2
$\begingroup$

Consider the problem of finding $u \in L^2(0,T;H^1)$ with $u' \in L^2(0,T;L^2)$ such that $$\int_0^T \int_{\Omega}u'(t)\varphi(t) + \int_0^T \int_{\Omega}\nabla (F(u(t)))\nabla \varphi(t) = \int_0^T \int_\Omega f(t)\varphi(t)$$ where $F$ is differentiable and $F'$ bounded above and below away from $0$, and $F(0)=0$.

I want to solve this problem without using a time-discretisation or semigroup approach.

Now the only issue I have is with showing that $u' \in L^2(0,T;L^2)$. I can get $u' \in L^2(0,T;H^{-1})$ (so the first term of the equation is a duality pairing for this case) by using a fixed point approach.

Does anyone have references to where this is shown? Again I want to avoid time discretisation or semigroup proofs. A fixed point proof would nice for example. Assume whatever smoothness of right hand side $f$ as needed (eg. $L^2(0,T;L^2)$)

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

Given your setting, you cannot hope to get the strong regularity $\partial_t u\in L^2(0,T;L^2)$ for free. Indeed you're solving the generalized Porous Media Equation $$ \partial_tu=\Delta F(u) $$ with zero Dirichlet boundary conditions. The fundamental reason is that $F(u)\in L^2(0,T;H^1_0)$ is really the energy space for this type of quasilinear equations, so the corresponding "natural" space for $\partial_tu$ is the dual $(L^2(0,T;H^1_0))'=L^2(0,T;H^{-1})$. In other words: since your weak formulation involves $\int_0^T\int_\Omega\left<\nabla F(u),\nabla\varphi\right>$ for test functions in $L^2(0,T;H^1_0)$, your time derivative can only be in the dual. Unless of course you can integrate by parts $\int_0^T\int_\Omega\left<\nabla F(u),\nabla\varphi\right>=-\int_0^T\int_\Omega (\Delta F(u))\varphi$ in order to act on $\varphi$ via $L^2(\Omega)$ product only, but in order to do so you have to prove that $\Delta F(u)\in L^2(0,T;L^2)$. This is highly non-trivial and also not true in general. For that you need further assumptions on your initial datum and nonlinearity.

Since you're considering here the non-degenerate case $F'(u)>cst>0$ you can basically compare with the heat equation, i-e the best case scenario $F(u)=u$. In general for the HE you only get $\partial_t u\in L^2(0,T;H^{-1})$ as a global regularity. Again for the HE, the global regularity strongly depends on your initial data: for example $\partial_tu\in L^2(0,T;L^2)$ essentially requires $u_0\in H^2$. But then you cannot simply use the energy methods or natural $H^1$ fixed point, since $H^2$ is not the natural space. The general strategy both for linear and non-linear equations is: 1) obtain existence of weak solutions in the natural energy space, and 2) prove that your energy solutions actually enjoy more regularity. Step 1 is usually the easy one (this is your fixed point), but step 2 is essentially a parabolic regularity result hence much more involved. There's no such thing as a free meal...

You should look at Vázquez's book [the porous medium equation] chapter 3 (in particular what he calls "generalized porous medium with "good" $\Phi$", which would be your $F$ here), or the classical reference [Ladyzenskaya, Solonnikov, Ural'ceva - Linear and quasilinear equations of parabolic type] (hard to read).

Improve this answer
$\endgroup$
8
  • $\begingroup$ 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.. $\endgroup$
    – markus
    May 30, 2014 at 9:37
  • $\begingroup$ 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 $F'(u)\geq cst>0$). $\endgroup$
    – leo monsaingeon
    May 30, 2014 at 13:07
  • $\begingroup$ 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. $\endgroup$
    – markus
    Jun 13, 2014 at 16:38
  • $\begingroup$ 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 $\partial_t u\in L^2H^{-1}$). $\endgroup$
    – leo monsaingeon
    Jun 13, 2014 at 17:22
  • $\begingroup$ @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. $\endgroup$
    – TheBook
    Jun 13, 2014 at 19:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.