Prove convergence of whole sequence $f_n$ of solutions to a differential problem to a limit $f$ (without uniqueness assumptions) Let $\{f_n\}_n \subset C^\infty \cap L^2(\mathbb R^N)$ be a sequence of functions that solves a linear differential equation $F_n(f_n, \nabla f_n) = 0$.  Suppose that there exists a subsequence $n_k$ such that $f_{n_k} \to f$ weakly in $L^2$ and that is the weak solution (not necessarily unique) of the limit problem $F(f,\nabla f) =0$.
If we had a uniqueness result for the limit problem, it would be trivial that the whole sequence $f_n \to f$. What strategy can one use to show this kind of result without relying on the uniqueness for the limit problem?
To fix ideas, consider
$$u^n_t + f(u^n)_x = \frac{1}{n}\Delta u^n$$
and
$$u_t + f(u)_x = 0.$$
How do you show that the whole sequence $u^{n}$ converges if we only have convergence up to subsequences and we don't want to exploit the uniqueness of the limit problem?
 A: I was tempted to post this as a comment since it is not more than a suggestion: however it is too long, thus here it is. The only paper I am aware of dealing with a problem of this kind is the following old paper [2] by Calogero Vinti. He deals with the following Cauchy problem for a single 1st order PDE in two variables:
$$ 
\begin{cases}
u_t (t,x)= f(u_x(t,x))\\
u(0,x)=u_0(x) 
\end{cases}.
\label{1}\tag{1}
$$
The initial data $u_0$ belongs to $C^1_b(\Bbb R)$ (the spaces of bounded $C^1(\Bbb R)$ functions) and satisfies the condition
$$
|u_0^\prime(x+h)-u_0^\prime(x-h)|\le |h| M_{u_0}(x)\quad\forall x, h\in\Bbb R,
$$
where $M_{u_0}(x)$ is a non-negative $L^1_\text{loc}(\Bbb R)$ function: let's call $\mathscr{M}$ the class of these functions.
He uses an approximation method for constructing solutions to \eqref{1} proposed by Emilio Bajada in [1] which works also for $f\in C_b^{0,1}(\Bbb R)$, the class of bounded Lipschitz functions where uniqueness of solution lacks, as shown by an example of Mauro Pagni (given in the same paper in an enhanced form): called $\mathscr{K}$ the class of $C^{0,1}([0,a]\times[0,b])$ ($a,b>0$) solutions to problem \eqref{1}, with initial data belonging to $\mathscr{M}$, he proves the following result:
Theorem ([2], "teorema" in §3, pp. 254-262) If $u(t,x)\in\mathscr{K}$ for a given $u(0,x)=u_0(x)\in\mathscr{M}$, however you choose an approximating sequence $\{u_0^n\}_{n\in\Bbb N}\subset\mathscr{M}$ satisfying the following supplementary conditions

*

*$|u_0^n-u_0|\underset{n\to\infty}{\longrightarrow} 0$ uniformly on the interval $\left[-\dfrac{a}{2}, b+ \dfrac{a}{2}\right]$ and

*$\displaystyle\int\limits_{-\frac{a}{2}}^{b+\frac{a}{2}} |{u_0^n}^\prime(x)-u_0^\prime(x)|\mathrm{d}x \underset{n\to\infty}{\longrightarrow} 0$
it is possible to find at least a sequence $\{u^n(t,x)\}_{n\in \Bbb N}\subset\mathscr{K}$ of solutions to \eqref{1}, $u^n(0,x)=u^n_0(x)$ for all $n\in \Bbb N$ such that

*

*$|u^n-u|\underset{n\to\infty}{\longrightarrow} 0$ uniformly on  $[0,a]\times[0,b]$ and


*$\displaystyle\int\limits_{0}^{b} |u^n_x(t,x)-u_x(t,x)|\mathrm{d}x \underset{n\to\infty}{\longrightarrow} 0$ uniformly on the interval $[0,a]$.
The proof goes on by checking that the sequence of solutions to \eqref{1} with the initial data $\{u_0^n\}_{n\in\Bbb N}$ constructed by using Bajada's method satisfies the statement of the theorem.
Appendix: the construction of solutions to the Cauchy problem \eqref{1}.
Here I recall the procedure for constructing the solutions introduced by Bajada in ([1], §2-4, pp. 5-10): obviously, for the proof, this last reference should be consulted.
Let's consider a strip $S= [0,a]\times\Bbb R$ and let $\{m_n\}_{n\in\Bbb N}$ be a monotone sequence of positive integers: define the positive real numbers
$$
\begin{split}
d_n &= \dfrac{a}{2^{m_n}}\\
a_{r_n}& = r_n d_n\quad\text{for all }r_n=0,1,2,3,\ldots, 2^{m_n}
\end{split}\quad n\in\Bbb N
$$
and use them in order to define the substrips
$$
S_{r_n}=[a_{r_n-1},a_{r_n}]\times \Bbb R\quad \text{for all }r_n=0,1,2,3,\ldots, 2^{m_n}
$$
Finally, define the sequence $\{v_{m_n}\}_{n\in\Bbb N}$ of functions $v_{m_n}:[0,a]\times \Bbb R\to\Bbb R$ as
$$
v_{m_n}(t,x)=\left\{
\begin{split}
\varphi_0(t,x)=&\dfrac{1}{2}u_0\left(x+\dfrac{t}{2}\right)\\
& +\dfrac{1}{2}u_0\left(x-\dfrac{t}{2}\right) \\
 & \quad + \displaystyle\int\limits_{x-\frac{t}{2}}^{x+\frac{t}{2}}\!\!\!f\big(u_0^\prime(y)\big)\mathrm{d}y &\quad (t,x)\in S_0\\
& \vdots \\
\\
\varphi_{r_n}(t,x)=&\dfrac{1}{2}\varphi_{r_n-1}\left(a_{r_n-1},x+\dfrac{t-a_{r_n}}{2}\right)\\
& +\dfrac{1}{2}\varphi_{r_n-1}\left(a_{r_n-1},x-\dfrac{t-a_{r_n}}{2}\right) \\
& \quad + \displaystyle\int\limits_{x-\frac{t-a_{r_n}}{2}}^{x+\frac{t+a_{r_n}}{2}}\!\!\!f\big(\partial_x\varphi_{r_n-1}(a_{r_n-1},y)\big)\mathrm{d}y & \quad(t,x)\in S_{r_n}\\
&\vdots
\end{split}\right.
$$
Bajada proves that $\{v_{m_n}\}_{n\in\Bbb N}$ is a sequence of bounded continuous functions, and using a compactness argument he is able to apply the classical Ascoli-Arzelà theorem and conclude that there is a subsequence $\{v_{m_n^s}\}_{n\in\Bbb N}\subseteq\{v_{m_n}\}_{n\in\Bbb N}$ such that $\lim_{n\to\infty} v_{m_n^s}(t,x)=u(t,x)$ is a solution to \eqref{1}.
Notes on the appendix

*

*I decided to explicitly add a description of Bajada's algoritm, since it seems to me that the brief sketch given by Vinti in ([2], §2, pp. 252-253) is flawed by many typos or at least written using an unfortunate notation: of course I hope that the "modernized" notation I used above does not hide the procedure in the same way.

*Bajada says that its method is "an application of the method of successive approximations": as a matter of fact, this is more similar to a multidimensional variant of Euler's method or more generally of Runge-Kutta methods.

Notes

*

*Applying to the question Vinti's result implies defining the approximating sequence as a sequence of solutions to particular Cauchy problems: while this is certainly possible in some (perhaps many) cases (including the example equation not involving the laplacian), I do not know if this holds true in general.

*The topology used by Vinti et al in order to define their concept of continuity is the basically the topology of uniform convergence applied to subspaces of the space of Lipschitz functions: is it possible to extend the result to more general/weak topologies? I do not know.

*Is it possible to extend the method of Bajada and Vinti in order to deal with higher dimension and/or higher order PDEs? My answer is unfortunately the same give to the preceding questions.

*Finally note that the paper [2] is not an easy read: apart from the fact that it is written in Italian, its notation is also not modern. in the above presentation I attempted to update the notation, but in the paper we could say it follows  Gaspard Monge's style: for example he uses $z$ as the independent variable and $p=z_x$, $q=z_y$.

References
[1] Emilio Baiada, "Considerazioni sull'esistenza della soluzione per un'equazione alle derivate parziali, con i dati iniziali nel campo reale (Considerations on the existence of a solution to a certain partial differential equation, with real valued initial data)" (in Italian), Annali di Matematica Pura e Applicata (IV), vol. XXXIV (1953), pp. 1-25, MR55541, Zbl 0051.07303.
[2] Calogero Vinti "Su una specie di dipendenza continua delle soluzioni dal dato iniziale, per l’equazione $p=f(q)$, in una classe ove manca l’unicità [On a kind of continuous dependence of solutions from the initial data, for the $p=f(q)$ equation, in a class where uniqueness lacks]" (in Italian), Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 3, Volume 19 (1965) no. 2, p. 251-263, MR185249, Zbl 0133.04602.
