Timeline for Existence of solution with $u' \in L^2(0,T;L^2)$ to a nonlinear parabolic PDE
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 13, 2014 at 20:53 | comment | added | leo monsaingeon | @ TheBook: ok, ok. But if you look closely at Evans' proof (which I just did) the "energy" method works to improve the regularity because the PDE is linear. Mainly to take $\partial_t u$ as a test function and derive $<\nabla u,\nabla \partial_t u>=d(|\nabla u|^2_{L^2})/dt$. In the nonlinear case this becomes $<\nabla F(u),\nabla\partial_tu>$, which cannot be written as $d/dt(something)$ without further assumptions on $F$. So in general the energy method does not give improved regularity for NL PDE's (at least not the regularity the OP is looking for, Hölder/DeGiorgi is something else). | |
| Jun 13, 2014 at 20:38 | comment | added | TheBook | @leomonsaingeon Well from the OP "assume whatever smoothness of right hand side as needed" and the first OP comment says "initial data can be as smooth as needed." So I read the intention as "how do I get $u_t \in L^2(L^2)$ using just energy/variational methods". | |
| Jun 13, 2014 at 19:45 | comment | added | leo monsaingeon | @TheBook: well, of course you can improve the regularity a posteriori as always. But if you read the above thread we were discussing the "optimal" regularity, i-e without any assumption on the initial data $u_0$ or inhomogeneity (except of course $u_0\in L^2(\Omega)$ and $f\in L^2H^{-1}$, which is the least one should ask). Well, at least I was talking about that, maybe I wasn't clear enough. | |
| Jun 13, 2014 at 19:12 | comment | added | TheBook | @leomonsaingeon Check out the next section 7.1.3. He gets $u' \in L^2(L^2)$ (of course we need $u_0 \in H^1$ and $f \in L^2(L^2)$) using Galerkin method. | |
| Jun 13, 2014 at 17:22 | comment | added | leo monsaingeon | Hummmm... Unless I'm mistaken when you construct solutions to the heat equation by Faedo-Galerkin you don't retrieve $\partial_tu\in L^2L^2$, but really $L^2H^{-1}$. The reason why is that the energy estimates essentially control $\nabla u$ in $L^2L^2$ thus $u$ in $L^2H^1_0$, this is why the time derivative must be in the dual $L^2H^{-1}$ in the end and no better than that. This goes a while back for me, but you should check Evan's book section 7.1.2, he goes trhough Faedo-Galerkin and energy estimates in great details (I just checked, he really gets $\partial_t u\in L^2H^{-1}$). | |
| Jun 13, 2014 at 16:38 | comment | added | markus | Dear Leo, in the case of heat equation ($F(u) = u$), we get the $\partial_t u \in L^2(0,T;L^2)$ if eg. we show the existence using a Faedo-Galerkin method. Then we can test with (the finite-dimensional version of) $u_t$ and rewrite the resulting bilinear form as a derivative of something and obtain an a priori bound on $u_t$ in the nice space. This I would call a functional analytic method. Do you happen to know of functional analytic methods to show the regularity in time in a fashion like this? I guess you would have mentioned it in your answer but just thought I might ask. | |
| Jun 10, 2014 at 9:31 | vote | accept | markus | ||
| May 30, 2014 at 13:07 | comment | added | leo monsaingeon | Yes, that's exactly what I'm saying: there's no easy way around... (at least not that I know of). You can always work out the regularity a posteriori, though. You should definitely get $C^{\alpha,\alpha/2}$ regularity in space-time. Assuming that $F$ is smooth you can probably use a bootstrap atgument to get $\mathcal{C}^{\infty}$ smoothness for $t>0$ (since your equation is non-degenerate $F'(u)\geq cst>0$). | |
| May 30, 2014 at 9:37 | comment | added | markus | Thanks. So you're basically saying that one cannot show this result without going to classical theory. Of course the PDE I am consider is slightly better than the PME as you acknowledged. Initial data can be as smooth as needed. I have read though in a paper that this result can be proved by standard time-difference schemes, or by $m-$ accretive operator theory. As you say one can use Ladyzenskaya, however I agree it is ugly to read so I wanted to avoid it. I was hoping there was a nice trick in the energy method.. | |
| May 30, 2014 at 9:22 | history | answered | leo monsaingeon | CC BY-SA 3.0 |