I am reading a paper where the authors prove $$ \frac{d}{dt} \Vert f \Vert_{H^4} (t)\leq C\Vert f \Vert^5_{H^4}(t) $$ Where $f=f(x,t)$ and $H^4$ stands for the usual Sobolev space. Using Gronwall's inequality, they obtain $\Vert f \Vert _{H^4}$ is bounded up to a time $T_*=T(\Vert f_0\Vert _{H^4})$. Now, they say applying energy method the local existence follows. I don't understand what the meant by 'energy method'.
Can someone explain what 'energy method' is here?
