From the energy functional, we can derive the Euler-Lagrange equation and its corresponding gradient flow equation. My question is, what is the physical unit for ``time'' in the gradient flow equation?


For example, the Oseen-Frank energy for liquid crystal is given by
$$\int\_{\Omega} \frac{1}{2} K |\nabla u|^2,$$
where $u$ is the angle of the director with the $x$-axis, and $K$ is an elastic constant with unit Newton. The corresponding gradient flow equation is 
$$\frac{\partial u}{\partial t} = K \triangle u.$$
But then what is the unit of time $t$? It seems that it does not have the unit of time. What am I missing here? 


Thanks!



Units of time in the gradient flow equation?
If you wish to interpret the gradient flow equation as an equation of physics (rather than mathematics), then you need to introduce a friction coefficient into your problem, which tells you how rapidly energy is dissipated in order to reach an equilibrium condition. The energy functional itself only tells you what the equilibrium state is, not how fast you will reach it if you start away from equilibrium. So just multiply the left-hand-side of your gradient flow equation by a friction coefficient $\gamma$ (units $Nsm^{-2}$) and integrate. You will then flow towards the equilibrium state having $\Delta u=0$, which minimizes the energy regardless of the value you chose for $\gamma$.


If you wish to learn more, see for example


<http://wwwx.cs.unc.edu/~mn/classes/comp775/doc/calculusOfVariations.pdf>


