Lecture 10 ed

lecture 10

10.0.1 Historical motivation: fluid mechanics

Considering my poor physics, we just quick review the Euler/Lagrangian formulation. (:з」∠)

10.0.1.1 Euler equation

one stares at a given, fixed point of space \(x\), then measure the velocity of the fluid going through this point at time \(t\).

The unknown is the velocity field of the fluid

\[ \underset{(t,x)}{\left[0, +\infty \right] \times \Omega} \underset{\mapsto}{\to} \underset{v(t,x)}{\mathbb{R}^3} \]

The incompressible Euler equation reads

\[ \begin{cases} \frac{\partial v}{\partial t} + (v \cdot \nabla) v = - \nabla p \\ \nabla \cdot v = 0 \end{cases} \]

Boundary condition: \(v\) tangent to \(\partial \Omega\).

10.0.1.2 Lagrangian formulation

one put a label on each fluid particle and study the motion of each labelled particle.

For every fixed \(x_0 \in \Omega\), denote \(m(t,x_0)\) be the position at time \(t\) of the particle that was located at \(x_0\) at time \(0\).

Then we have for every fixed \(t\), the map \(m(t,\cdot) : \Omega \to \mathbb{R}^3\).

Then \[ \begin{cases} v(t,m(t,x_0)) = \frac{\rm{d}}{\rm{d} t} m(t,x_0) \\ m(0,x_0) = x_0\ \end{cases} \]

Since the physical property, we have for every fixed \(t\), \(m(t, \cdot): \Omega \to \Omega\) is a measure-preserving map. \(\implies \rm{det}\left( \nabla m(t, \cdot) \right)= 1\).

Then view every \(m(t, \cdot)\) as an element of \(G(\Omega): \underset{t}{\left[ 0, \infty\right]} \underset{\mapsto}{\to}\underset{m(t, \cdot)}{G(\Omega)}\), and replace \(m\) by \(g\).

the purpose of replacing \(m\) by \(g\) is only want to keep in harmony with the book.

\(g^{-1}_t\) physical meaning is the original position of the particle.

Then we have the Lagrangian formulation

\[ \begin{align} \implies v(t, g(t,x)) &= \frac{\rm{d}}{\rm{d} t} g(t,x)\\ v &= \partial_t g \circ g^{-1}_t\\ \end{align} \]

10.0.1.3 Arnold’s interpertation

Euler equation \(\iff\) the geodesic on \(G(\Omega)\),

endowed with the Riemannian structure inherited from \(L^2(\Omega, \mathbb{R}^3)\)

Recall that \(g: [-t, t] \to G(\Omega)\) is a geodesic iff the acceleration \(\frac{\rm{d}^2 g}{\rm{d}^2 t} \bot\) the tangent space \(T_{g(t)} G(\Omega)\) in \(L^2(\Omega, \mathbb{R}^n)\).

Let us determine the tangent space:

Recall from the preceding discussion that: \[ \text{a curve, starting from} \; g(0)\text{, stays in} \; G \iff \begin{cases} \frac{\partial g}{\partial t} \bot \; \text{boundary} \\ \nabla \cdot ( \partial_t g \circ g^{-1}_t ) = 0 \end{cases} \]

Then the tangent vectors in \(T_gG\) are all vector fields \(h\) satisfied \(\nabla \cdot ( h \circ g^{-1})= 0\), or equivalently \(h = w_0 \circ g\), where \(w_0 \in D_0 = \{ \text{the space of divergence-free vector field}\}\)

Then \(D_0^\bot = \{ -\nabla p , p: \Omega \to \mathbb{R} \}\) under some regularity conditions on \(\Omega\).

Brenier’s Polar Factorization Theorem

So the equation for geodesic becomes

\[ \frac{\rm{d}^2 }{\rm{d}^2 t} g(t) = - \nabla p(t, g(t) ) \]

Then we have the following result: