In the definition of a well-posed problem it states that a problem is well posed if:
1.A solution exists. 2.The solution is unique. 3.The solution's behaviour changes continuously with the initial conditions.
Could someone explain how to interpret condition 3.
Thanks for any assistance
$\endgroup$2 Answers
$\begingroup$Small deviations at the input should cause only small deviations at the output.
$\endgroup$ 2 $\begingroup$Suppose you are solving the heat equation on a thin metal rod (a 1D PDE). Imagine some initial temperature distribution $T_{1}(x,t=0)$ on the rod. That's your initial condition, and the heat equation will govern the time evolution of the temperature distribution on that rod: $T_{1}(x,t)$. Now imagine that, instead of that temperature distribution as your initial condition, you raise the initial temperature uniformly by some small amount $dT$: $T_{2}(x,t=0)=T_1(x,t=0)+dT$. The solution to the heat equation (its time evolution) should not change substantially (i.e. hit a singularity) as you slowly vary the inputs: $T_{2}(x,t) \sim T_{1}(x,t) + F(x,t,T) \;dT$, where $F(x,t,T)$ is continuously differentiable.
If instead you made a small change to the initial condition and the solution (evolution in time) suddenly blew up or changed in a significant way, then something went wrong: your problem was not well-posed to begin with.
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