I found this wikipedia article about well-posedness of PDEs and the energy method. They consider the following problem the advection problem $$ u(t,x):[0,T]\times [0,1] \rightarrow \mathbb{R} , \quad s.t. \begin{cases} \displaystyle \partial_t u + \alpha \partial_x u = 0,\\ u(0,x) = f(x),\\ u(t,0) = u(t,1) = 0.\\ \end{cases} $$
From here they multiply the equation by $u$, integrate over $[0,1]$ and use the BC: $$ \partial_t \frac{1}{2}\int_0^1 u^2 dx = -\alpha \int_0^1 u \partial_x u = -\alpha \int_0^1 \partial_x \left ( \frac{u^2}{2} \right ) dx = - \alpha \left. \frac{u^2}{2} \right|_0^1 = 0, $$
which shows boundedness of the $L^2$ norm (or i guess "energy"?) of $u$. In fact, if we now integrate over $[0,T]$, we obtain $$\begin{aligned} \int_0^T \partial_t \Vert u \Vert_2 dt &= 0\\ \Vert u(T,x) \Vert_2 - \Vert\underbrace{ u(0,x) }_{=f(x)}\Vert_2 &= 0. \end{aligned}$$
But the wikipedia article states an inequality as the result: $$\Vert u(T,x) \Vert_2 \leq \Vert f(x)\Vert_2$$
Where is the $\leq$ coming from?
My idea: It might be, because they suddenly consider more general, non-zero boundary conditions for which $$\Vert u(T,x) \Vert_2 = \Vert f(x)\Vert_2 - \alpha \left. \frac{u^2}{2} \right|_0^1.$$
Now let $u(t,0)^2<u(t,1)^2$, and $\alpha>0$, meaning that intuitively we have a "net outflow of high solution values" at $x=1$. In this case, the estimate becomes: $$\Vert u(T,x) \Vert_2 = \Vert f(x)\Vert_2 - \underbrace{\alpha [u(t,1)^2-u(t,0)^2]}_{>0}\leq \Vert f(x)\Vert_2$$
The same works in the case of a net outflow at $x=0$, but not for net inflows at either $x=0$ or $x=1$.
- Is my idea correct?
- Why does it show well-posedness, i.e. why does it show uniqueness or existence of solutions?
EDIT:
- the wikipedia article changed not long after i posted the question here (maybe even because of my question?). The article linked above now discussed the same procedure for the diffusion equation, and explains the concepts much better.
- The problem i have discussed above is problematic, as I assume an outflow BC. This BC is too much and should not be posed. In the proof of uniqueness of solutions, and in the proof of continuous dependence on IC and BC, this turns out to be problematic.