Lower Bounds of Decay Rates for Solution to the Single-Layer Quasi-Geostrophic Model ()
1. Introduction
We study the initial value problem of a single-layer quasi-geostrophic model with viscosity (see e.g. [1] [2] ):
(1)
where
denotes the stream functions, F is the rotational Froude number,
is the Coriolis parameter,
is the Reynolds number,
is the viscosity terms, and the nonlinear term
is defined by
The quasi-geostrophic
-plane model is considered as a simplification of the shallow-water equations when the Rossby number is small and the magnitude of bottom topography variations is comparable to the Rossby number.
A generalization of the single-layer quasi-geostrophic model (1) is the two-layer (multi-layer) quasi-geostrophic model. There are many research results on the existence, uniqueness and long-time behavior of the solutions. For results in this regard, please refer to [3] [4] [5] [6] .
In 2010, X. Pu and B. Guo [7] proved global existence of weak solutions for the fractional quasi-geostrophic equation with
replaced by
in Equation (1) and they also obtained long-time behavior
of the solution when
.
In 2023, H. Li, J. Li and J. Zhang [8] showed that the existence and uniqueness of the global smooth solution to the Cauchy problem of Equation (1). They also obtain the upper bound of the decay estimates of the solution in
. And show that if
and
be an integer, then for any multi-index
, the solution
satisfies
and
Following the work of [8] , we study the lower bound of the decay estimate of the solution for the initial-value problem (1) equipped with the initial data
(2)
To obtain the lower bound of the decay estimate of the solution, we not only use the Fourier splitting method which is originated from Schonbek [9] [10] and is improved by Zhang [11] , we also need to construct a new expression
where
and
are the solutions of Equation (1) itself and the linear part of Equation (1) with the given initial data
, respectively. In addition, in order to ensure that this method works, we must show the nonlinear solution
decays faster than the linear part. To obtain the lower decay bound, we define the set for
,
. We have the following theorem.
Theorem 1. If the initial data
and
be an integer, then for any multi-index
, the solution
of Equation (1) satisfies
This paper is organized as follows. In Section 2, we review the notations used throughout the paper. In Section 3, we consider the upper and lower bounds of the linear part of Equation (1). In Section 4, we estimate the upper bounds of the difference between the solution of linear part of Equation (1) and the solution of Equation (1). Furthermore, we show that the decay rate to the lower bounds of Equations (1)-(2).
2. Notations and Preliminaries
The Fourier transform
of a tempered distribution
on
is defined as
where
.
For
, we denote by
the Lebesgue space equipped the norm
and
For
,
denotes the nonhomogeneous Sobolev space defined by
where
is the Fourier transform of u.
3. Decay Estimates for the Linear Part
In this section, we first study the lower and upper decay bounds on the decay rates for the following linear quasi-geostrophic model:
(3)
(4)
To solve Equations (3)-(4), we take the Fourier transform for the above equations to get
then we get
(5)
Since F is not zero, we note the following facts
(6)
and
(7)
where
is an arbitrary positive number.
We now recall some estimates for the solutions of the liner equation which will be useful later.
Lemma 2. Let
and
be an integer. Then the solution
of Equations (3)-(4) satisfies
(8)
where the constant C depends on
and the initial data.
Proof. When
, from Equation (5), we have
where the following observation is used in the last step
Therefore Lemma 3.1 is proved. □
We recall the set for
,
(9)
Lemma 3. Let
and
be an integer. Then the solution
of problem (3)-(4) satisfies
(10)
where the constant C depends on
and initial data.
Proof. From Equations (5), (6) and (9), we have
where the following observation is used in the last step
Therefore Lemma 3.2 is proved. □
Lemma 4. Let
and
be an integer. Then the solution
of problem (3)-(4) satisfies
(11)
where the constant C depends on
and initial data.
Proof. When
, we use Hausdorff-Young inequality to obtain
where the following observation is used in the last step
Therefore Lemma 3.3 is proved. □
4. Lower Bound on the Decay Rates
In this section, we study the lower bound of the decay rate of the solution of the nonlinear quasi-geostrophic equation. In [8] , the existence of smooth solutions has been obtained, and the following upper bound is also proved.
Lemma 5. [8] Let
,
is an integer, and
is the solution of Equation (1), then for any multi-index
, we have the decay estimates
(12)
(13)
Combining equation
. Since we have already obtained the lower decay bound for
in Section 3, the key point is to estimate estimate the upper bound for
. The equations for
can be written in the form
(14)
(15)
which implies that
(16)
This involves inequalities with respect to
, please refer to [8] for detailed proof.
Proof of Theorem 1.1. We first show
(17)
We multiply Equation (14) with
, then integrate it in
with
, we obtain
(18)
Applying Plancherel’s theorem to the Equation (18), we have
(19)
Let
Then the second term of Equation (19) becomes
(20)
Inserting Equation (20) into Equation (19), using the estimate (16), we have
(21)
And both sides of Equation (21) are multiplied by
, we get
Integrating the above inequality over interval
, we get
Therefore
where the constant C depends on
and initial data.
Next, we want to prove
(22)
We multiply Equation (14) with
, then integrate it in
with
, we obtain
(23)
Applying Plancherel’s theorem to the equations Equation (23), we have
(24)
Let
Then the second term of Equation (24) becomes
(25)
Inserting Equation (25) into Equation (24), using the estimate (16), we have
(26)
And both sides of Equation (26) are multiplied by
, we get
Integrating the above inequality over interval
, we get
Therefore
Then we will prove
(27)
We multiply Equation (14) with
, then integrate it in
with
, we obtain
(28)
Applying Plancherel’s theorem to the equations Equation (28), we have
(29)
Let
Then the second term of Equation (29) becomes
(30)
Inserting Equation (30) into Equation (29), using the estimate (16), we have
(31)
And both sides of Equation (31) are multiplied by
, we get
Integrating the above inequality over interval
, we get
Therefore
Finally, applying the same processing method as above, we can get
Because the way of proof is similar to Equations (17), (22) and (27), it is omitted here. □