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Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required. Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b < \infty\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times \Omega).$$

We know that there exists a function $u \in L^2(0,T;H^1)\cap H^1(0,T;L^2(\Omega))$ satisfying $u(0) = u_0$ given $u_0 \in L^2(\Omega)$ such that $$\int_0^T \int_\Omega u'(t) \varphi(t) + \int_0^T \int_\Omega g(t)\nabla u(t) \nabla \varphi(t) = \int_0^T\int_\Omega f(t)\varphi(t)$$ for all $\varphi \in L^2(0,T;H^1(\Omega))$.

My question is, if we know that $u_0 \in L^\infty(\Omega)$, does it follow that $u' \in L^\infty((0,T) \times \Omega)$? I.e. do we have the expected regularity of $u$? We expect this because $f$ is appropriately regular and as is $u_0$ so we expect the same for $u'$.

If so can anyone refer me to a citation for this result? Thank you. If not, what additional smoothness do I need? I posted this on MSE too but did not receive any answers.

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  • $\begingroup$ What can you say in the special case $n=1$, $\Omega=[-1,1]$, and $g$ is a $t$-independent step-function in $x$: $g(t,x)= a$ if $x<0$, and $g(t,x)=b$ if $x\geq 0$? In this case the equation looks like a parabolic problem with distributional coefficients. $\endgroup$
    – Liviu Nicolaescu
    Mar 10, 2014 at 9:45
  • $\begingroup$ Same question with a more complicated $g$: $g(t,x)=a$ for $x\leq t$, $=b$ for $x>t$. In any case, I find your question intriguing. $\endgroup$
    – Liviu Nicolaescu
    Mar 10, 2014 at 9:55
  • $\begingroup$ Good questions. I'm trying to find literature that address bounded coefficients.. $\endgroup$
    – student
    Mar 10, 2014 at 10:15
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    $\begingroup$ Note that in the first of above examples the coefficient of $\partial_x$ is a distribution, $(b-a)\delta_0(x)$, where $\delta_0(x)$ denotes Dirac's delta at $0$ in the $x$-direction. $\endgroup$
    – Liviu Nicolaescu
    Mar 10, 2014 at 11:18
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    $\begingroup$ @student in elliptic case there is a difference for regularity between $n=2$ and higher dimensions. In particular solutions of equations in divergence form with measurable coefficients belong to $C^{1+\alpha_0}$ locally. In higher dimensional case to $C^{\alpha_0}$ for some $\alpha_0\in (0,1)$. And two dimensions in elliptic case corresponds to one space variable for parabolic equations. So the answer may differ for $n=1$ and $n\ge2$. $\endgroup$
    – Andrew
    Mar 10, 2014 at 18:05

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Your variational formulation is for some parabolic equation in divergence form. The kind of regularity your are looking for can be deduced by time differentiating the equation and then applying a Moser-Alikakos-like iteration scheme for $L^p$-powers of the time derivative. Such arguments don't usually carry a reference since every parabolic equation is different and so it's yours. Plus you want an estimate that shows regularity for times $t>0$. On the other hand, there may be other ways to get what you want but it all depends for what purpose you wish to use this regularity result. In any event, I have performed many of such estimates for systems that were somewhat similar to yours, and the details are usually not straight-forward. This is again to emphasize the fact that you don't a priori know what assumptions these procedures will require although they will be close to what your are asking.

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