Uniqueness of the solution to a PDE by means of a regularization

Question

I have a technical question, I'm trying to solve a nonlinear parabolic PDE. For $\epsilon>0,$ I have considered a regularized PDE instead that I can solve. Which resulted in a sequence of solutions of the regularized PDE that we denote by $(u_\epsilon)_\epsilon$ depending on a parameter $\epsilon>0.$ I was able to derive uniform estimates of the sequence $(u_\epsilon)_\epsilon$ in convenable functional spaces. which allowed me by letting $\epsilon \rightarrow 0$ to prove that the solution to the original PDE exists. My question is the following.

Suppose further that I can prove that the solution $u_\epsilon$ of the regularized PDE is unique $\forall \epsilon>0$. Can I deduce the uniqueness of the solution of the original PDE by letting $\epsilon \rightarrow 0$?

Answer 1

I don't think that uniqueness of the regularized PDE implies uniqueness of the original PDE. It is easy to imagine that the original PDE has two solutions, and that the way you regularize gives a sequence converging to one of the solutions.

A trivial example to illustrate what I am thinking about : consider the equation $\epsilon x = 0$ on $\mathbb{R}$. It has a unique solution $x = 0$, which is also a solution of the "limit equation" $0 = 0$. But obviously the "limit equation" does not have a unique solution.

Answer 2

If we supposed that the original PDE has two solutions say $u_1$ $u_2$. If you're working in normed spaces and such that the solution to the regularized PDE and the original PDE blong to the same normed space. Then if we suppose that $u_1$ and $u_2$ are two solutions to the original PDE. By definition they are limits of the sequences $u_1^\epsilon$ and $u_2^\epsilon$ solutions to the regularized PDE, moreover $\forall \epsilon >0$ $$ 0 \leq \Vert u_1-u_2 \Vert=\Vert u_1 - u_1^\epsilon+u_1^\epsilon-u_2+u_2^\epsilon-u_2^\epsilon\Vert \leq \Vert u_1^\epsilon - u_2^\epsilon \Vert+\Vert u_1^\epsilon-u_1\Vert+\Vert u_2^\epsilon-u_2 \Vert $$ and $$\Vert u_1^\epsilon - u_2^\epsilon \Vert \rightarrow 0 \;\; \text{by uniqueness of the solution of the regularized pde}.$$ $$\Vert u_1^\epsilon - u_1 \Vert \rightarrow 0 \;\; \text{by definition of the solution of the original PDE (as a limit)}.$$ $$\Vert u_2^\epsilon - u_2 \Vert \rightarrow 0 \;\; \text{by definition of the solution of the original PDE (as a limit)}.$$ So the solution to the original PDE is unique "in a sense", the latter being as a limit of a sequence of solutions to the regularized PDE.