In this section, we complete the proof of Theorem 1.1 and suppose that (or if with ).
First step: approximate solution
By using the standard iterative process, we construct a sequence of smooth solutions . Assume that , by induction we define a sequence of smooth functions by solving the following linear transport equation:
(3.1)
(3.2)
By using the fact that belong to , from Lemma 2.4, for all , we can show by induction that problem (3.1)-(3.2) has a global solution .
Second step: uniform bounds
For (or if with ) and , we prove that
(3.3)
with .
From (2.3) of Lemma 2.3 and (3.1), we derive that
(3.4)
where
(3.5)
By using the multiplier property of and the fact with (or if with ) is a Banach algebra, we have
(3.6)
Inserting (3.6) into (3.4) yields (3.3).
Let us fix such that
(3.7)
and suppose that
(3.8)
Since , by using (3.8), we have
(3.9)
When in (3.9), we have
(3.10)
Inserting (3.9), (3.10) into (3.3) yields
(3.11)
Thus, is uniformly bounded in . By using (3.9) and the fact that is a Banach algebra and (3.5) and using the -multiplier property of , we have that
(3.12)
and
(3.13)
Consequently,
(3.14)
Remark Inserting (3.7) into (3.8) yields
(3.15)
for . From (3.11) and (3.7), , we have that
(3.16)
and
(3.17)
with the aid of Fatou’s lemma. We define
(3.18)
Thus,
(3.19)
for .
Third step: convergence
We prove that is a Cauchy sequence in . For , we have
(3.20)
Combining (2.3) with (3.20), we have that
(3.21)
where
By using (3) and (7) of Lemma 2.2, we have that
(3.22)
Inserting (3.22) into (3.21) yields
(3.23)
From (3.2) and Lemma 2.1, we can easily obtain that
(3.24)
Obviously,
(3.25)
Inserting (3.24) and (3.25) into (3.23) leads to
(3.26)
We define
(3.27)
Inserting (3.27) into (3.26) leads to
(3.28)
We define
(3.29)
and
(3.30)
Combining (3.30) with (3.29), (3.28), by using Fatou’s lemma, we have that
(3.31)
Applying Gronwall’s inequality to (3.31) yields
(3.32)
for . According to the definition of , we can easily obtain that
Combining (3.32) with (3.33), we have that
Hence, is a Cauchy sequence in .
Fourth step: existence in
Now we prove that and satisfies (1.4)-(1.5) since is uniformly bounded in . From (6) in Lemma 2.2, we have that . From (1.4), we can easily prove that . It is easily checked that u is indeed a solution to (1.4)-(1.5) by passing to the limit in (3.1)-(3.2).
Now we prove that . Since , , there exists such that
(3.35)
From the definition of Besov spaces, we have that
(3.36)
By using the mean value theorem, we have that
(3.37)
where . Inserting (3.37) into (3.36) yields
(3.38)
We may choose δ sufficiently small such that
(3.39)
Inserting (3.39) into (3.38) leads to
(3.40)
Thus, we derive that
(3.41)
Combining (3.41) with (1.4), we can easily obtain
(3.42)
From (3.41) and (3.42), we have that .
Fifth step: uniqueness of solution
The uniqueness of the solution to the Cauchy problem for (1.4) can be proved similarly to Proposition 3.1 of [14].
Sixth step: continuity in with (or if with )
The continuity of the solution to the Cauchy problem for (1.4) can be proved similarly to the continuity of the solution to the Degasperis-Procesi equation which can be seen in [14].