Gevrey Regularity and Time Decay of Fractional Porous Medium Equation in Critical Besov Spaces ()
1. Introduction
In this paper, we consider existence and regularity of mild solutions for the initial value problem of the following fractional porous medium equation (FPME) in
for
:
(1)
where u and
denote the density and gas pressure respectively, while
denotes the given initial data. And
. The positive operator
can be defined by
where
is a normalization constant. By the Fourier transform, we can also get a very simple alternative representation, as
, where
is the Fourier transform and
is the inverse Fourier transform, respectively. To simplify the notation, we consider
. For FPME research, there have been many results. For example, in 2019, Feng and Liu [1] used generalized Riccati transformation technique and the differential inequality method to obtain the oscillation criteria of a class of nonlinear fractional differential equations.
If
,
,
,
, the system (1) reduces to the classical Keller-Segel model
(2)
The Keller-Segel system has been established as the model of chemotaxis by Keller and Segel [2]. Biler and Karch [3] demonstrated local and global solutions with small initial data of the equation in critical Lebesgue space
for
. Biler and Wu [4] studied global well-posedness of the equation with small initial data in the critical Besov spaces
for
. Zhao, Cui and Liu [5] proved small data global existence and large data local existence of solutions in critical Besov space
for
and
.
The purpose of this paper is to prove the well-posedness and Gevrey analyticity of Equation (1) in the Besov spaces. When
,
,
, the equation degenerates to the classic form of the Porous Media Equation. We refer it by PME, and the classical properties of this equation can be found in literature [6]. It can describe the movement of an ideal gas flowing through a porous medium or be regarded as a kind of non-local quadratic evolution problem. This equation is widely used in describing Brownian diffusion, gas particle interaction, and biotaxis. When
,
,
, the equation was first proposed by Caffarelli and Vázquez [7]. It has been proved by them that, when the initial value
is a bounded function and exponentially decays at infinity, the equation has a weak solution. In [8], they studied the
regularity of the weak solution of the equation when
,
. For the detailed information about the solution of this equation, please refer to literature [9].
Secondly, we present some results of Gevrey analyticity in recent years. In 1989, Foias and Temam discovered this method and employed it for the first time to study the analyticity of the Navier-Stokes equations with space periodicity boundary condition, see [10] [11]. In the following periods, a few more authors made full use of this method, and extended it to various functional spaces and equations. For instance, in 2001, Zhan [12] obtained the Gevrey regularity of the solution of the superconducting phase-locked equation. Ferrari and Titi in [13] studied the regularity of solutions to a large class of analytic nonlinear parabolic equations on the two-dimensional sphere. Chueshov and Polat in [14] studied the Gevrey regularity of the global attractor of the dynamical system generated by the generalized Benjamin-Bona-Mahony equation with periodic boundary conditions. Recently, the well-posedness and spatial regularity of the classic K-S equation have been completed by Zhao [15].
In this paper, we will consider well-posedness and Gevrey analyticity of the fractional porous medium Equation (1) with initial data in critical Besov spaces
for
and
. To address the equations, we consider the following integral equations:
(3)
where
. We use the following the contraction mapping to get the solution of (3):
Then we utilize the Gevrey class regularity to certify analyticity of solutions. As an application, we get time decay rates in Besov spaces for global solutions.
Let us denote by
the Fourier multiplier whose symbol is given by
.
The overall structure of the article is shown below. In Section 2, we review the Littlewood-Paley dyadic decomposition theory and the definition of Besov spaces. In Section 3 and Section 4, we prove Theorem 3.3 and Theorem 3.5 respectively by the standard fixed point argument.
2. Notations and Preliminaries
First of all, let’s introduce some notations mentioned in the paper. We will consider the solution of system (1) in
. For two constants A and B, if there is a finite constant C whose value of each line may vary such that
, we denote it as
. For a quasi-Banach space X and for any
, we use standard notation
to denote the quasi-Banach space of Bochner measurable functions f from
to X endowed with the norm
Especially, if
, we still use
rather than
.
Let us introduce some basic knowledge on Littlewood-Paley theory and Besov spaces.
Let
be a radial positive function such that
Define the frequency localization operators as follows:
here
and
.
By Bony’s decomposition, we can split the product
into three parts:
with
Let us now define the Besov spaces as follows.
Definition 2.1 For
(or
if
),
,
and
, we define the homogeneous Besov space
as
Here the norm changes normally when
or
, and
is the set of all polynomials. If
and
(or
if
), then
is defined as the subset of distributions
so that
when
.
Definition 2.2 For
,
,
, we set (with the usual convention if
)
We then define the space
as the set of temperate distributions f over
such that
in
and
.
Lemma 2.3 [16] Let
be a ball, and
a ring in
. There exists a constant C such that for positive real number
, any nonnegative integer k and any couple of real numbers
with
, we know
(4)
(5)
Lemma 2.4 [16] Let f be a smooth function on
which is homogeneous of degree m. Then for any
,
, and
, or
and
, the operator
is continuous from
to
.
Lemma 2.5 [17] Let
be a ring in
. There exist two positive constants
and
such that for any
and any couple
of positive real numbers, we have
(6)
3. The Case
: Well-Posedness, Regularity and Time Decay
In this chapter we will demonstrate well-posedness and Gevrey analyticity of Equation (1) in critical Besov spaces
for
. Let
,
. Suppose
. When
, let us prove the above properties.
3.1. The Case
: Well-Posedness
First of all, we give a priori estimate for the following fractional dissipative equation:
(7)
Proposition 3.1 ( [4] ) Let
,
and
. There exists a constant
depending only on
and n such that for any
, we have
(8)
Next we present the following vital bilinear estimates.
Lemma 3.2 Let
,
,
,
,
with
. There holds
(9)
Proof. According to Bony’s paraproduct decomposition, we find that
Firstly, we estimate
. Applying Hölder’s inequality and Lemmas 2.3, 2.4, there holds
(10)
Multiplying (10) by
and taking
-norm, we get
(11)
Similarly, for
, utilizing Hölder’s inequality and Lemmas 2.3, 2.4, when
, we find that
(12)
Now we deal with the term
. For one thing, we premeditate the case of
, firstly using Bernstein’s inequality, then by Hölder’s inequality with
,
(13)
For another case:
,
,
, firstly using Bernstein’s inequality, then by Hölder’s inequality, and exploiting Bernstein’s inequality again,
(14)
Under the assumption of Lemma 3.2, we have
Therefore we deduce from the estimate (13)-(14) that for all
,
Now we are in a position to prove well-posedness of the system (1) in the case that
and
.
Theorem 3.3 Let
,
,
and
. Then we have a
such that the system (1) has a unique solution
, where
with
If
, we have
Furthermore, if the initial value
is small enough in
, then
.
Proof. Define the map
(15)
in the metric space
:
with
Exploiting Proposition 3.1 and Lemma 3.2 by choosing
, for any
, we get
(16)
and
(17)
Now by the standard contraction mapping argument ( [16] ), there exists a unique solution
for T small appropriately. And from Proposition 3.1, we get
Thus the solution u can be extended to the maximum time
, we have
If
and
, we need to premeditate the following integral equation
Similarly, we show that
Using the contraction mapping argument as in (16), the solution exists on
. Taking T closed to
and by the local existence, the solution exists on some time larger than
, which contradicts the maximum time
. Besides, we can take
in (16) and (17) if
is small enough. Thus we proved Theorem 3.3.
3.2. The Case
and
: Well-Posedness
In this part, we study the endpoint case
. The crucial bilinear estimates are as follow.
Lemma 3.4 For
,
, we have
(18)
Proof. We consider the following the estimation of
:
Therefore, according to Definition 2.2, there holds
(19)
In the same way, for
, when
, we obtain
(20)
Now we treat with
.
(21)
Under the hypothesis
, we have
. Hence considering the estimate (21), and multiplying
to the resulting inequality, then taking
norm yields
(22)
Due to (19), (20) and (22), we come to the (20). The proof of Lemma 3.9 is done.
Theorem 3.5 Let
and
. Assume that
is small enough. When
, then there is a unique solution to this system (1) that satisfies
Proof. To proving Theorem 3.5, we consider the space
. From the mapping (15), Proposition 3.1 and Lemma 3.4, one has
(23)
Applying the standard contraction mapping argument as before, we can show that system (1) confesses a unique solution in
if
is sufficiently small.
3.3. The Case
: Gevrey Analyticity
In this part, we calculate analyticity of system (1) with initial data in
for
and
. At the very beginning, let’s review the following three results.
Lemma 3.6 (Lemma 3.2 in [18] ) We consider the operator
for
. Then
is either the identity operator or is the Fourier multiplier with
kernel whose
-norm is bounded independent of s and t.
Lemma 3.7 (Lemma 3.3 in [18] ) Assume that the operator
for
. Then
is the Fourier multiplier which maps boundedly
for
, and its operator norm is uniformly bounded with respect to
.
Lemma 3.8 (Proposition 3.5 in [15] ) Let
,
,
and
. Assume that
and
. Then (7) has a unique solution
. In addition, there exists a constant
depending only on
and n so that for any
, we see
(24)
Let’s take
represent for
. Now we employ the operator
to certificate a result similar to Lemma 3.2.
Lemma 3.9 Let
,
,
with
. Then for any
,
, there holds
(25)
In addition, if we choose
, then (25) also holds for
.
Proof. Set
. Then according to the idea of Lemma 3.2, firstly, by Bony’s paraproduct decomposition to get
To estimate the items
, we draw support a thought in [19] and [20] and find the coming bilinear operator
of the descriptor
Based on the symbols
,
, and
, we can split the integration domain into subdomains. For
,
,
such that
,
,
, we define
is a characteristic function on the domain D. Next we redefine
as
By this way, we bring in the monodimensional operators:
and
Furthermore for
, we have the operator
(26)
The above tensor product (26) means that the j-th operator acts on the j-th variable of the function
. By calculation we have the following identity:
Observing that for
,
and
,
have to be a member of the following set:
Notice that
,
,
and the elements in
are Fourier multipliers in
for
, and the operators
and
are combination of identity operators and those Fourier multipliers. It is clear that the operators
and
are bounded linear operators on
with
, and the relevant operator norm of
is bounded independent of
. Moreover, for
, we have
Because of the new restriction of the bilinear operator
, we can do prove Lemma 3.9 based on Lemma 3.2. In fact, let’s take the example of
:
The rest of terms are similarly estimated. Therefore, we get the estimated formula (25).
Theorem 3.10 Let
. Then the solution obtained in Theorem 3.3 satisfies
(27)
What is more, if
is small enough, then
.
Proof. Taken together with Lemma 3.8 and Lemma 3.9, using the mapping (15), we get
Based on estimate the above estimate and the contraction mapping argument, we accomplish the proof.
3.4. The Case
and
: Gevrey Analyticity
Theorem 3.11 Let
. Then the solution obtained in Theorem 3.5 satisfies
(28)
Proof. Suppose
. Then
meets the following integral equation
(29)
Since the symbol
is uniformly bounded for all
and decays exponentially for
, the Fourier multiplier
maps uniformly bounded from
to
for all
. Using Young’s inequality, there holds
For the nonlinear part, although the operators
and
don’t map
to
bounded, these operators are bounded in
when localized in dyadic blocks. Based on the calculations line from (19) to (22) and by the continuing the same line as the proof of Lemma 3.9, and notice that Lemma 3.9, we have
This finishes the proof, as expected.
3.5. Decay Rate of Solution
In this part, we concentrate on the decay rate estimates of solutions gained in Theorem 3.3, Theorem 3.5, Theorem 3.10 and Theorem 3.11. The proof is based on the following consequence.
Lemma 3.12 ( [15] ) For all
and
, the operator
is the convolution operator with a kernel
for all
. Moreover
(30)
where
.
Now we have show that if the initial data
is small enough in critical Besov spaces
for either
,
and
or
,
and
, then the solution is in the Gevrey class. As a result, we obtain the time decay of global mild solution for all
in Besov spaces:
This proof is done.
4. The Case
: The Proof of Theorem
In this part, we will prove the case
for the system (1) with initial data in critical spaces
.
4.1. The Case
: Well-Posedness
For any initial data
, we think the resolution space
. Slightly modifying the proof Lemma 3.4, we obtain the consequence as follows.
Lemma 4.1 For any
, when
, we find
(31)
Proof. Firstly, we calculate the estimation of
:
(32)
Multiplying
to (32), then taking
norm to the resulting inequality, we obtain
Meanwhile, for
, we obtain
Now we treat with
. We discuss in two situations. One is the case of
:
the other is that
, and satisfies that
and
, that is
Multiplying
by
and taking
norm to them, we get
Theorem 4.2 Let
. Then there exists a sufficiently small
such that the system (1) has a unique solution
.
Proof. According to the Proposition 3.1, Lemma 4.1 and the mapping (15), there holds
By the contraction mapping argument as before, the system (1) admits a unique solution in
for small
.
4.2. The Case
: Well-Posedness
In the case
, the resolution space
is not able to be adjusted to the system (1). We use the resolution space
.
Theorem 4.3 Let
. Assume that
is small enough. Then there is a unique solution to the system (1) that satisfies
Proof. By Proposition 3.1,
(33)
Next, through Lemma 3.4, we obtain
(34)
Therefore think over the mapping (15), we infer from Proposition 3.1, (33) and (34) that
(35)
Thus by the standard contraction mapping argument, system 1 admits a unique solution in
for small initial data
.
4.3. The Case
: Gevrey Analyticity
Theorem 4.4 Let
. Then the solution acquired in Theorem 4.2 meets
Proof. The dissipation term
is not sufficiently strong to overcome the operator
. Thus, we demand to define a more precise Gevrey operator. Because of
, we choose
So
fulfills the integral equation as follows:
(36)
Because the symbol
is uniformly bounded and decays for any
, the operator
is a Fourier multiplier map
to
(
) with uniformly bound independent of t. By Proposition 3.1, there holds
(37)
With regard to the nonlinear term, we define
Hence by the new boundedness properties of the operator
and the bilinear operator
of the form
we can get the Gevrey analyticity of the global solution as before. In fact, the corresponding operators
and
are bounded independent of
, hence, in regard to
,
, we have
4.4. The Case
: Gevrey Analyticity
In order to resolve the Gevrey analyticity of the solution in the case
, the following a priori estimate is true:
(38)
Theorem 4.5 Let
. Then the solution is given in theorem 4.3 which satisfies
(39)
Proof. Because from previous analysis, for
, we know
(40)
With respect to the nonlinear part, we find out the bilinear operator
of the form
Although the above operator is not bounded from
to
, by the similar analysis as before, we can conclude the results line by line.
4.5. Decay Rate of Solution
In this part, we hold up the decay rate estimates of solutions gotten in Theorem obtained in the fourth part. On the strength of Lemma 3.12, we know that for all
, the operator
is the convolution operator with a kernel
for all
. What’s more,
(41)
where
. Theorem obtained in fourth part shows that if
is sufficiently small, then the solution is in the Gevrey class. As a result, for
, utilizing (41), we acquire
5. Conclusions
In summary, we can get the following conclusions:
Conclusion 5.1 Let
,
. Suppose
. When
, there exists the following results:
(1) (Well-posedness for
) Let
,
,
and
. Then we have a
such that the system (1) has a unique solution
, where
with
If
, we have
Furthermore, if the initial value
is small enough in
, then
.
(2) (Well-posedness for
) Let
and
. Assume that
is small enough. When
, then there is a unique solution to this system (1) that satisfies
(3) (Analyticity for
) Let
. Then the solution obtained in (1) satisfies
What is more, if
is small enough, then
.
(4) (Analyticity for
) Let
,
. Then the solution obtained in (2) satisfies
(5) (Decay rate for
) With any
,
or
and
, the global solution acquired in (1) and (2) satisfies
where
.
Corresponding to Conclusion 5.1, when
, we can get the results as follows.
Conclusion 5.2 Let
and suppose
. When
,
, there exists results as follows:
(1) (Well-posedness for
) Let
. Then there exists a sufficiently small
such that the system (1) has a unique solution
.
(2) (Well-posedness for
) Let
. Assume that
is small enough. Then there is a unique solution to the system (1) that satisfies
(3) (Analyticity for
) Let
. Then the solution acquired in (1) meets
(4) (Analyticity for
) Let
. Then the solution is given in (2) which satisfies
(5) (Decay rate for
) With any
,
or
and
, the global solution got in (1) and (2) fits
where
.
Acknowledgements
The author would like to thank collaborator, editors and reviewers for many helpful discussions and suggestions. This work was partially supported by National Natural Science Foundation of China (No. 11601223).