1. Introduction
In what follows, we are concerned with the local well-posedness of the Cauchy problem for the Novikov system:
Equation (
1) was first proposed in [
1,
2], it can be derived as the reduction of the following multi-component Novikov system
by setting
.
Equation (
2) was derived from the zero-curvature equation for the vector prolongation of Lax pair representation for the Geng–Xue equation [
3]. It can be reformulated as a bi-Hamiltonian system with the following Hamiltonian functionals:
While
, Equation (
2) is reduced to the Degasperis-Procesi (DP) equation [
4]
It is a model for nonlinear shallow water dynamics [
5]. The DP equation was proved to be formally integrable by constructing a Lax pair and the direct and inverse scattering approach [
6]. Moreover, it was also shown that it has a bi-Hamiltonian structure and an infinite number of conservation laws [
7] and admits exact peakon solutions, which are analogous to Camassa–Holm peakons. It is noted that the peakons replicate a feature that is characteristic for the waves of great heights or waves with largest amplitudes, which are exact solutions for the governing equations of irrotational water waves. The local well-posedness of DP equation in Sobolev and Besov spaces has been investigated in many papers, see [
8,
9,
10].
While
, Equation (
2) is reduced to the Novikov equation
The Novikov equation shares similar properties with the Camassa–Holm equation, such as a Lax pair in the matrix form, a bi-Hamiltonian structure, infinitely many conserved quantities, and peakon solutions given by the formula
[
11].
While
, Equation (
2) is reduced to the Geng–Xue equation [
3]
The integrability [
3,
12], dynamics and structure of the peaked solitons of the Geng–Xue system [
13,
14] were considered recently. In [
15], the well-posedness and wave breaking phenomena of the Cauchy problem were discussed.
The well-posedness and ill-posedness of Camassa–Holm type equations in Besov spaces have recently been intensively explored [
16,
17,
18,
19]. The main difference between the Novikov equation and the Camassa–Holm equation is that the Novikov equation contains terms of cubic nonlinearity, while the Camassa–Holm equation only contains quadratic terms. In [
20], the local well-posedness of the Novikov equation was researched in Besov spaces
for
with
and
; meanwhile, the ill-posedness of the Novikov equation was also considered in
by using peakon solutions. The ill-posedness result was lately studied by Li et al. [
21] in supercritical Besov spaces
with
and
, in the sense that the solution map starting from
is discontinuous at
in the metric of
. For the local well-posedness of Novikov equation in critical Beosov spaces, it is worth mentioning that Ni and Zhou [
22] obtained local well-posedness in
, later, the local well-posedness result was extended to the broader range of Besov spaces
with
based on the Lagrangian coordinate transformation by Ye et al. [
23]. The continuous dependence of the solution to Novikov equation on the initial data in the space
with
was also researched in [
18], and the solution map was proved to be weakly continuous with
.
For the two-component Novikov system (
1), the local well-posedness result in super critical Besov spaces
for
with
and
has already been established [
24]. However, whether (
1) is well-posed or not in the critical Besov spaces
with
is still open to debate, therefore, it would be reasonable to study the local well-posedness of Equation (
1) in such spaces, we aim to construct the local well-posedness result in the present paper, the methods adopted in this paper are also applicable to two-component Camassa–Holm equations and Geng–Xue system etc.
The motivation to use the methods adopted in each step can be stated as follows: We construct a sequence of approximate solutions using an iterative scheme for the purpose of showing the existence of the solution. In the first step, we demonstrate the uniform boundedness of the solution sequence via induction; then, in step 2, we extract two subsequences of the above mentioned approximate solution sequence, the norm of the difference between which in the space
is then shown to converge to zero, thus we obtain the compactness of the approximate sequence and get a solution
which solves Equation (
3) in the sense of distributions. The Moser-type inequality is always needed in the proof of uniqueness; however, the critical index restriction
makes it impossible to apply the Moser-type inequality any longer. Fortunately, inspired by [
23], we devise the Lagrangian scale of (
1) to overcome this difficulty and succeed in proving the uniqueness in step 3. In order to prove the continuous dependence of the solution on the initial data in step 4, we decompose the solution components, and afterwards, by making use of transport theories and the energy method, we manage to show that if the initial data sequence converges to a fixed initial data, then the corresponding approximate solution sequence converges to the solution corresponding to the fixed initial data.
For
and
, we define
Before proceeding any further, we need to reformulate (
1) into a workable form of transport equation-type. By applying the inverse of Helmholtz operator
to both sides of the equations in (
1), we obtain
We introduce our main result regarding local well-posedness of (
3) in critical Besov spaces as follows.
Theorem 1. Let with , then there exists a time such that the Cauchy problem (3) has a unique solution , and the solution map is continuous from a neighborhood of into . The rest of this paper is organized as follows. The properties of Besov spaces, as well as the transport theories, which provide a basis of the study of the associated local well-posedness problem is given in
Section 2. In
Section 3, we split the proof of Theorem 1 into four steps, which cover uniform boundedness, convergence and regularity, uniqueness, and continuous dependence, respectively.
Notation 1. For given Banach space X, we denote its norm by . Since all function spaces are over , we drop in our notations of function spaces for simplicity if there is no ambiguity.
3. Proof of Theorem 1
We split the proof of the main theorem into four steps. First we prove the uniform boundedness of the approximate solutions, then we show its convergence; afterward we prove the uniqueness of solution; finally, we establish the continuous dependence of the solution on the initial data.
Step 1. Uniform boundedness of approximate solutions constructed with an iterative scheme.
Suppose that
, we define by induction a solution sequence
of the following transport equation:
Assume that
belongs to
for all
, we know from Lemma 1 that
are algebras and the embedding
holds. Thus, we have
It can be deduced from Lemma 2 that there exists a unique solution
of (
5). Moreover, by applying Lemma 3, one obtains
where
.
Similarly, we can infer that
Let
, it follows by adding (
6) and (
7) that
Fix a
such that
, and suppose by induction that
Since
substituting the above inequality into (
8) yields
Therefore, is uniformly bounded in .
Step 2. Convergence and regularity of solutions.
Next, we will prove that
is a Cauchy sequence in
. It follows from (
5) that
Among which
By applying the algebraic property of Besov spaces
and
, we obtain
Thanks to Lemmas 2 and 3, we see that for any
,
Therefore, by making use of the induction, one can see a positive constant
exists, which is independent of
n and
m, such that
which indicates that the sequence
is a Cauchy sequence and converges to
. Therefore, via Fatou’s Lemma, we obtain
Next, we need to show that
solves the Equation (
3) using the following steps.
In view of (
5), we consider the following system
where
is a test function. By virtue of Proposition 1, we can take the limit as
in the first equation of (
10) to check its convergence
which converges to 0 as
, where
, deduced from which we find that
The convergence of other terms can be computed in a similar way. Thus we can derive for any
,
Similarly, for the second equation of (
10), we can derive that
Consequently,
solves the Equation (
3), Lemma 4 then shows that
. Notice that
and
, we conclude that
.
Step 3. Uniqueness with respect to the initial data.
To prove the uniqueness result, we follow the structure of [
23] and introduce the associated Lagrangian scale of (
3) as follows:
By introducing the new variable
, then (
3) becomes
By Denoting
, we may derive from (
11) that
and
Similar to the proof of Theorem 1.1 in [
23], we can find that
,
and
for sufficient small
T.
We estimate
next. The main difficulty lies in estimates of coupling terms containing derivatives. For the terms containing the sign function, we choose to estimate the following term as an example
Notice that
is monotonically increasing, therefore
, and thus,
If
(or
), then
(or
, therefore we can derive
By using a similar technique and applying the inverse trigonometric inequality, we have
Substituting estimates (
17) and (
18) back into (
16) yields
Similar to (
19), we can derive the estimates of other terms containing the sign functions as follows:
For the other two terms which do not contain the sign function, we estimate them as follows:
Combining (
19)–(
25), we find that
Similarly, we can also obtain
Combining (
26) and (
27), we have
Next, we are ready to prove uniqueness. Suppose that
and
are two solutions to (
3), then
satisfies (
3) and the following equations for
With the help of (
28), we see
where we use the fact that
.
Applying the Gronwall Lemma to (
29) yields
Therefore, it follows from (
30) that
Similarly, we can obtain
(
31) together with (
32) implies that
Then, from the embedding theorem, it follows that
Thus, if , we can prove the uniqueness result immediately.
Step 4. Continuous dependence on the initial data
Let
in
as
, and
are the solutions with initial data
, respectively. By the above discussion, we find that
are uniformly bounded in
and
which indicates that
in
. By applying the interpolation inequality we can find
Next, in order to prove the continuous dependence result, we just need to prove
in
.
For this purpose, we decompose the solution components as follows:
where
solves the following problem:
and
solves the following problem
and
are defined as follows:
among which,
Note that is bounded in , we can deduce that and are also bounded in . As in , Lemma 5 guarantees that in .
It can also be derived from Lemmas 2 and 3 that, for any
,
. In view of (
34):
Notice that
is bounded in
, by applying the algebraic property of Besov space
, we have
It follows that for all
,
In view of the fact that
in
,
in
,
in
, by applying the Gronwall Lemma, one can deduce that
in
. Therefore,
tends to 0 as
, which implies
Combining (
33) with (
36), we find
Theorem 1 is a direct consequence of step 1–step 4.