In my studies on the Ricci flow, I was faced with a problem. To prove the existence and uniqueness of solutions to the Ricci flow, it is proved that the Ricci flow is a Parabolic PDE type. Then one can find that it is weakly parabolic, so short-time existence does not follow from standard parabolic theory and use the DeTurck trick.
I've sought to understand the difference between strongly parabolic and weakly parabolic equations, But did not get a good result. Please guide me.
Thanks!
Best Answer
Here is a toy model: Consider a function $u=u(t,x,y,z)$. Then the standard heat equation $\partial_tu=(\partial_x^2+\partial_y^2+\partial_z^2)u$ is strongly parabolic, while e.g. the equation $\partial_tu=(\partial_x^2+\partial_y^2)u$ is only weakly parabolic (more commonly called degenerate parabolic).
Similarly, the Ricci flow it only degenerate parabolic. When you write down the symbol of its linearization you will find some null-directions (this corresponds to the $z$-direction in the above toy example). It is geometrically obvious, that there must be such null-directions, since the equation is invariant under diffeomorphisms. Anyway, after you take care of the diffeomorphisms using DeTurck's trick (how this works is explained in great detail in the books you read) the equation becomes strictly parabolic and you can apply standard theory.