We divide the proof of Theorem 1.1 into several steps.
Step 1. A priori estimates.
Taking the operation on both sides of the first equation of (1.1), we have
(3.1)
Let be the solution of the following ordinary differential equations:
(3.2)
Then it follows from (3.1) that
(3.3)
which implies that
(3.4)
Multiplying , taking the norm on both sides of (3.4), we get by using the Minkowski inequality
(3.5)
Next, taking the norm with respect to on both sides of (3.5), we get by using the Minkowski inequality that
(3.6)
Using the fact that is a volume-preserving diffeomorphism due to , we get from (3.6) that
(3.7)
Thanks to Proposition 2.3, the last term on the right side of (3.7) is dominated by
(3.8)
and thus
(3.9)
where we used (1.4) and the boundedness of the Calderón-Zygmund singular integral operator on .
Now from (1.1) we have immediately
(3.10)
for all , since div . Summing up (3.9) and (3.10) yields
(3.11)
which together with the Gronwall inequality gives
(3.12)
Step 2. Approximate solutions and uniform estimates.
We construct the approximate solutions of (1.1). Define the sequence by solving the following systems:
(3.13)
We set and let be the solutions of the following ordinary differential equations:
(3.14)
for each . Then, following the same procedure of estimate leading to (3.11), we obtain
(3.15)
where we used the fact that , Sobolev embedding theorem for , (1.4) and the boundedness of the Calderón-Zygmund singular integral operator on . Equation (3.15) together with the Gronwall inequality implies that
(3.16)
for some independent of n. Thus, if we choose such that
we have, for any ,
(3.17)
by the standard induction arguments. Then, . Moreover, it follows from Proposition 2.1 that
by Sobolev embedding and the boundedness of the Calderón-Zygmund singular integral operator on , and then
(3.18)
which implies that . This together with (3.17) gives the uniform estimate of in n.
Step 3. Existence.
We will show that there exists a positive time () independent of n such that and are Cauchy sequences in . For this purpose, we set
Then, it follows that satisfies the equations
(3.19)
Applying to the first equation of (3.19), we get
(3.20)
Exactly as in the proof of (3.7), we get
(3.21)
where we used Proposition 2.1, Proposition 2.3, the embedding , and the boundedness of the Calderón-Zygmund singular integral operator on . Thanks to the Fourier support of , we have
(3.22)
Now, we estimate the norm of . Multiplying on both sides of the first equation of (3.19), and integrating the resulting equations over , we obtain
which together with (3.21) and (3.22) gives
(3.23)
Equation (3.23) together with (3.17) yields
(3.24)
where . Thus, if , then
This implies that
Thus, is a Cauchy sequence in . By the standard argument, for , the limit solves (1.1) with the initial data . The fact that follows from the uniform estimate (3.18).
Step 4. Uniqueness.
Consider is another solution to (1.1) with the same initial data. Let and . Then δθ satisfies the following equation:
In the same way as the derivation in (3.24), we obtain
for sufficiently small T. This implies that , i.e., .
Blow-up criterion.
For the a priori estimate (3.12), we only need to dominate and . From Proposition 2.2 and the boundedness of the Calderón-Zygmund operator from into itself, we have
Similarly,
Thus, the a priori estimate (3.12) gives
By the Gronwall inequality
Therefore, if , then .
On the other hand, it follows from the Sobolev embedding for that
Then implies .