I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am considering the hyperbolic equation of the form $$w_t+\frac{x}{T-t}w_x=0, x\in[0,1]$$ and some continuous initial data $w(x,0)=w_0(x)$. I would like to claim that this is a well posed problem on $[0,T]$. One can analytically solve this equation and obtain the unique solution but I am stuck with "continuous dependence" on the initial data. So I did the following. Choose $L^2$ space and use energy methods for an equation of the form $w_t-a(x)w_x=0, \; y \in [0,1]$: $$ \frac{d||w||^2}{dt}=(w_t,w)+(w,w_t)=(a(y)w_x,w)+(w,a(x)w_x) $$ $$ =(a(x)w_x,w)-(w_x,a(x)w)-(w,a_x(x)w)+a(x)ww|^1_0 $$ $$ \leq max_{x\in [0,1]}|a_x(x)|||w||^2+a(x)ww|^1_0 $$ Thus, the energy estimate boils down to the bound of $a_x$ provided boundary conditions are bounded. In my case $a(x)=-\frac{x}{T-t}$ and $a_x=\frac{1}{T-t}$. Thus, I can't show that inequality for any $t$ only for $t\in [0,T-\epsilon]$. However, I do know the solution at $t=T$ being $u(x,T)=u_0(0,0)$ and it is constant so the $L^2$ norm is perfectly bounded. Thus, please suggest me some ways to make a complete statement about this problem, is that well posed in $L^2$?

If I take $L^{\infty}$ instead to show such a dependence on initial data it directly follows from method of characteristics and clearly it is well posed in $L^{\infty}([0,T])$ but I am not sure I can have some "limiting statement" in $L^2$ as so far I have only on $[0,T-\epsilon]$

  • $\begingroup$ You get to define what exactly you mean by "continuous dependence on initial data". If a unique solution really does exist on $[0,T]$ for, say, arbitrary smooth initial data, then it is highly likely that there are norms on the space of initial data and the space of solutions with respect to which the solution is a continuous function of the initial data. You just have to figure out what norms work. $\endgroup$ – Deane Yang Jan 13 '13 at 22:08
  • $\begingroup$ "continuous dependence on initial data" means to have an estimate $||w(t,x)||^2\leq C ||w(0,x)||^2$ in some norm. I have to choose a norm and I happen to like $L^2$ spaces(because all these integration by parts tricks work), so I would like to have such an estimate there. But at the moment, I can't show it. But, I have a suspicion it is true even in $L^2$ since I know all about solution and the equation just "flattens" any initial data I provided, so it should not be a problem, but energy estimates show there is a big problem at $t=T$ regardless how smooth or integrable the initial data is. $\endgroup$ – Kamil Jan 13 '13 at 22:26
  • $\begingroup$ Although I suspect that some kind of $L^2$ energy norm should work, you might have to add a weighting factor depending on $t$ or $T-t$. In general, you have to let the PDE tell you what the right norm is. $\endgroup$ – Deane Yang Jan 14 '13 at 3:15
  • $\begingroup$ I am staring at this pde for a while but I still can't make it clear. It is seems to be straightforward to show that $||w(t,x)||_2\leq ||w(0,x)||_{\inty}$, just by observing the underlying characteristics. However, does it imply that it is $L^2$ stable? If so, how to explain the fact that energy methods tell me the opposite? $\endgroup$ – Kamil Jan 15 '13 at 2:38
  • $\begingroup$ a transport equation that flattens initial data? really? $\endgroup$ – Delio Mugnolo Jan 17 '13 at 8:28

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