Who remembers the Buddha?

The Buddha did not, in the canon (eg MN63), place any weight on knowledge with regards to whether the world is eternal or not, whether the Buddha 'exists' or 'does not exist', 'both' or 'neither' after parinibbana - there are 14 questions to which the Buddha provides no answers on purpose (his objective is only the cessation of stress/suffering).

I prostrate to the Perfect Buddha, the best of teachers,
Who taught that whatever is dependently arisen is
Unceasing, unborn,
Unannihilated, not permanent,
Not coming, not going,
Without distinction, without identity,
And free from conceptual construction. - Nagarjuna

In discussing dependent-origination as a whole, is as close to a description of Self as you will get. But local realism (to use the modern Physics term for Self, or any theory of things which have self-referential properties - properties independent of observation), is unlikely.. here's a recent experiment! local-realism

If you are interested in understanding a philosophical, logically consistent argument against the validity of Self or Essential existence, as formulated within the Buddhist sphere, Mulamadhyamakakarika from part 2, with the commentary, is very educational!

Plato's Parmenides is the exact mirror of the above linked text, as it argues for One or Self from within logic, so is a useful counterbalance. It is interesting Plato himself dismembers his argument so effectively that it is hard to see why he dos not abandon it at the end of his writing the text - it all ends up being so contradictory!!

So my question is who remembers the past existences of Buddha ?

You're assuming there is a "who".

People do that with "God", e.g. "Who made the world? That "who" must have been God! The world exists, therefore God exists!"

Buddhism teaches that [almost] everything is impermanent, e.g. it comes into being and ceases. This (list of impermanent phenomena) includes all contacts with sense-objects -- sights, sounds, and ideas.

Buddhism also teaches that you should avoid seeing these impermanent things as "self" because that view (view-of-self or identity-view) leads to suffering.

You state some ideas which aren't proven. You just postulate them as axioms. On such basis it's possible to prove anything you want.

For example:

White is a color that is not colored. White can be of different luminosity (brightness). When its luminosity is low enough, people call that color black. Therefore, what people call "black" is actually white.

So we can prove with reasoning that white is black, or that black is white. I did that in various ways in childhood. Likewise, we can prove anything else.

It all depends on how we choose meanings of terms, their borders, and which axioms we postulate.

In your model, yes, there is Self, and it must be immortal.

OK. Congratulations.

Now the question is, how practical that system is.

It is just like: should we really call white "black"?.. or call black "white"?..

Buddhism explains a system quite different from yours. It says that "existence" and "non-existence" are mental constructions, and both are somehow illusory.

For example, does the shape of a waterfall exist?

If we say it exists, then the next moment streams of water slightly changed, drops of water moved, and that previous shape doesn't exist now.

If we say that the shape of a waterfall doesn't exist, then what would we see on a photograph?

So in reality it does not absolutely exist, but it is not absolutely non-existent.

Likewise, everything else both "exists" and "doesn't exist" at the same time.

Buddhism explains that existence is illusory.

Memories are impermanent. So if "self" is based on memories, it is impermanent.

Buddhism explains such things very thoroughly, not for the desire to claim or to prove something, but in order to uproot the habits of attachment to illusory self. Those habits cause suffering. To cease creating suffering, we study "self", and we find there is no permanent tangible self.

If you can postulate some imaginary idea of constant self, OK, but why does it matter? There are ideas like Truth or Justice, which must be "constant", otherwise we couldn't refer to such ideas repeatedly (in the same way). But having constant imaginary ideas of Truth or Justice, it doesn't mean we actually experience constant perfect Truth or Justice.

Buddhism is about experience.

If there is some imaginary constant idea, but the self you experience is impermanent, then what's the sense?

Isn't it better to study Buddhism and discover the illusoriness of ideas of existence and non-existence, which enslaved us in dissatisfaction?

The purpose of Buddhist teachings is liberation from dissatisfaction. These teachings work, by guiding us to seeing the reality beyond mental constructions. In order to do that, it's useful to put our obsession with constructions aside, and practice meditation - to develop direct experience of the reality.

And we know that past Knowledge , past experience ,past actions are remembered by Self.

This is false because the mind only can remember via memory a small proportion of past actions.

Self must also be immortal because Buddha is immortal.

This is false because the ancient scriptures show the Buddha heavily rebuked & admonished deluded monks of wrong view that regarded the Buddha to be a self (e.g. SN 22.81; SN 22.87).

The Buddha is actually the Dhamma (truth of nature; truth of impermanence, unsatisfactoriness & not-self). Buddha is emptiness (sunnata). The Buddha is reported to have said:

Enough, Vakkali! What is there to see in this vile body? He who sees Dhamma, Vakkali, sees me; he who sees me sees Dhamma. Truly seeing Dhamma, one sees me; seeing me one sees Dhamma.

Vakkali Sutta