On the Cauchy Problem for Mildly Nonlinear and Non-Boussinesq Case-(ABC) System ()
1. Introduction
In this paper, we consider the following Cauchy problem for Mildly Nonlinear and Non-Boussinesq case-(ABC) system (see [1] ):
(1.1)
where
is the averaged weighted vorticity, and
is the interface displacement.
are the small parameters with
and
, i.e. the Mildly Nonlinear (MNL) case. Moreover,
are the nonnegative parameters, and B is a parameter. For more details, we can refer to [1] [2] .
Fluid waves are a ubiquitous phenomenon in marine and atmospheric science. An important cause of fluid internal waves is density stratification. In density-stratified flows, the displacement of a fluid mass from its neutrally buoyant position results in internal wave motion. The dynamics of these internal waves has been of great interest and has been the subject of much research (see e.g. [3] [4] ). The study of these long-wave limit currents approaches the existence of a physical environment of rapidly varying density and produces a variety of mathematical models depending on the relative strength of the different influences. The models obtained can be dispersive or non-dispersive and are weakly or fully nonlinear. Physically, dispersion is controlled by the relative magnitude of the horizontal length scale with respect to the height of the domain, however, the nonlinearity is controlled by the wave amplitude with respect to the height of the fluid domain.
Strongly nonlinear, non-dispersive approximations take the form of hyperbolic or mixed-type first-order PDEs, first derived in this context by Long [5] . Weakly nonlinear dispersive approximations result in Korteweg-de Vries type models [6] and fully nonlinear dispersive approximations lead to the so-called Miyata-Camassa-Choi system [7] [8] . However, in this paper, we focus on the Hamiltonian structure of 2-layer dispersed stratified fluids in the non-Boussinesq case under mildly nonlinear assumptions (i.e. the Mildly Nonlinear and Non-Boussinesq case-(ABC) system (1.1)).
Moreover, in Ref. [1] , solutions of special forms of Equation (1.1), i.e. traveling wave solutions and unidirectional waves were considered and the dispersion relation was calculated.
For the equivalent form of (1.1), we set
and
with
, then we get:
and
where
and s is a integer.
In the sequel, we will, for notational convenience, demonstrate local well-posedness of the following initial-value problem with more general coefficients:
(1.2)
with
. Where u is the averaged weighted vorticity, and
is the interface displacement. Moreover,
(
) are all parameters, the operator
is a S-multiplier with
.
Inspired by the argument of Danchin [9] [10] in the study of the local well-posedness to the CH equation, we first establish the local well-posedness of (1.2) in Besov spaces.
Theorem 1.1. Suppose that
and
and the initial data
. Then, there exists a time
such that the Cauchy problem Equation (2.2) has a unique solution
(see Definition 2.3). Moreover, the map
is continuous from a neighborhood of
in
into:
for every
when
and
when
.
Remark 1.2. In the proof of Theorem 1.1, we use the transport equations theory and the transport diffusion equations theory to establish the local well-posedness of Equation (1.2) in nonhomogeneous Besov spaces. It is well-known that the Besov spaces
coincide with the Sobolev spaces
. Theorem 1.1 implies that under the condition
with
, we can obtain the local well-posedness for the data-to-solution map in Sobolev spaces.
In Theorem 1.1, we have proved that for every
when
and
when
, the data-to-solution map of system (1.2) is continue in
. Now, another natural question raised: whether or not the data-to-solution map of system (1.2) is continue in
for
and
. In our next theorem, we shall further show that this data-to-solution map is ill-posedness in
in the sense that the solutions starting from
are discontinuous at
in the metric of
, this means that:
Theorem 1.3. Suppose that
and
. Then, the system (1.2) is ill-posed in the Besov spaces
. More precisely, there exist
and a positive constant
for which the Cauchy problem of system (1.1) has a unique solution
for some
while
Motivation by [11] [12] , we use the generalized Ovsyannikov theorem to solve the Gevrey and analytic regularity for (1.2). To begin with, we introduce the Sobolev-Gevery spaces [11] [12] , a suitable scale of Banach spaces, as follows:
It is easy to cheek that
equipped the norm
is a Banach space by the completeness of
. Then, we can define the Gevrey and analytic regularity as follows:
Definition 1.4. Denoting Fourier multiplier
by
, i.e.
. If
, it is called ultra-analytic function. If
, it is usual analytic function and
is called the radius of analyticity. If
, it is the Gevrey function.
Now, we establish that solutions of (1.2) are analytic in both space and time variables.
Theorem 1.5. Let
and
. Assume that
. Therefore, for every
, there exists a
such that the system (1.2) has a unique solution
, which is holomorphic in
with values in
.
The rest of our paper is organized as follows. In Section 2, we recall several results in the Littlewood-Paley theory and some properties of Besov spaces are reviewed. In Section 3, we establish the local well-posedness result for Equation (1.2). Moreover, the ill-posedness result for this system is presented in Section 4. Finally, we give the Gevrey regularity in Section 5.
2. Preliminaries
In this section, for the convenience of the readers, we will recall some facts on the Littlewood-Paley theory, which will be frequently used in the following arguments. Then, we introduce some properties of the Besov spaces which will play a key role in proving the local well-posedness and other properties for the system (1.2). One may check [10] [13] for more details. First, we introduce some notations.
Notation. For simplicity, the norm
means
in the following, the symbol
means that there is a uniform positive constant C independent of A and B such that
.
Proposition 2.1. (See Proposition 2.10 in [13] ) Let
and
. There exist two radial functions
and
such that:
Moreover, let
and
. Then, for all
, the dyadic operators
and
can be defined as follows:
Therefore,
and the right-hand side is called the nonhomogeneous Littlewood-Paley decomposition of f.
Definition 2.2. (See Definition 2.68 in [13] ) Let
,
. The inhomogenous Besov space
(
for short) is defined by:
where
If
,
.
Definition 2.3. For
and
, we define:
Proposition 2.4. (See Corollary 2.86 in [13] ) For any positive real number s and any
in
, the space
is an algebra and a constant C exists such that:
If
or
,
, then we have:
Proposition 2.5. (See Proposition 1.3.5 in [10] ) Suppose that
,
(
). We have:
1) Topological properties:
is a Banach space and is continuously embedded in
.
2) Density:
is dense in
.
3) Embedding:
↪
, if
and
.
↪
locally compact, if
.
4) Algebraic properties: for all
,
is an algebra. Moreover,
is an algebra
↪
(or
and
).
5) Complex interpolation:
for all
, for all
.
6) Fatou lemma: If
is bounded in
and
in
, then
and
7) Let
and let f be an
-multiplier (i.e.
is smooth and satisfies that for all
, there exists a constant
, s.t.
for all
). Then, the operator
is continuous from
to
.
Lemma 2.6. (See Lemma 2.8 in [14] or [13] ) Suppose that
and
. Assume
,
, and
If
solves the following 1-D linear transport equation:
(2.1)
then exists a constant C depending only on
, such that the following statements hold:
1) If
or
,
or
with
2) If
, then for all
, 1) holds true with
.
3) If
, then
. If
, then
for all
.
Lemma 2.7. (See Theorem 3.3.1 in [10] or [15] ) Let
, and let
with
. Assume that
,
. Let v be a time-dependent vector field such that
for some
,
and
and
Then, Equation (2.1) has a unique solution
and the inequalities in Lemma 2.6 hold true.
Moreover, if
, then we have
.
Lemma 2.8. (See [13] ) Let
,
, then we have:
where
.
3. Local Well-Posedness in Besov Space
In this section, we shall discuss the local well-posedness of the Cauchy problem (1.2) in nonhomogeneous Besov spaces, and prove Theorem 1.1. In the following, we denote
a generic constant only depending on
. Moreover, uniqueness and continuity with respect to the initial data
are an immediate consequence of the following lemma.
Lemma 3.1. Let
and
. Suppose that:
be two given solutions of the initial-value problem (1.2) with the initial data
, and denote
,
. Then, for every
, we have:
(3.1)
for
, where
For the critical case
,
where
.
Proof (Proof of Lemma 3.1) It is easy to see that
and
which implies that the solution pair
, and
solves the following transport equations:
(3.2)
where
and
Integrating the first equation of (3.2) with respect to variable t, it is easy to see that:
Note that the operator
is a S-multiplier. Applying Proposition 2.4, Proposition 2.5 and the algebraic property for
for
, we have:
Therefore, we obtain:
(3.3)
Applying the Lemma 2.6 for
and
, we get the following inequality:
(3.4)
with
Using Proposition 2.5 (3) (
↪
locally compact, if
) and Proposition 2.5 (4) (
↪
), for
, we get:
(3.5)
For
and
, applying Proposition 2.4, we can obtain that:
(3.6)
Submitting (3.5)-(3.6) into (3.4), we may derive:
(3.7)
for
and
.
Consequently, combining (3.3) and (3.7), we have:
Applying Gronwall’s lemma to the above inequality leads to (3.1).
For the case
, choosing
, then we have:
and
And according to the interpolation formula in Proposition 2.5 (5) and the obtained result (3.1), we can see that:
which yields the Lemma 3.1.
Now, let us start the proof of Theorem 1.1, which is motivated by the proof of local existence theorem about Camassa-Holm type equations in [15] [16] . Next, by using the classical Friedrichs regularization method, we construct the approximate solutions to (1.2).
Lemma 3.2. Let p, r and s be as in the statement of Lemma 2.6. Assume that
. There exists a sequence of smooth functions
solving:
(3.8)
where
and
Moreover, there is a positive time T such that the solutions satisfy the following properties:
1)
is uniformly bounded in
.
2)
is a Cauchy sequence in
.
Proof (Proof of Lemma 3.2) Since all the data
and
belong to
, Lemma 2.7 indicates that for all
, Equation (3.8) has a global solution in
.
Obviously, we can get the following inequality:
Using Lemma 2.6 again, for
, we can see that:
with
And with the help of Proposition 2.4 again, we have:
Combining the above inequalities, we get:
Therefore, if we define
and
, then
(3.9)
and
(3.10)
Choosing
, by induction, we show that:
(3.11)
In fact, suppose that (3.11) is valid for k, then for
, we have:
(3.12)
and it is easy to see that:
Submitting (3.11) and (3.12) into (3.9) and (3.10), we obtain:
in the last inequality we used that
.
Therefore,
is uniformly bounded in
. Using Equation (3.8) and the similar argument in the proof Lemma 3.1, one can easily prove that
is uniformly bounded in
, i.e. the sequence
is uniformly bounded in
.
Then, let us show that
is a Cauchy sequence in
. For all
, from (3.8), we can obtain that:
and
with
Using the fact that
is an algebra and the operator
is a S-multiplier, and let
, for every
, we have:
moreover, using Lemma 2.6 again, we get:
By using Proposition 2.1, we can obtain that:
and
In view of
being uniformly bounded in
, and combining the above inequalities, one may find a positive constant
independent of
such that:
for all
. Moreover, using the induction procedure with respect to the index k, we have:
Since that
may be bounded independently of j, we conclude to the existence of some new constant
independent of
such that:
Thus, we have proved that
is a Cauchy sequence in
. Hence, the proof of Lemma 3.2 is complete.
Finally, we prove the existence and uniqueness for (1.2) in Besov space.
Proof (Proof of Theorem 1.1) By using lemma 3.2,
is a Cauchy sequence in
, so it converges to some limit function
. Next, we have to prove that
and solves (1.2). Using lemma 3.2 again, we can see that
is uniformly bounded in
. Fatou property for Besov spaces (Proposition 2.5 (6)) insures that
also belongs to
.
On the other hand, as
converges to
in
, an interpolation argument guarantees that the convergence holds in
, for any
.
Passing limit in (3.8) reveals that
satisfy system (1.2). In view of the fact that
belongs to
, for
,
and
are algebras, we obtain that the right-hand side of the equation:
belongs to
, and the right-hand side of the second equation:
belongs to
.
In particular, for the case
, Lemma 2.6 implies that
for any
. Finally, by using the equation again, we see that
if
, and in
otherwise. Therefore, the pair
.
The continuity with respect to the initial data in:
for all
, can be obtained by Lemma 3.1 and a simple interpolation argument. The case
can be proved through the use of a sequence of viscosity approximation solutions
for System (1.2) which converges uniformly in:
gives the continuity of solution
in
. Hence, the proof of Theorem 1.1 is complete.
4. Ill-Posedness in Besov Space
Next, we select the appropriate initial data to complete the proof of the Theorem 1.3. And we choose the initial data:
here
and
is an non-negative, even and real-valued function satisfying:
It is easy to prove that:
then it can be verified for
,
(4.1)
so we have:
Similarly, we have:
(4.2)
Next, we give several estimates that play an important role in the proof of the Theorem 1.3.
Lemma 4.1. Let
. Then, for the above constructed initial data
, we have:
(4.3)
and
(4.4)
for n large enough.
Proof We only show (4.4), because (4.3) can be obtained in a similar way.
According to (4.1), we get:
therefore, one has:
Since
is a real valued continuous function on
, then there exists
, we have:
(4.5)
for any
.
Therefore, we derive from (4.5) that:
By choosing n large enough such that
, then we can yield (4.4). So, Lemma 4.1 has been proved.
Lemma 4.2. Let
. For the above constructed initial data
, then there exists some
, for
, we have:
(4.6)
Proof Since
and according to the local existence result (see Theorem 1.1), hence the system (1.2) has a unique solution
for some
, and
(4.7)
Using the differential mean value theorem, the Minkowski inequality, Proposition 2.4 together with (4.7) for
, we have:
and
Therefore, we have completed the proof of Lemma 4.2.
Lemma 4.3. Under the assumption of Theorem 1.3, for all
, we have:
(4.8)
where
Proof We denote that:
Firstly, using the differential mean value theorem and the Minkowski inequality for
, using Proposition 2.4, Proposition 2.5 and (4.7), we get:
(4.9)
and
(4.10)
Thus, the proof of Lemma 4.3 is completed.
Proof (Proof of Theorem 1.3.) Based on the definition of the Besov norm, we have:
(4.11)
Since
Using Proposition 2.5, Lemmas 2.8 and (4.2), we deduce that:
and
Taking above estimates into (4.11), we have:
And using Lemma 4.1 and Lemma 4.3, we have:
Choosing n large enough such that
, then we get:
Picking
when
, we have:
Likewise, we have:
This completes the proof of Theorem 1.3.
5. Gevrey Regularity for System (1.2)
In this section, we will apply nonlinear Cauchy-Kowalevski theory to establish the existence of analytic solutions to system (1.2). To this purpose, it is necessary to introduce the Cauchy-Kowalevski theorem.
Theorem 5.1. (See [11] [12] ) Let
be a scale of decreasing Banach spaces, such that for any
, we have
with
. Consider the Cauchy problem:
(5.1)
Let
and
. For given
, assume that F satisfies the following conditions:
(H1) If for any
, the function
is holomorphic on
and continuous on
with values in
and
then
is a holomorphic function on
with values in
.
(H2) For any
and
, there exists a positive constant L depending on
and R such that:
(H3) There exists a
depending on
and R such that for any
,
Then, there exists a
and a unique function
to the Cauchy problem (5.1), which is holomorphic in
with values in
for every
.
We would like to include four properties of the spaces
, which will be used in the proof of Theorem 1.5 (the proofs of these properties can be find in [12] ).
Proposition 5.2. Let
,
and
. From Definition 1.3, one can check that
↪
,
↪
and
↪
.
Proposition 5.3. Let s be a real number and
. Assume that
. Then, we have:
(5.2)
Proposition 5.4. (Product acts on Sobolev-Gevrey spaces with
) Let
and
. Then,
is an algebra. Moreover, there exists a constant
such that:
(5.3)
Proposition 5.5. Let
and
. There exists a constant
such that”
(5.4)
Now, we use all the above tools to prove Theorem 1.5.
Proof (Proof of Theorem 1.5) Assume that:
By using Definition 1.3 and Proposition 5.2, we can see that
is a scale of decreasing Banach spaces for a fixed
,
and
. Moreover, for
,
,
and the estimates (5.2)-(5.4), we have:
and
where
is a constant depends only on
. Therefore, we can obtain that:
Similarly, for any
, we also can obtain that:
Therefore, these spaces and
satisfy condition (H1) and (H3) in Theorem 5.1.
Next, in order to prove our desire result, it suffices to show that
satisfies the condition (H2) in Theorem 5.1. Assume that
and
with
, for
,
, we can see that:
and
Thus,
where in the last inequality we apply the fact that
and
. From the above inequality, for
,
,
, we verify that
satisfies the condition (H2) of Theorem 5.1 with
. This completes the proof of Theorem 1.5.