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Suppose I have a PDE, for example the Fokker-Planck one, in which I am mostly interested: $$ \frac{\partial}{\partial t}u(x,t)=-\frac{\partial}{\partial x}(\mu(x,t)u(x,t))+\frac{1}{2}\frac{\partial^2}{\partial x^2}(D(x,t)u(x,t)). $$ 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 better).

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    $\begingroup$ It depends very much on the type of PDE. The keyword is "stability". For linear evolution equations, look into "semigroup theory" and consult books such as Pazy, Goldstein, or Engel-Nagel (roughly, in order of comprehensiveness). For linear elliptic equations, the standard reference is Gilbarg-Trudinger's book. For nonlinear elliptic equations, I know of this recent book: amazon.fr/Compactness-Stability-Nonlinear-Elliptic-Equations/dp/… $\endgroup$ Commented May 19, 2016 at 9:03
  • $\begingroup$ @GiuseppeNegro: Helpful comment. +1 What if $D(x,t)$ is a constant and I would like to know the continuity of $u(x,t)$ with respect to $D$? Is there a straightforward answer? $\endgroup$
    – Hans
    Commented May 10, 2018 at 8:04
  • $\begingroup$ @Hans: see below. $\endgroup$ Commented May 10, 2018 at 10:17

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WORK IN PROGRESS (This is a community wiki to record progress in this problem).

Easy case. If $\mu$ and $D$ are constants, there is an explicit representation of the solution: $$u(t, x)=\sqrt{\frac{1}{2\pi Dt}}\int_{-\infty}^\infty \exp\left(\frac{(x-y-\mu t)^2}{Dt}\right)\, u_0(y)\, dy.$$ From this formula one can easily establish all the continuity properties one wants.

Constant $D$ case. We assume that $D$ is a constant and we ask whether $$\lim_{D\downarrow 0} u(t, x) = w(t, x),$$ where $\partial_t w = -\partial_x (\mu(t, x)w(t, x))+\frac{D}2\partial^2_x w,$ and $w(0, x)=u(0, x)$. (The convergence is intended in an appropriate sense to be specified).

If $\mu=\mu(x)$ the answer is affirmative: see Hans comment below for a transformation that reduces the problem to the constant $\mu$ case. The abstract framework of semigroup theory also applies; see Pazy, Semigroups of linear operators and applications to PDEs, section 3.4: "The Trotter approximation theorem".

If $\mu=\mu(t)$ the answer is affirmative in the following sense. Taking the spatial Fourier transform termwise, the equation reduces to $\partial_t \hat{u} = \mu(t)(-i\xi \hat{u}(t, \xi))-\frac{D}2\xi^2\hat{u}(t, \xi).$ This is a family of ODEs indexed by $\xi$, and by Gronwall's lemma arguments, as $D\to 0$ its solutions converge uniformly for $t$ in compact intervals. (There is some work to be done here, to show that there is uniformity in $\xi$. This seems true, though).

If $\mu=\mu(t, x)$ is analytic in time, the problem can be reduced to the study of the sequence of PDEs $$\partial_t u_n=-t^n \partial_x(\mu_n(x)u_n(t, x))+\frac{D}2\partial_x^2 u_n.$$

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  • $\begingroup$ Thank you for the answer. You misunderstood me, though. I am well aware of the analytic solution of the standard heat equation and thus its analyticity with respect to its constant coefficient. I specifically NOT specializing $\mu(x,t)$ to constant. I am leaving $\mu(x,t)$ as a general function as it is, maybe specializing to be an analytic function if that gives any nice result. As such, I am particularly interested in the continuity of $u(x,t)$ with respect to $D$ at $D=0$. $\endgroup$
    – Hans
    Commented May 10, 2018 at 15:06
  • $\begingroup$ Well, this formula already shows a non-trivial phenomenon: $$\lim_{D\to 0} u(t, x) = u_0(x-\mu t), $$ where in the right-hand side you see the solution to the transport equation $\partial_t u + \mu\partial_x u=0$. It should be proven that this remains true if $\mu=\mu(t, x)$. This shouldn't be hard. $\endgroup$ Commented May 10, 2018 at 15:23
  • $\begingroup$ Are you still talking about a constant $\mu$ which is NOT I want? By "this formula" are you referring to the standard constant coefficient Gaussian integral you wrote in your answer? $\endgroup$
    – Hans
    Commented May 10, 2018 at 16:29
  • $\begingroup$ Let me rephrase. If $\mu$ is constant, we see from the Gaussian integral that $\lim_{D\to 0} u(t, x)$ is the solution to the equation $\partial_t u + \mu \partial_x u=0$. I conjecture that, even if $\mu=\mu(t, x)$, this is still true. (With some assumptions on $\mu$, of course). $\endgroup$ Commented May 10, 2018 at 16:43
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    $\begingroup$ For $\mu(x,t)=\mu(x)$ i.e. depending only on x, we can make a transformation $x=t+g(y)$ for some function $g$ so as to turn the PDE into the standard heat constant coefficient heat equation $\frac{\partial v(t,y)}{\partial t}=\frac12D\frac{\partial^2 v(t,y)}{\partial y^2}$. The hard problem is when $\mu(t,x)$ is a nontrivial function of both $t$ and $x$. $\endgroup$
    – Hans
    Commented May 10, 2018 at 20:15

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