Getting an estimate of the form $\lVert u(t+h)-u(t) \rVert_{L^1(\Omega)} \leq \frac{Ch}{t}$ on solution of PDE

Question

Let $u$ be a weak solution (i.e. $u \in C([0,T];L^1(\Omega))$ of some degenerate or nondegenerate parabolic equation $u' - Au = f$ on a bounded domain. (For my purpose it is enough to have this for the porous medium equation, in which also $u^m \in L^2(0,T;H^1(\Omega))$.)

I am seeking a estimate of the form $$\lVert u(t+h)-u(t) \rVert_{L^1(\Omega)} \leq \frac{Ch}{t}$$ for $h >0$ small (want to send $h$ to zero later). Here $C$ is some constant not depending on $h$, $t$ or $u$. Note that the power of $h$ must be one, not a half.

This type of estimate is useful in order to show time regularity.

My question is, what techniques are available and can one use to in order to derive such an estimate, apart from the one I describe below?

I know of one way, which is detailed in the book on PME by Vazquez. This way comes out of rescaling the solution so that it satisfies the PME again with a rescaled initial data and then using the comparison principle after picking the scaling parameter appropriately.

Is there some other method? I'd prefer variational methods and not semigroup.

Why I want to do this? Because I work on a similar such equation where the rescaling I described above doesn't work so I am looking for different methods.

Answer 1

The answer is given by Ph. Bénilan & M. G. Crandall in "Regularizing effects of homogeneous evolution equations", Contributions to analysis and geometry (Baltimore, Md., 1980), pp. 23–39, Johns Hopkins Univ. Press, Baltimore, Md., 1981.

They prove this estimate for three PDEs, namely $$\partial_tu=\partial^2_x(|u|^{\alpha-1}u),\qquad\partial_tu=\partial_x(|u|^{\alpha-1}u),\qquad\partial_tu=\partial_x(|\partial_x u|^{\alpha-1}\partial_x u).$$ A crucial point is that the equation admits a scaling invariance group. I presume that it is the way it is presented in Vazquez' book.