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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. $L^2(0,T;L^2)$)

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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).

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  • $\begingroup$ 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
    Commented 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
    Commented 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
    Commented 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
    Commented 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
    Commented Jun 13, 2014 at 19:12

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