## Bootstrap arguments

A bootstrap argument is an analogue of mathematical induction where the natural numbers are replaced by the non-negative real numbers. This type of argument is a powerful tool for proving long-time existence theorems for evolution equations. For instance, it plays a central role in the proof of the stability of Minkowski space by Christodoulou and Klainerman and the theorem on formation of trapped surfaces by Christodoulou discussed in previous posts. The name comes from a story where someone pulls himself up by his bootstraps, leather attachments to the back of certain boots. This story is often linked to the name of Baron Münchhausen. In another variant he pulls himself out of a bog by his pigtail. This was a person who really lived and was known for telling tall tales. In later years people wrote various books about him and incorporated many other tall tales from various sources. The word ‘booting’ applied to computers is derived from ‘bootstrapping’ in the sense of this story. There are also bootstrap methods used in statistics. They involve analysing new samples drawn from a fixed sample. In some sense this means obtaining more knowledge about a system without any further input of information. It is this aspect of ‘apparently getting something for nothing’ which is typical of the bootstrap. In French the procedure in statistics has been referred to as ‘méthode Cyrano’. Unfortunately is seems that in PDE theory the French have just adopted the English term. I say ‘unfortunately’ because of a fondness for Cyrano de Bergerac. As in the case of Münchhausen there was a real person of this name, this time a writer. However the name is much better known as that of a fictional character, the hero of a play by Edmond Rostand. The non-fictional Cyrano wrote among other things about a trip to the moon. There is also a Münchhausen story where he uses a kind of inverse bootstrap (could there be a PDE analogue here?) to return from the moon. He constructs a rope which he attaches to one of the horns of the moon but it is much too short to reach down to the ground. He climbs to the bottom of the rope, reaches up and cuts off and detaches the part ‘which he does not need any more’ and ties it onto the bottom. He then repeats this process. Returning to Cyrano, he describes seven methods for getting to the moon of which the sixth is the one relevant to the bootstrap. He stands on an iron plate and throws a magnet into the air. The iron plate is attracted by the magnet and starts to rise. Then he rapidly catches the magnet and throws it into the air again. I should point out that Cyrano does not believe in the nonsensical stories he is telling – his aim is a practical one, holding the attention of the Duc de Guiche so as to delay him for a very specific reason.

Now I return to the topic of bootstrap arguments for evolution equations. I have given a discussion of the nature of these arguments in Section 10.3 of my book. Another description can be found in section 1.3 of Terry Tao’s book ‘Nonlinear dispersive equations: local and global analysis‘. A related and more familiar concept is that of the method of continuity. Consider a statement $P(t)$ depending on a parameter $t$ belong to the interval $[0,\infty)$. Let $S$ be the subset consisting of those $t$ for which the statement $P$ is true on the interval $[0,t)$. If it can be shown that $S$ is non-empty, open and closed then it can be concluded that the statement holds for all $t$, by the connectedness of the interval. The special feature of a bootstrap argument is the way in which openness is obtained. Suppose that, starting from $P(t)$, we can prove a string of implications which ends again with $P(t)$. This is nothing other than a circular argument and proves nothing. Suppose, however, that in addition this can be improved so that the statement at the end of the string is slightly stronger than that at the beginning. This improvement is something to work with and is a typical way of proving the openness needed to apply the continuity argument. It is more convenient here to work with the open interval $(0,\infty)$ since we want to look at properties of solutions of an evolution equation defined on the interval $(0,t)$. Let $P(t)$ be the statement that a certain inequality (1) holds on the interval $(0,t)$ and suppose that $P(t)$ implies the statment $Q(t)$ that a stronger inequality (2) holds on the same interval. Things are usually set up so that $Q(t)$ implies by continuity that (2) holds at $t$ and that the the property of being ‘stronger’ then shows that $P(t')$ holds for $t'$ slightly greater than $t$. This shows the openness property. I think the best way to really understand what a bootstrap argument means is to write out a known example explicitly or, even better, to invent a new one to solve a problem which interests you. The key thing is to find the right choice of $P$ and $Q$. What I have described here is only the simplest variant. In the work of Christodoulou mentioned above he uses a continuity argument on two-dimensional sets.