Getting existence for $L^1$ data given existence for $L^\infty$ data and $L^1$ continuous dependence result

Question

Let $F:\mathbb{R} \to \mathbb{R}$ be locally Lipschitz, monotone and continuous. For the sake of concreteness only let us suppose it is of porous medium type (eg. $F(r) = r^{\frac 1m}$.)

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $Q=(0,T)\times\Omega$.

Given $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty(Q)$, I have $F(u) \in L^\infty(Q) \cap L^2(0,T;H^{-1})$ with $u \in L^\infty(Q) \cap L^2(0,T;H^1)$ such that $$\int_0^T \langle (F(u))_t, \varphi \rangle + \int_0^T\int \nabla u \cdot \nabla \varphi = \int_0^T \int f\varphi$$ for all test functions $\varphi$. I also have the continuous dependence result for two solutions corresponding two two data: $$\lVert F(u_1) - F(u_2) \rVert_{L^1(0,T;L^1)} \leq C\left(\lVert u_{01} - u_{02}\rVert_{L^1} + \lVert f_1 - f_2 \rVert_{L^1(0,T;L^1)}\right).$$ Now I wish to extend my existence result to $L^1$ data satisfying a weaker formulation $$-\int_0^T \int F(u) \varphi_t - \int_0^T\int u \Delta \varphi = \int_0^T \int f\varphi$$ for smooth $\varphi$. Are there any standard tricks to do this using this continuous dependence result?

Of course we can approximate the $L^1$ data by $L^\infty$ data and using the above estimate (we obtain a Cauchy sequence and so) we find $F(u_n) \to F$ in $L^1(L^1)$ for some $F$ where $F(u_n)$ is the solution with data that approximates the $L^1$ data. From this I can obtain $u_n \to u$ for some $u$ pointwise a.e., but this is not enough pass to the limit.

Edit: I am aware of the book on PME by J Vazquez. If I recall correctly he handles $L^1$ data rather differently and I would like to know whether the above approach can work.

Answer 1

Hy

First I have two remarks. The dependence continuous should be modified : it depends on $F(u_{01})-F(u_{02})$ and not on $u_{01}-u_{02}$. As far as the "weaker formulation" you do not precise the space where $\varphi$ is lying and there is no initial data. A formulation with $\varphi\in C_c([0,T)\times \Omega)$ can include the initial data term.

For your question, you need to have some compactness result in $L^1_{loc}$ (at least) for $u_\varepsilon$. Your approach can work

1) if $|F(r)|>C|r|$ for large value of $r$. But it is too restrictive

2) if $F(r)$ verifies a growth assumption $|F(r)|>C |r|^{\alpha}$, using the Boccardo-Gallouët estimates technique you can obtain a lower bound of $\alpha$ depending upon the dimension which insures and $L^1$ bound. But it is too restrictive.

The two previous methods do not use strongly the linear character of the operator :

3) use the regularity properties of $-\Delta$ operator and integrate with respect to time, like in the paper you cite in your question A question about PDE argument involving monotone convergence theorem and Sobolev space