I am reading the article, and I am more or less halfway through it. I have some questions though on some parts I am not understanding, so I wanted to ask about these here. I apologize for listing the questions like this, but I thought it might be more useful to post this once in order to have everything together, rather than a series of separate questions.
So I am doing a list here trying to explain briefly and as clearly as I can what I don't get. I'll repeat this when needed, but the "crucial equation" I may refer to is the following
$$\partial_t u - b \cdot \nabla_xu + cu =0$$
1) Right at the introduction at page 512, he considers $\bar{\lambda}$ the push-forward through $X(t)$ of the Lebesgue measure, where $X(t): x \mapsto X(t,x)$ and $X$ is the v.f. given by $\dot{X}= b(X)$. Now, he says that $\bar{\lambda}$ satisfies $\partial_t \bar{\lambda} - div(b \bar{\lambda}) =0 $, but testing with a $\phi$ I get:
$$ \int(- \partial_t \phi + b \cdot \nabla \phi )d \bar{\lambda}= \int - \partial_t \phi (X) + b(X) \cdot \nabla \phi (X) d \lambda $$
and I don't know what to do, if the equation was $\partial_t \bar{\lambda} + div(b \bar{\lambda})$ I'd have a total derivative inside and I'd conclude by compactness of the support of $\phi$. Maybe a typo?
2. The crucial equation in the article is $\partial_t u - b \cdot \nabla_xu + cu =0$. In proposition 2.1 it says it follows by standard arguments that (formally) that $||u(t)||_\infty \leq ||u_0||_\infty + \int_0 ^t ||cu||_\infty$. Now, I can see this if I first estimate $||u(t)||_p$ and then pass to the limit in $p$, but not directly. Maybe I'm missing something trivial?
3. At page 515, around the end of the proof of Proposition 2.1, he considers weak-$\ast$ sequences of approximate solutions of the equation (obtained by mollification as usual) in $L^ \infty (0,T; L^p)$ when $p>1$. Fair enough, this is a dual space only if $p>1$. So he uses another approach for the case $p=1$. Namely, he wants to show that $u_\epsilon$ is relatively weakly compact in $L^\infty (0,T; L^1)$ and to do so he considers $u_n ^0$ smooth functions with compact support converging in $L^1$ to the initial condition $u^0$, and then considers $u_{\epsilon , n}$ the correpondent approximated solutions with initial condition $u^0 _n$. Then it says that one can see:
$$
||u_{n, \epsilon}||_{L^\infty (0,T; L^p (\mathbb{R}^n))} \leq C(n,p) \text{ for all p>1, and } ||u_\epsilon - u_{\epsilon , n}||_{L^\infty(0,T; L^1(\mathbb{R}^n))} \leq C_0 ||u^0 - u^0 _N||_{L^1}
$$
and this gives the desired weak compactness. But why?
4. In Corollary 2.2, it's proved that under certain assumptions on $b,c$ the solution lies in $C([0,T];L^p)$. Now, at the beginning of page 520 he uses that
$$ \frac{d}{dt} \int _{\mathbb{R}^n} |u|^p + \int_{\mathbb{R}^n} (pc + div(b)) |u|^p =0 \text{ a.e. on }(0,T).$$
Then he concludes, I guess by exploiting the linearity in $|u|^p$ and Gronwall (but I wouldn't bet on it), that $||u(t)||_p \in C([0,T])$. Then using the equation $\partial_t u - b \cdot \nabla_xu + cu =0$, it says we can easily see that $u \in C([0,T];L^p)$. Now, in general continuity of the norm does not imply continuity of $u$ itself. So how do you get the latter? He also says that "the case $p=1$ is more delicate", so for some reason whatever argument works for $p>1$ doesn't work for $p=1$.
5. The crucial idea in the paper is the notion of "renormalized solution", basically he considers a class of $C^1$ functions with certain properties and proves that under some assumptions on $b,c$ we have a solution if and only if we have a renormalized solution, i.e. a solution for
$$ \partial_t \beta (u) - b \cdot \nabla _x \beta (u) + cu \beta ' (u)=0 \text{ in } (0,T) \times \mathbb{R}^n$$
for all such admissible $\beta$. To be precise, $\beta$ needs to be $C^1$, vanish near 0, and both $\beta$ and $\beta' (1 +|t|)^{-1}$ are bounded in $\mathbb{R}$. He then states Theorems 2.3 and 2.4 where issues like existence and uniqueness of solutions are treated. Now, before doing this there is the definition of the following subset of measurable functions (you can find this at the end of pag. 520).
$$ L^0 = \{ u \text{ measurable } | \mathcal{L} ^n (\{ |u|> \lambda \} < \infty \text{ for every } \lambda \}.$$
They say that:
$$ u^n \to u \text{ in } L^0 \text{ if } \beta(u^n) \to \beta (u) \text{ in } L^1 \text{ for every admissible } \beta$$
and
$$ u^n \text{ is bounded in } L^0 \text{ if } \beta (u^n) \text{ is bounded in } L^1 \text{ for every admissible } \beta.$$
Then they say we're authorized to talk about functions in $C([0,T]; L^0)$ and $L^\infty([0,T] ; L^0)$, and I am completely lost. I mean, to talk about continuous functions you need, at least, a topology. In general a notion of convergence should "topologize" a space only if this space is metrizable, so is this the case? The definition of $L^0$ reminds me of that of weak $L^p$ spaces, but they are not quite the same thing. Also for the definition of $L^\infty (0,T; L^0)$, what is it? I'd think of it as the set of functions $u$ with $ \sup_{t \in [0,T]} ||u||_{L^0} < \infty$ , but what is such norm?