For the non-linear time evolution PDEs modeling 3D fluid motion, one does NOT know time-$T$ WELL-POSEDNESS, that is one does not know for a describable dense set of initial conditions in some space there exists in that space a unique set of solution paths up to time $T$.
This talk will discuss how to use ALGEBRAIC TOPOLOGY to construct CONSISTENT finite dimensional models/algorithms for $3D$ fluids which are structurally similar in their algebraic and geometric aspects to the continuum model. This is done with an eye toward achieving TYPICAL stability of computation. The IDEA is that algorithms with typical stability might be useful for $3D$-fluid computations whether or not the continuum PDEs are known to be WELL-POSED or indeed perhaps these algorithms would be useful for prediction even if the continuum PDEs were known NOT to be well posed. This idea is motivated by the Lax-Richtmeyer Equivalence Theorem.
The continuum PDEs for fluid motion can be expressed in the language of differential forms using exterior d, the wedge product and the hodge star operator. We replicate these structures combinatorially using the ideas of algebraic topology. Indeed the first two structures gave birth and development to algebraic topology via their discrete analogues. We need to discuss the analogue of the star operator which works quite well for the cubical cell decomposition of $3$-space.
The Combinatorial Star in our case is a bijection between $k$-chains and $[3-k]$-cochains and visa versa. It will be the composition of the Poincare bijection between cells and dual cells with a fortuitous cellular bijection that exists between the cell decomposition and the dual cell decomposition in the case of the 3-space subdivision into cubes. The combinatorial star converts the boundary operator on chains into the coboundary operator on cochains and visa versa. Pre-composing Combinatorial Star with the coboundary on one cochains leads to the Combinatorial Curl operator $D$ on one cochains in $3D$.Post-composing yields the Combinatoral Divergence operator.
The next step in $3D$ is the construction of an alternating three form $\{ , , \}$ on one cochains. This
$3$-form is non zero on a triple of edges if they can be arranged to form a path between diagonally opposite vertices for some cube in the subdivision. Using the inner product $( , )$ which makes the cells an orthonormal basis, using the alternating three form on one cochains $\{ , , \}$ and using the combinatorial curl operator $D$, one can write the finite grid version of the $3D$ fluid evolution. We can show that this model at each grid level satisfies analogues of the known invariances of the continuum model [e.g. constancy of circulation around any transported closed curve] and is characterized by these invariances.These arguments depend on the fortuitous fact
which is true in this case that the combinatorial curl is self adjoint and invertible on the configuration space of combinatorial divergence free fields.