Hi I am looking at the proof of theorem 2 energy estimates in Evans PDE. I have some difficulties regarding the estimate for each term. First for the first term. Q1 I am a little vague how (23) is obtained. Well, what I see is that $$2(u'_m,u_m)+2\beta\|u_m\|_{H_0^1(U)}\le 2(u'_m,u_m)+2 B[u_m,u_m;t]+2\gamma\|u_m\|_{L^2(U)}^2$$
Then by (21) the above becomes $$2(u'_m.u_m)+2\beta\|u_m\|_{H_0^1(U)}\le2(f,u_m)+2\gamma\|u\|_{L^2(U)}^2$$, i.e. $$\frac{d}{dt}(\|u_m\|_{L^2(U)}^2)+2\beta\|u_m\|_{H_0^1(U)}\le2(f,u_m)+2\gamma\|u\|_{L^2(U)}^2$$ Now, I believe, the following fact (stated in the book) $$|(f,u_m)|\le\frac{1}{2}\|f\|_{L^2(U)}^2+\frac{1}{2}\|u_m\|_{L^2(U)}^2$$ is used to conclude $$\frac{d}{dt}(\|u_m\|_{L^2(U)}^2)+2\beta\|u_m\|_{H_0^1(U)}\le C_1\|u_m\|_{L^2(U)}^2+C_2\|f\|_{L^2(U)}^2$$
I guess $C_1$ is $1+2\gamma$ and $C_2$ is 1. But I am really vague about this, because, the inequality for $|(f,u_m)|$ is not quite the same as the inequality for $(f,u_m)$.
Q2 I do not get the inequality sign on $\eta(0)=\|u_m(0)\|_{L^(U)}\le\|g\|_{L^2(U)}.$ According to (15), i.e. $d_m^k(0)=(g,w_k)$, I have $$\|u_m(0)\|_{L^2(U)}^2=\|\sum_k (g,w_k)w_k\|_{L^2(U)}^2=(\sum_k (g,w_k)w_k,\sum_j (g,w_j)w_j)=\sum_k\sum_j|(g,w_k)|^2(w_k,w_j)$$
Now since $w_k$ is an orthonormal basis in $L^2$, I think I should end up with an equality rather than inequality. Please clarify.
Next, for the estimate of the 2nd term, I think, all I need is to integrate †he estimate $\max_{t\in[0,T]}\|u_m(t)\|_{L^2(U)}^2$ over $[0,T]$, right?
Finally, for the 3rd term. I cannot see the 3rd last line, i.e. $$|<u'_m,v>|\le C(\|f\|_{L^2(U)}+\|u_m\|_{H_0^1(U)})$$ is deduced from the previous line. Basically, my understanding is $$\left|<u'_m,v>\right|\le\left|<f,v^1>-B[u_m,v^1;t]\right|$$...
Please help.
