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.
MSC: 76S05, 76D03.
Keywords:well-posedness; incompressible porous media equation; blow-up criterion; Fourier localization; Bony’s paraproduct decomposition; Triebel-Lizorkin space
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
Similarly, in 3D case, applying the curl operator twice to Darcy’s law, we get
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
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.
if and only if
We refer to  for more details.
Lemma 2.1 (Bernstein’s inequality) 
The classical Calderón-Zygmund singular integrals are operators of the form
The general version (1.4) of the relationship between u and θ is in fact ensured by the following result (see e.g.).
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.
Then it follows from (3.1) that
which implies that
Thanks to Proposition 2.3, the last term on the right side of (3.7) is dominated by
Now from (1.1) we have immediately
which together with the Gronwall inequality gives
Step 2. Approximate solutions and uniform estimates.
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
Step 3. Existence.
Exactly as in the proof of (3.7), we get
which together with (3.21) and (3.22) gives
Equation (3.23) together with (3.17) yields
This implies that
Step 4. Uniqueness.
In the same way as the derivation in (3.24), we obtain
Thus, the a priori estimate (3.12) gives
By the Gronwall inequality
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.
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).
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