Asymptotic Stability of Singular Solution for Camassa-Holm Equation ()
1. Introduction and Main Results
1.1. Introduction
Consider the well-known Camassa-Holm equation as follows (see [1] ):
(1.1)
where
,
is the velocity of fluid, m is the momentum given by
is the critical speed and
relates to the length scale. Thus,
(1.2)
Given the initial value as
for
.
The Camassa-Holm equation describes unidirectional propagation of surface water waves in shallow water area. For the global well-posedness and stability of solutions, we recommend that the reader refers to [2] - [9], etc. For the wave breaking analysis, we refer the reader to [6] [10] - [15], etc. When
and
, the Camassa-Holm equation becomes to the classical Camassa-Holm equation, which admits a bi-Hamiltonian structure [1] [5]. Moreover, the explicit peakon solution and its stability have been established in [12] [16] [17] [18] [19], etc.
Since it is rare to see the explicit stable blowup solutions of Camassa-Holm equation, in this paper, we study the stability of the explicit solution of (1.2) as follows (see [20] ):
(1.3)
where
is a constant.
1.2. Main Results
Now, we state our main result of this paper.
Theorem 1.1. Let
be an integer and
is a sufficiently small constant. Then the explicit solution (1.3) of the Camassa-Holm Equation (1.2) is asymptotic stable, i.e., if the initial data
satisfies
then there is a solution
of (1.2) satisfying
where C and
are positive constants that depend on s.
1.3. Notations
Denote
and
by the Lebesgue spaces and Sobolev spaces with norms
and
, respectively. * denotes the convolution.
stands for the commutator.
2. Proof of Theorem 1.1
Let
(2.1)
be the solution of (1.2), where
is the explicit solution. Substituting (2.1) into (1.2), we get
(2.2)
with the initial condition
for
.
For the singular coefficients in (2.2), let
by
and
, then (2.2) becomes to
(2.3)
Let
and
. Then (2.3) becomes to
(2.4)
Let the operator
. Since
admits a fundamental solution
, we have
for all
. Let
, then
, where
. Furthermore, we have
,
and
. Then (2.3) can be rewritten as
(2.5)
with the initial data
(2.6)
and the boundary condition
(2.7)
Before making a priori estimate of the solutions to problems (2.5)-(2.7). We recall the following commutator estimate.
Lemma 2.1 ( [21] ). Let
. Then it holds
(2.8)
where C is a positive constant that depends on s.
Now, we derive a priori estimate of the solutions for (2.5).
Lemma 2.2. Let
and
. Assume that w be a solution of (2.5), then
(2.9)
where C is a positive constant depending upon s.
Proof. Applying
to both sides of (2.5) and taking the
-inner product with
, we get
(2.10)
Next, we estimate each of terms in (2.10).
(2.11)
(2.12)
(2.13)
(2.14)
In addition, using (2.8), we have
(2.15)
similarly,
(2.16)
where C is a positive constant depending upon s.
Substituting (2.11)-(2.16) into (2.10), we get
, and then
. Integrating this inequality above with respect to
from 0 to
, we get
(2.17)
This completes the proof of Lemma 2.2. o
Proof of Theorem 1.1. Now, we study the well-posedness for (2.5)-(2.7). Define the linear operator L as
(2.18)
then (2.5) becomes to
(2.19)
where f is the nonlinear terms:
(2.20)
Lemma 2.3. Let
. Then
·
for
.
· L is a closed and densely defined linear operator in
.
Proof. It is a direct verification by the definition of L. o
Lemma 2.4. Let
. Then L is a dissipative operator in
, i.e.,
.
Proof. Using (2.11)-(2.14), a direct calculation shows that
(2.21)
This completes the proof. o
Lemma 2.5 (Young inequality with
, see [22] ). Let
and
. If
satisfy
. Then
(2.22)
where
.
Lemma 2.6. Let
. Then the operator L is invertible in
. Furthermore, it generates a
-semigroup
in
.
Proof. Firstly, we show that the existence of
. Indeed, we need to prove L is injective and surjective. On the one hand, let
such that
, then
(2.23)
This combining with the boundary condition (2.7) gives that
. So the operator L is injective. On the other hand, for all
, put
(2.24)
Applying
to (2.24) and multiplying the result by
, and then integrating over
, we get
(2.25)
It follows from the Young inequality with
in Lemma 2.5 that
(2.26)
Note that
, then by the standard theory of elliptic equations (see [22] ), there exists a unique weak solution
, moreover, we have
if
. Thus, the operator L is surjective. Secondly, by the Lumer-Phillips theorem (see [23] ), the operator L generates a
-semigroup
in
. This completes the proof. o
As a consequence, we have
Proposition 2.7. Let
. Then the Cauchy problem
(2.27)
with zero boundary condition exists a unique solution
, where
is the initial data defined in (2.6).
Using the Duhamel’s principle, the solutions of (2.19) satisfies the integral equation:
(2.28)
To show this integral equation exists a solution, we define the solution space as
(2.29)
and the map
as
(2.30)
We need to prove that
has a fixed point in the space
.
Lemma 2.8 ( [21] ). Let
. Then
is an algebra, and
(2.31)
where C is a positive constant depending upon s.
Lemma 2.9. Let
be an integer. Assume that
for some sufficiently small
. Then
is a self-mapping on
. Moreover,
is a contraction mapping.
Proof. By Lemma 2.8, we have
(2.32)
where
is a positive constant.
Note that
and
, then using Lemma 2.2, we have
(2.33)
for sufficiently small
. Thus,
is a self-mapping on
.
To show
is a contraction mapping, we choose
, by Lemma 2.8 and a direct calculation show that
(2.34)
Thus,
(2.35)
Since
is sufficiently small,
is a contraction mapping. o
Thus, we have the following existence results.
Proposition 2.10. Let
be a fixed integer and
is a sufficiently small constant. Then
· if
, there exists a unique solution
to (2.5) with the initial data (2.6) and the boundary condition (2.7).
· there exists a global solution
to (2.3) with the initial data (2.6) and the boundary condition (2.7). Moreover, if the initial data
satisfies
, then
(2.36)
Here C and
are two positive constants that depend on s.
Proof. By Lemma 2.9 and the Banach fixed point theorem, the map
has a fixed point in
, which is a solution of Equation (2.5). Thus, there exists a global solution of (2.3) as
(2.37)
Furthermore, we have
(2.38)
Thus, by Lemma 2.2, we get
(2.39)
where we have used
in the last inequality. This completes the proof. o
As a consequence, we obtain that the global well-posedness of the initial value problem (2.2). This implies that the asymptotic stability of the explicit singular solution (1.3) for the Camassa-Holm Equation (1.2). Hence, we complete the proof of Theorem 1.1.
3. Conclusion
In this paper, the Semigroup theory of linear operators has been used to study the asymptotic stability of the explicit blowup solution of Camassa-Holm equation. This result shows that the explicit solution is a meaningful physical solution. However, this explicit solution does not depend on the wavelength (i.e., it does not depend on
). Thus, further studies are needed to construct the explicit solutions that depend on
, and then prove their stability.