On the abstract bootstrap principle in the book "Nonlinear Dispersive Equations" by Terence Tao

Question

In Terence Tao's book "Nonlinear Dispersive Equations", he gives the following "Abstract bootstrap priciple":

"Let $I$ be a time interval, and for each $t \in I$ suppose we have two statements, a "hypothesis" $H(t)$ and a "conclusion" $C(t)$. Suppose we can verify the following four assertions:

(a) (Hypothesis implies conclusion) If $H(t)$ is true for some time $t \in I$, then $C(t)$ is also true for that time $t$.

(b) (Conclusion is stronger than hypothesis) Is $C(t)$ is true for some $t \in I$, then $H(t')$ is true for all $t'\in I$ in a neighborhood of $t$.

(c) (Conclusion is closed) If "$t_1, t_2,\ldots$" is a sequence of times in $I$ which converges to another time $t\in I$, and $C(t_n)$ is true for all $t_n$, then $C(t)$ is true.

(d) (Base case) $H(t)$ is true for at least one time $t \in I$.

Then $C(t)$ is true for all $t\in I$."

My question is about (a) and (b), according to his remarks and examples, I noticed that (a) is not the real "Hypothesis implies conclusion" . I mean $H(t)$ is actually not stronger than $C(t)$, that is if we want to deduce $C(t)$ from $H(t)$, we must employ some other hypothesis or conditions in this system which we are investigating, not just $H(t)$ ($H(t)$ is actually not sufficient!). Is this understanding of this principle right? I want to be sure.

Also in his remarks and examples, he makes (b) truely hold, I mean we don't need to exploit another hypothesis to prove that. I also want to ask: can we make (b) also not very exactly? I mean just like above I discussed about (a), we need also introduce some other hypothesis to prove (b) not just $C(t)$. Is that still correct? I consider this because in the proof of this principle, I didn't see any necessity to make the implication of (a) and (b) hold strictly. The main purpose of (a) and (b) in the proof is to ensure $\Omega$ open.

The following appendix is from Tao's book.

Appendix:

Answer 1

The abstract bootstrap principle says nothing more that if you can verify (a) - (d), then $C(t)$ is true in all of the interval $I$. It is entirely agnostic to how you can verify that (a)-(d) are true.

You should compare this to the tool of induction, which we can write in the following form:

Induction: Suppose you have a function $P$ mapping the natural numbers $\mathbb{N}$ to $\{0,1\}$ (or boolean true/false values). Let the hypothesis $H(t)$ and $C(t)$ both represent the statement $P(t) = 1$.

1. (Hypothesis implies conclusion) $H(t)$ implies $C(t)$.
2. (Conclusion is stronger than hypothesis) $C(t)$ implies $H(t+1)$.
3. (Conclusion is closed) If $t_1, \ldots$ is a sequence in $\mathbb{N}$ such that $t_i \to t_\infty$, and if $C(t_i)$ are all satisfied, then $C(t_\infty)$ is also satisfied.
4. (Base case is true) There exists $t_0\in\mathbb{N}$ such that $H(t_0)$ is true.

then $C(t)$ is true for all $t \geq t_0$.

Note that in this presentation of mathematical induction, the first and third items are trivial.

Note further in the statement of mathematical induction: 1, 2, 3, 4 are assumed to hold. Given an arbitrary function $P$ they do not have to hold always. But when they do hold, the conclusion can be reached.

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To return to your question:

Do not take the splitting of (a) and (b) too seriously.

Let us return to the case of mathematical induction: You can redefine $H(t)$ to be the statement $P(t) = 1$, while $C(t)$ to be $P(t+1) = 1$. In this formulation, that the conclusion is stronger than hypothesis is trivial, while the hard step becomes proving that hypothesis implies conclusion.

So perhaps a slightly better way to think about the bootstrap principle is the following three-condition description:

Abstract bootstrap principle. Let $C:I \to \{0,1\}$ be a mapping, where $I\subset \mathbb{R}$ is some interval. Suppose we can show that

1. (Conclusion implies better than conclusion) If $C(t_0) = 1$ for $t_0\in I$, then there exists some $\epsilon > 0$ such that $C(t) = 1$ for $t\in (t_0 - \epsilon,t_0 + \epsilon) \cap I$.
2. (Conclusion is continuous) If $t_1,\ldots$ is a sequence in $I$ such that $t_i \to t_\infty$. If furthermore $C(t_i) = 1$ we can conclude $C(t_\infty) = 1$.
3. (Conclusion holds somewhere) There exists some $t_0\in I$ such that $C(t_0) = 1$.

then we can conclude that $C\equiv 1$ on $I$.

(Note that here I have cleverly hidden any notion of "implication". The three conditions, for the purpose of the bootstrap principle, are hypotheses you impose on the function $C$. The bootstrap principle is agnostic to how those hypotheses are satisfied: as long as you are given a function $C$ such that those conditions hold, you obtain the final conclusion. In applying the abstract bootstrap principle you have to of course verify that the three statements hold. The degree to which each one is obvious or universal depends on the precise problem you study and the definition of the function $C$ of course. But quite often both the first and second need to be shown using the properties of the PDE you are examining, while the third you can often get away by postulating it in the initial data.)

This now brings us to the fundamental form of the bootstrap principle, which is nothing more than the continuity principle from analysis and topology.

Continuity principle Let $X$ be a connected topological space. If $Y\subset X$ is a non-empty, open, and closed subset, then $Y = X$.

To derive the bootstrap principle from the continuity principle, we observe that $I$ is a connected topological space with the usual topology. Let $B = \{0,1\}$ with the discrete topology. Let $Y = C^{-1}(1)$. By the first condition $Y$ is open. By the second condition $Y$ is closed. By the third condition $Y$ is nonempty. Hence $Y = X$.

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Going back to the splitting of (a) and (b): generally to prove that "conclusion implies better than conclusion" what one does is to

1. Start with $C(t)$.
2. Prove (here on the word prove implicitly means that you are using the properties of the system/PDE you are studying!) $C(t) \implies$ some other statement $C_1(t)$.
3. Prove $C_1(t) \implies C_2(t) \implies\ldots \implies C_k(t)$.
4. From $C_k(t)$ you can appeal to local existence of your PDE to get that $C$ holds on a neighbourhood of $t$.

Note that from steps 1 - 3 we can immediately conclude that $C_1,\ldots C_k$ all hold on a neighbourhood of $t$.

That is to say: the "circular" structure of the argument makes it not necessary to identify any one of the $C_i$ as the "hypothesis" and any one of the $C_i$ as the "conclusion". So you do not have to force yourself to think in terms of "hypotheses" and "conclusions".

Answer 2

All four assertions are proven using properties of the system. A proof supporting part (a) will be along the lines of "Suppose $H(t)$ is true at time $t$. Then, because ...(much stuff about the details of the system in question)..., it turns out $C(t)$ is also true."

The order these things come into existence is:

• There is a system.
• Someone makes a model for the system which is unfortunately nonlinear.
• Someone proposes a property/condition/predicate pair $(\mathbf{H}_1(t),\mathbf{C}_1(t))$ and hopes that their understanding is sufficient to show, using the properties of the model, that the four assertions are true for this pair. Frequently, this doesn't quite work. As observed, it is usually straightforward to find a pair that satisfy (b), (c), and (d), but getting (a) can be hard.
• Repeat prior step: $(\mathbf{H}_2,\mathbf{C}_2), (\mathbf{H}_3,\mathbf{C}_3), (\mathbf{H}_4,\mathbf{C}_4), \dots$ until someone actually succeeds at finding a pair, $(\mathbf{H}_n,\mathbf{C}_n)$, that can be shown to satisfy the four assertions.
• Now reap all the consequences from a successful bootstrap.

The use of the term "weaker" is to distinguish the statements:

• $\mathbf{C}(t)$ is just true at all times $t$.
• At those times, $t$, where $\mathbf{H}(t)$ happens to be true, it turns out that $\mathbf{C}(t)$ is also true.

It should be clear that the first one is a much more assertive claim than the second.