I am trying to understand a method that is used to show well-posedness of nonlinear systems near equilibria.
What I often see is that a system of nonlinear PDEs (usually from physics) is first nondimensionalized, then written as a operator equation. For example , let $$ F(f_1(t,x),\ldots,f_n(t,x))= \begin{bmatrix} \text{PDE in }(f_1(t,x),\ldots,f_n(t,x)) \\ \text{PDE in }(f_1(t,x),\ldots,f_n(t,x)) \\ \vdots \\ \text{another PDE in }(f_1(t,x),\ldots,f_n(t,x))\end{bmatrix}=0\tag{*} $$ be such an operator equality with some initial data: $$ (f_1(t,x),\ldots,f_n(t,x))\vert_{t=0}=(f_{1,0}(x),\ldots,f_{n,0}(x)). $$
Then, apparently, one needs to find equilibrium solutions to $F$. Therefore, we let $$(\bar{f}_1(x), \bar{f}_2(x),\ldots,\bar{f}_n(x))$$ be an equilibrium solution. Then, $$ F(\bar{f}_1(x), \bar{f}_2(x),\ldots,\bar{f}_n(x))=0. $$
In the next step, the following happens (I think it's called perturbation?):
$$ \phi_1={f}_1(x)-\bar{f}_1(x)\\ \phi_2={f}_2(x)-\bar{f}_2(x)\\ \vdots\\ \phi_n={f}_n(x)-\bar{f}_n(x).\\ $$ Apparently, we have now reduced the original problem to finding $(\phi_1, \phi_2,\ldots,\phi_n)$.
Then, we use Taylor series to write $F$ as $$ F(\phi_1, \phi_2,\ldots,\phi_n)=F(\bar{f}_1(x), \bar{f}_2(x),\ldots,\bar{f}_n(x))+F'(\bar{f}_1(x), \bar{f}_2(x),\ldots,\bar{f}_n(x))[(\phi_1, \phi_2,\ldots,\phi_n)]+R, $$ where $R$ is the remainder term in an integral form. This is equal to $$ F(\phi_1, \phi_2,\ldots,\phi_n)=F'(\bar{f}_1(x), \bar{f}_2(x),\ldots,\bar{f}_n(x))[(\phi_1, \phi_2,\ldots,\phi_n)]+R, $$ So, finding $(\phi_1, \phi_2,\ldots,\phi_n)$ such that (*) holds is equivalent to solving $$ F'(\bar{f}_1(x), \bar{f}_2(x),\ldots,\bar{f}_n(x))[(\phi_1, \phi_2,\ldots,\phi_n)]=-R. \tag{1} $$ It seems that, if one can show optimal regularity of (1), then there exists a solution operator $\Phi$ that solves (1).
Now, the interesting part is that we, for some reason, can use this solution operator to show well-posedness using the Banach fixed-point theorem in a $\epsilon$-ball close to the equilibrium. And this seems to imply that the original nonlinear system (from physics) has a unique solution near equilibria.
My questions:
1: If one can show optimal regularity for (1), why can't we then use this solution operator to conclude well-posedness of the original system? Why is the contraction (Banach contraction principle) step necessary? In other words, why is $\Phi$ not a unique solution to the original nonlinear equation?
2: Why do we need to stay close to the equilibrium?