RE: Research diaries #2: Singularities and regularizations
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This time, I won't comment on the cat (too much data in it).
2 . or you can regularize the governing equations themselves and then apply standard theory to show the existence of the desired solution.
Does this include changing the governing equations by possibly very different ones? For instance, we may have a primary system of equations that apply in a given domain or under certain circumstances. During the course of the evolution, we may however leave the domain of definition of the primary system, so that the whole set of equations need to be replaced by something else.
Of course, I have physics in mind. For instance, Newtonian mechanics needs to be upgraded to special relativity for fastly moving systems. In this case however, we can recover the former by taking the right limit of the latter.
In a sense this is what I do but then in a local sense. At the singularity the equations are not defined so I construct regularized equations which are actually like a new set of equation which can be used in a neighbourhood of the singularity.
So I have something like a state space manifold where I have a special chart in the neighbourhood of the singularity.
Oh I see. So it is really like the analytical continuation of the sinc function that you took as an example. Thanks for answering!
An example of these class of problems would be:
under what conditions on the parameter a do there exist solutions to
dx/dt= ax/t + bt such that \lim_{t \to 0} x(t)/t^2 exists (where b is a free parameter, which here acts like a perturbation term)
with some straightforward calculus you can show that if a \neq 2 the desired solution exists.
The b-term acts like a perturbation term to the ax/t part so the more general version of the above you would replace the b-term by some very general perturbation.
Ok I (think I) see.
However, in your example, if b=0 then the
a \neq 2condition is not necessary, is it? In this maybe pathological case, the solution is trivially proportional to ta. In this sense, the b-term has in my taste deeper implications than just being a perturbation.Am I missing something trivial or am I putting too much emphasis on an irrelevant configuration?
For b=0 you have a zero solution. That's correct. Casting it in the more general framework you want to find something that holds for all b. So b defines her a space of functions.
More general you would look at
dx/dt= ax/t + g(x,t)
with g in a specific space. So you would want to find results for all g in that specific space.
These problems are motivated by classical problems of the type:
dx/dt = Ax + f(x)
with f(x) in O(x^2)
As always, I focus too much on the pathological cases. Thanks for the clarifications ;)