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This may be too general a question so please let me know if I need to make it more specific.

I am a first year graduate student in PDEs, and as such have not had much exposure to non-linear PDEs. I am starting to look at research papers, in many of which the PDE at hand is non-linear, and the 'linearization' of such a problem has been considered without much justification as to why this is useful. For instance, on page 4 here, after the authors have established that we wish to show the existence of a solution to a fully non-linear PDE, they suddenly shift to talking about the linearized problem and I have little idea of

1) How this linearized problem is derived, and

2) How the existence of a solution to the linearized problem relates to the existence of a solution to the original problem.

From what I gather in the short document here, we can only linearize 'about a known solution', but what if we have no known solution? Even if we do have a known solution, question 2 above still stands.

I guess more specifically I am asking for either a reference on the linearization of non-linear PDEs (i.e. how it is done, why it is useful), or if possible an answer explaining this concept, either in relation to the paper I linked or more generally. I have done two courses in elliptic PDE theory (covering chapters 5 and 6 in Evans in the first course, and going quite far beyond that in the second course) and have still never encountered linearization - hopefully that gives some indication of my background. Thank you.

jl2
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1 Answers1

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Basically, near an equilibrium point, the solution to a non linear PDE is qualitatively the same as its linearization. This is shown formally in the Hartman Grobman theorem. So if we are interested in the qualitative behavior of the non linear PDE it is useful to first look at the linearized version.

https://en.wikipedia.org/wiki/Hartman%E2%80%93Grobman_theorem

Here is an old question which has an example of linearizing a simple PDE to give you an idea of what you do. The idea is you first need a solution which you then 'perturb' and keep the linear part.

What is the 'linearization' of a PDE?

Jürgen Sukumaran
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    That old question happens to be one of mine! I understood the explanation by example but still didn't find it to be overly illuminating as to why this was all being done, or how we can linearize 'about a solution' when we don't even know a solution exists (of course this existence is the problem in the first place!). I will look at the Hartman-Grobman theorem too. – jl2 Dec 23 '17 at 22:43
  • Oh wow I didn't realize it was your question, not a very useful reference then. – Jürgen Sukumaran Dec 23 '17 at 22:47
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    Linearization is done to gain insight into a nonlinear PDE/ODE which is in general difficult to get in closed form. This is why it is done. As mentioned in the answer Grobman theorem justifies the linearization of a nonlinear problem near a fixed point (I believe only true when the eigenvalues are not 0).

    If you can't get a solution with which to linearize about, then there is not much you can do in the way of analytics.

    *EDIT:* This is assuming you can't do asymptotic arguments to neglect certain terms and reduce the system to something more tractable.

     – Gregory Dec 23 '17 at 23:21
  • @Gregory I don't suppose you have any good literature on asymptotic methods for PDEs? – Matthew Cassell Dec 24 '17 at 00:57
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    @mattos bender and orszag is pretty good. Good reference for intro to perturbation methods. It’s what was given to me in grad school. – Gregory Dec 24 '17 at 21:39