On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin spaces
Boundary Value Problems volume 2014, Article number: 95 (2014)
In this paper, we prove the local well-posedness for the incompressible porous media equation in Triebel-Lizorkin spaces and obtain blow-up criterion of smooth solutions. The main tools we use are the Fourier localization technique and Bony’s paraproduct decomposition.
In this paper, we are concerned with the incompressible porous media equation (IPM) in ( or 3):
where , , θ is the liquid temperature, u is the liquid discharge, p is the scalar pressure, k is the matrix of position-independent medium permeabilities in the different directions, respectively, divided by the viscosity, g is the acceleration due to gravity, and is the last canonical vector . For simplicity, we only consider .
By rewriting Darcy’s law we obtain the expression of velocity u only in terms of temperature θ[1, 2]. In the 2D case, thanks to the incompressibility, taking the curl operator first and the operator second on both sides of Darcy’ law, we have
thus the velocity u can be recovered as
Through integration by parts we finally get
where the kernel is defined by
Similarly, in 3D case, applying the curl operator twice to Darcy’s law, we get
where , thus
We observe that, in general, each coefficient of (with t as parameter) is only the linear combination of the Calderoń-Zygmund singular integral (with the definition see the sequel) of θ and θ itself. We write the general version as
where , , , are all operators mapping scalar functions to vector-valued functions and equals a constant multiplication operator whereas means a Calderón-Zygmund singular integral operator. Especially the corresponding specific forms in 2D or 3D are shown as (1.2) or (1.3).
We observe that the system (IPM) is not more than a transport equation with non-local divergence-free velocity field (the specific relationship between velocity and temperature as (1.4) shows). It shares many similarities with another flow model - the 2D dissipative quasi-geostrophic (QG) equation, which has been intensively studied by many authors [3–8]. From a mathematical point of view, the system (IPM) is somewhat a generalization of the (QG) equation. Very recently, the system (IPM) was introduced and investigated by Córdoba et al. In , they treated the (IMP) in 2D case and obtained the local existence and uniqueness in Hölder space for by the particle-trajectory method and gave some blow-up criteria of smooth solutions. Recently, they proved non-uniqueness for weak solutions of (IPM) in . For the dissipative system related (IPM), in , the authors obtained some results on strong solutions, weak solutions and attractors. For finite energy they obtained global existence and uniqueness in the subcritical and critical cases. In the supercritical case, they obtained local results in , and extended to be global under a small condition , for , where c is a small fixed constant.
Recently, Chae studied the local well-posedness and blow-up criterion for the incompressible Euler equations [10, 11], and quasi-geostrophic equations  in Triebel-Lizorkin spaces. As is well known, Triebel-Lizorkin spaces are the unification of several classical function spaces such as Lebegue spaces , Sobolev spaces , Lipschitz spaces , and so on. In , the author first used the Littlewood-Paley operator to localize the Euler equation to the frequency annulus , then obtained an integral representation of the frequency-localized solution on the Lagrangian coordinates by introducing a family of particle-trajectory mappings defined by
where v is a divergence-free velocity field and is a frequency projection to the ball (see Section 2). He also used the following equivalent relation:
to estimate the frequency-localized solutions of the Euler equations or quasi-geostrophic equations in Triebel-Lizorkin spaces. However, it seems difficult to give a strict proof for the above equivalent relation (1.6) and its related counterpart due to the lack of a uniform change of the coordinates independent of j. To avoid this trouble, Chen et al. introduced a particle trajectory mapping independent of j defined by
and then established a new commutator estimate to obtain the local well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces.
In this paper, we will adapt the method of Chen et al. to establish the local well-posedness for the incompressible porous media equation (1.1) and to obtain a blow-up criterion of smooth solutions in the framework of Triebel-Lizorkin spaces.
Now we state our result as follows.
Theorem 1.1 (i) Local-in-time existence. Let, . Assume that, then there existssuch that (1.1) has a unique solution.
Blow-up criterion. The local-in-time solutionconstructed in (i) blows up atin, i.e.
if and only if
Let and . Choose two nonnegative smooth radial functions χ, φ supported, respectively, in ℬ and such that
We denote , and . Then the dyadic blocks and can be defined as follows:
Formally, is a frequency projection to the annulus , and is a frequency projection to the ball . One easily verifies that with our choice of φ
With the introduction of and , let us recall the definition of the Triebel-Lizorkin space. Let , , the homogeneous Triebel-Lizorkin space is defined by
and denotes the dual space of and can be identified by the quotient space of with the polynomials space .
For and , we define the inhomogeneous Triebel-Lizorkin space as follows:
We refer to  for more details.
Lemma 2.1 (Bernstein’s inequality) 
Let. There exists a constant C independent of f and j such that, for all, the following inequalities hold:
For any, there exists a constantsuch that the following inequality holds:
Let, , or, then there exists a constant C such that
Letwith. Suppose, then there exists a constant C such that the following inequality holds:
Let, or, and f be a solenoidal vector field. Then for
The classical Calderón-Zygmund singular integrals are operators of the form
where Ω is defined on the unit sphere of , , and is integrable with zero average and where . Clearly, the definition is meaningful for Schwartz functions. Moreover if , is bounded, .
The general version (1.4) of the relationship between u and θ is in fact ensured by the following result (see e.g.).
Lemma 2.3 Letbe a homogeneous function of degree 0, andbe the corresponding multiplier operator defined by, then there existandwith zero average such that for any Schwartz function f,
Remark 2.1 Since , the Fourier multiplier of the operator is rather clear. In fact, each component of its multiplier is the linear combination of the term like , , which of course belongs to and is homogeneous of degree 0.
3 Proof of Theorem 1.1
We divide the proof of Theorem 1.1 into several steps.
Step 1. A priori estimates.
Taking the operation on both sides of the first equation of (1.1), we have
Let be the solution of the following ordinary differential equations:
Then it follows from (3.1) that
which implies that
Multiplying , taking the norm on both sides of (3.4), we get by using the Minkowski inequality
Next, taking the norm with respect to on both sides of (3.5), we get by using the Minkowski inequality that
Using the fact that is a volume-preserving diffeomorphism due to , we get from (3.6) that
Thanks to Proposition 2.3, the last term on the right side of (3.7) is dominated by
where we used (1.4) and the boundedness of the Calderón-Zygmund singular integral operator on .
Now from (1.1) we have immediately
for all , since div . Summing up (3.9) and (3.10) yields
which together with the Gronwall inequality gives
Step 2. Approximate solutions and uniform estimates.
We construct the approximate solutions of (1.1). Define the sequence by solving the following systems:
We set and let be the solutions of the following ordinary differential equations:
for each . Then, following the same procedure of estimate leading to (3.11), we obtain
where we used the fact that , Sobolev embedding theorem for , (1.4) and the boundedness of the Calderón-Zygmund singular integral operator on . Equation (3.15) together with the Gronwall inequality implies that
for some independent of n. Thus, if we choose such that
we have, for any ,
by the standard induction arguments. Then, . Moreover, it follows from Proposition 2.1 that
by Sobolev embedding and the boundedness of the Calderón-Zygmund singular integral operator on , and then
which implies that . This together with (3.17) gives the uniform estimate of in n.
Step 3. Existence.
We will show that there exists a positive time () independent of n such that and are Cauchy sequences in . For this purpose, we set
Then, it follows that satisfies the equations
Applying to the first equation of (3.19), we get
Exactly as in the proof of (3.7), we get
where we used Proposition 2.1, Proposition 2.3, the embedding , and the boundedness of the Calderón-Zygmund singular integral operator on . Thanks to the Fourier support of , we have
Now, we estimate the norm of . Multiplying on both sides of the first equation of (3.19), and integrating the resulting equations over , we obtain
which together with (3.21) and (3.22) gives
Equation (3.23) together with (3.17) yields
where . Thus, if , then
This implies that
Thus, is a Cauchy sequence in . By the standard argument, for , the limit solves (1.1) with the initial data . The fact that follows from the uniform estimate (3.18).
Step 4. Uniqueness.
Consider is another solution to (1.1) with the same initial data. Let and . Then δθ satisfies the following equation:
In the same way as the derivation in (3.24), we obtain
for sufficiently small T. This implies that , i.e., .
For the a priori estimate (3.12), we only need to dominate and . From Proposition 2.2 and the boundedness of the Calderón-Zygmund operator from into itself, we have
Thus, the a priori estimate (3.12) gives
By the Gronwall inequality
Therefore, if , then .
On the other hand, it follows from the Sobolev embedding for that
Then implies .
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This work was supported by the National Natural Science Funds of China for Distinguished Young Scholar under Grant No. 50925727, The National Defense Advanced Research Project Grant Nos. C1120110004, 9140A27020211DZ5102, the Key Grant Project of Chinese Ministry of Education under Grant No. 313018, and the Fundamental Research Funds for the Central Universities (2012HGCX0003).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Yu, W., He, Y. On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin spaces. Bound Value Probl 2014, 95 (2014). https://doi.org/10.1186/1687-2770-2014-95