In Brezis's Functional Analysis book through chapters 3-4, I've seen the $\sigma(L^\infty,L^1)$ topology on $L^\infty$ but did not see (so far) any application of it in differential equations. Is there an example of a differential equation where this topology might be helpful in analyzing the $L^\infty$ norm of the solution for example? Or maybe some other thing I don't know about?



Usefulness of the $\sigma(L^\infty,L^1)$ topology in the context of differential equations
This topology is ubiquitous in modern analysis of PDEs, typically in conjunction with the [Banach-Alaoglu theorem](https://en.wikipedia.org/wiki/Banach%E2%80%93Alaoglu_theorem). (Bounded sets in $L^\infty$ are relatively compact for the predual $\sigma(L^\infty,L^1)$ / weak-$\ast$ topology.)


Here is one (not so) specific scenario: When trying to prove existence of solutions to non-linear PDEs, one possible and classical approach is to approximate the equation by a sequence of better-behaved ones, for which one can construct a solution $u\_n$. If the approximation is well constructed (typically the "structural" properties that one expects formally from the original equation should be reflected in the approximation procedure) then the approximate solutions usually satisfy some "nice" estimates. For example if $\{u\_n\}$ satisfies Lipschitz bounds uniformly in $n$ then one can conclude that the derivatives $ \nabla u\_n\overset{\ast}{\rightharpoonup} \nabla u$ for the weak-$\ast$ ($\sigma(L^\infty,L^1)$) topology. The Arzelà-Ascoli theorem moreover gives strong uniform convergence $u\_n\to u$, which helps passing to the limit in non-linear terms.


For example, if the PDE is quasi-linear elliptic of the form $div(A(u) \nabla u)=f(u)$ then typically one approximates $A\_n,f\_\approx A,f$ (e.g. by truncation $A\_n(u):=A(\min(n,u))$). The above procedure allows to pass to the limit in the weak formulation
$$
\int A\_n(u\_n)\nabla u\_n\cdot \nabla\varphi \to \int A(u)\nabla u\cdot \nabla\varphi
$$
and
$$
\int f\_n(u\_n)\varphi\to \int f(u)\varphi
$$
for any fixed reasonably smooth test-function $\varphi$.


Another such example is the theory of viscosity solutions Due to Crandall-Lions.
Here the Hamilton-Jacobi PDE
$$
\partial\_t u+H(x,u,\nabla u)=0
$$
is usually approximated by a "viscous" Hamilton-Jacobi-Bellman equation
$$
\partial\_t u\_\epsilon+H(x,u\_\epsilon,\nabla u\_\epsilon)=\epsilon\Delta u\_\epsilon,
$$
and the solution is constructed as $u=\lim\limits\_{\epsilon\to 0}u\_\epsilon$.


