Continuity of PDE solutions with respect to coefficients

Question

(Cross-posted from Math.SE)

Suppose I have the Fokker-Planck PDE: $$ \frac{\partial}{\partial t}u(x,t)=-\frac{\partial}{\partial x}(\mu(x,t)u(x,t))+\frac{1}{2}\frac{\partial^2}{\partial x^2}(D(x,t)u(x,t)). $$ Denote the solution $u$ as $u^\mu$ to strength the fact that, in particular, it depends on $\mu$.

Where can I find results that guarantee the continuity of the PDE solution with respect to the coefficients?

That is, results guaranteeing that $$ \mu_n(x,t)\to \mu(x,t) \implies u^{\mu_n}(x,t)\to u^{\mu}(x,t), $$ under some conditions and appropriate definitions for the above convergences. The simpler the result, the better.

Answer 1

Maybe the use of the implicit function theorem could be helpful for this, as it could be used in Banach space and works for some PDE.

See for example section II.3.3 of Hamilton for an application to elliptic PDE, Richard S. Hamilton, MR 656198 The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65--222.

In your case, I also believe that you have much more than just continuity, provided of course the right assumptions and $D$ and $\mu$ to guarantee the well-posedness of the PDE.

Answer 2

Let me make a suggestion. I will assume $u = 0$ on $\partial \Omega$, since everything can be generalised. The existence of this kind of problem is tricky, and even more so the continuity.

I will suggest you use the weak formulation, you have that for every $\eta \in \mathcal D ([0,T]), \varphi \in H_0^1 (\Omega)$: $$ \int_0^T \int_\Omega u \eta' \varphi + \int_0^T \int_{\Omega} \eta D \nabla u \nabla \varphi = \int_0^T \int_{\Omega} \mu u \nabla \varphi + \int_\Omega u(0) \eta(0) \varphi$$ You should be able to show existence of $u^{\mu_n}$ for every $n$ (maybe going to the mild formulation). Then you want to uniform boundary $u^{\mu_n}$ in some $W^{1,p}$. Then try to pass to the limit in that equation. I would suggest you try $\mu_n \to \mu$ in some $L^p$, since this is most natural setting.