I've been having a look at the Gage and Hamilton's The Heat Equation Shrinking Convex Plane Curves (here). In particular I've been working in the Lemma 4.4.2 and some further results where they find bounds for the higher derivatives of $\kappa$. The lemma states the following:
If $\kappa$ and $\kappa'$ are bounded, then $\int_0^{2\pi} (\kappa'')^4$ is bounded.
In the proof, they calculate $${\partial\over\partial t}\int_0^{2\pi} (\kappa'')^4=-12\int_0^{2\pi} \kappa^2(\kappa'')^2(\kappa''')^2+2\kappa\kappa'(\kappa'')^3(\kappa''')+3\kappa^2\kappa'(\kappa'')^2\kappa''',$$ using that $${\partial\kappa\over\partial t}=\kappa^2{\partial^2\kappa\over\partial\theta^2}+\kappa^3.$$ Then, for the first expresion, they use the Peter-Paul inequality $ab\leq 4a^2/\varepsilon+\varepsilon b^2$ (a version of the Young's inequality for products) to "bound the second and third terms by the first term and some additional penalty terms" to obtain $${\partial\over\partial t}\int_0^{2\pi} (\kappa'')^4\leq \int_0^{2\pi} C_1\kappa'^2(\kappa'')^4+C_2\kappa^2\kappa'^2(\kappa'')^2.$$
The proof continues a little bit, but I'm struggling to understand the last step. I don't really get how they use the previous inequality to obtain the estimate. Any idea or hint is more than welcome. Thanks in advance.
This kind of computations show up a lot in geometric analysis. The point is that you have a negative term
$$-12\int_0^{2\pi} \kappa^2(\kappa'')^2(\kappa''')^2$$
on the right hand side, which you want to take advantage of. So you wish to make up the term $\kappa\kappa''\kappa'''$. For the second term, write
$$2\kappa\kappa'(\kappa'')^3(\kappa''') = 2\big(\kappa\kappa''\kappa''' \big) \big(\kappa'(\kappa'')^2\big)\le \epsilon{\big(\kappa\kappa''\kappa''' \big)^2} + \frac{1}{\epsilon} (\kappa'(\kappa'')^2)^2$$
Similarly for the last term,
$$3\kappa^2\kappa'(\kappa'')^2\kappa''' = 3\big(\kappa\kappa''\kappa''' \big) \big(\kappa \kappa'\kappa''\big) \le \frac{3}{2} \bigg( \delta\big(\kappa\kappa''\kappa''' \big)^2 + \frac{1}{\delta} \big(\kappa \kappa'\kappa''\big)^2\bigg)$$
Then get $\epsilon, \delta$ small so that $-12$ kills off the terms.