This paper studies the wave-breaking criterion for the generalized weakly dissipative two-component Hunter-Saxton system in the periodic setting. We get local well-posedness for the generalized weakly dissipative two-component Hunter-Saxton system. We study a wave-breaking criterion for solutions and results of wave-breaking solutions with certain initial profiles.
In recent years, the Hunter-Saxton equation [
The Camassa-Holm equation admits many integrable multicomponent generalizations. So many authors studied the two-component Camassa-Holm system [
The authors of [
In general, avoiding energy dissipation mechanisms in a real world is not so easy. Wu and Yin [
Our major results of this paper are Theorems
Throughout this paper,
In this part, we will establish the local well-posedness for the Cauchy problem of system (
We now provide the framework in which we will reformulate (
Integrating both sides of (
Next, we apply Kato’s theory to establish the local well-posedness for the system (
Given the evolution equation (
Moreover, the map
Given any
Moreover, the solution depends continuously on the initial data, that is, the mapping
Now, discuss the initial value problem for the Lagrangian flow map as follows:
Let
Since
Similarly, we have
Let
On the one hand, integrating the second equation in (
On the other hand, multiplying (
Multiplying the second equation in (
Adding the above two equations, we get
We acquire
This completes the proof of Lemma
Let
By computing directly, we have
Multiplying (
By Gronwall's inequality, we get
This completes the proof of Lemma
Assume that
By assumption, there is
Then, for
This implies
In this section, by using transport equation theory, we obtain the wave-breaking criteria for solutions to (
The following estimates hold:
where
Suppose that
Then
or
Let
Then
The above proposition was proved in [
Let
Our next result describes the necessary and sufficient condition for the blow-up of solutions to (
Suppose that
The approach one takes here is the method of characteristics. Applying the following lemma, we may carry out the estimates along the characteristics
Let
Let
The constants above are defined as follows:
By Theorem
Let
We can consider
Hence,
Take the trajectory
Now, let
Therefore, along the trajectory
We first compute the upper and lower bounds for
Since
When
Combining (
Since
Hence,
Hence,
For any given
Observing that
We now claim that
Assume the contrary that there is
Let
To derive (
Hence,
Using previous arguments, we take the characteristic
Let
Hence, along the trajectory
Define
For any given
We now claim that
Suppose not, then there is
Then,
Therefore,
Let
We first compute the upper and lower bounds for
Now, we turn to the lower bound of
Combining (
We know
Hence,
Therefore, we have
Integrating (
To obtain a lower bound for
Since
Because of
This means that
Then,
Integrating (
Suppose that
Moreover, if there exists
Then
Differentiating the left hand side of (
This completes the proof of (
By Lemma
To obtain (
Suppose that
It now follows from Lemma
Conversely, the Sobolev embedding theorem
The authors would like to thank the referee for comments and suggestions. This work is supported by the National Nature Science Foundation of China (no. 11171135), the Nature Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003), and the high-level talented person special subsidizes of Jiangsu University (no. 05JDG047).