derivatives. To prove the second, we need the following lemma

Lemma 1

$$\frac{d}{dϵ}\det {\partial }_x{\Phi }^{t+ϵ,t}(x)=\nabla \cdot u(t,x)$$

the use of $ϵ$ is necessaty, since we want to differentiate ${\Phi }^{t,t}$ only for the first $t$.

To prove this, we need the Determinant near identity lemma. With this lemma the proof of lemma $1$ is straightforward. $\square$