# Geometric flow equations which are second order in time derivative

All examples about geometric flow equations given in Wikipedia's Geometric flow article are first order in time derivative. Would it make sense to have a geometric flow equation which was second order in time derivative and is there examples of those? If we replace the first order time derivative with second order time derivative in some geometric flow equation (e.g Ricci flow) then does the new equation make sense?

• Do you have a specific model in mind? – András Bátkai Sep 11 at 11:59
• No, I'm just curious whether such equations have been studied. – Professor Kirby Sep 11 at 12:36
• Indeed, the analog of Ricci flow with a second order time derivative has been studied by D-X Kong and K Liu, see e.g. arxiv.org/pdf/math/0610256.pdf arxiv.org/pdf/0709.1607.pdf and more papers available on Mathscinet – YangMills Sep 14 at 13:46

Geometric flows are modeled on fundamental flow equations, namely the heat and wave equations. The heat equation, which uses only a first order derivative in time, diffuses the function, so when $$t \rightarrow \infty$$, the solution not only converges to a limit, usually zero, but its derivatives all converge to zero. On the other hand, solutions to the wave equation, which uses a second order derivative in time, are oscillatory and do not ever settle down to a stable solution.