On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin spaces
© Yu and He; licensee Springer. 2014
Received: 22 January 2014
Accepted: 24 March 2014
Published: 6 May 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.
Keywordswell-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 .
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.
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.
- (ii)Blow-up criterion. The local-in-time solutionconstructed in (i) blows up atin, i.e.
and denotes the dual space of and can be identified by the quotient space of with the polynomials space .
We refer to  for more details.
Lemma 2.1 (Bernstein’s inequality) 
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.).
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.
where we used (1.4) and the boundedness of the Calderón-Zygmund singular integral operator on .
Step 2. Approximate solutions and uniform estimates.
which implies that . This together with (3.17) gives the uniform estimate of in n.
Step 3. Existence.
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.
for sufficiently small T. This implies that , i.e., .
Therefore, if , then .
Then implies .
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).
- Castro A, Córdoba D, Gancedo F, Orive R: Incompressible flow in porous media with fractional diffusion. Nonlinearity 2009, 22: 1791-1815. 10.1088/0951-7715/22/8/002MathSciNetView ArticleGoogle Scholar
- Córdoba D, Gancedo F, Orive R: Analytical behaviour of the two-dimensional incompressible flow in porous media. J. Math. Phys. 2007, 48: 1-19.View ArticleGoogle Scholar
- Abidi H, Hmidi T: On the global well posedness of the critical quasi-geostrophic equation. SIAM J. Math. Anal. 2008, 40: 167-185. 10.1137/070682319MathSciNetView ArticleGoogle Scholar
- Caffarelli L, Vasseur V: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equations. Ann. Math. 2010, 171: 1903-1930. 10.4007/annals.2010.171.1903MathSciNetView ArticleGoogle Scholar
- Chen Q, Miao C, Zhang Z: A new Bernstein’s inequality and the 2D dissipative quasigeostrophic equation. Commun. Math. Phys. 2007, 271: 821-838. 10.1007/s00220-007-0193-7MathSciNetView ArticleGoogle Scholar
- Córdoba A, Córdoba D: A maximum principle applied to the quasi-geostrophic equations. Commun. Math. Phys. 2004, 249: 511-528.View ArticleGoogle Scholar
- Kislev A, Nazarov F, Volberg A: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 2007, 167: 445-453. 10.1007/s00222-006-0020-3MathSciNetView ArticleGoogle Scholar
- Wu J: Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces. SIAM J. Math. Anal. 2004, 36: 1014-1030.MathSciNetView ArticleGoogle Scholar
- Córdoba D, Faraco D, Gancedo F: Lack of uniqueness for weak solutions of the incompressible porous media equation. Arch. Ration. Mech. Anal. 2011, 200: 725-746. 10.1007/s00205-010-0365-zMathSciNetView ArticleGoogle Scholar
- Chae D: On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces. Commun. Pure Appl. Math. 2002, 55: 654-678. 10.1002/cpa.10029MathSciNetView ArticleGoogle Scholar
- Chae D: On the Euler equations in the critical Triebel-Lizorkin spaces. Arch. Ration. Mech. Anal. 2003, 170: 185-210. 10.1007/s00205-003-0271-8MathSciNetView ArticleGoogle Scholar
- Chae D: The quasi-geostrophic equations in the Triebel-Lizorkin spaces. Nonlinearity 2003, 16: 479-495. 10.1088/0951-7715/16/2/307MathSciNetView ArticleGoogle Scholar
- Chen Q, Miao C, Zhang Z: On the well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces. Arch. Ration. Mech. Anal. 2010, 195: 561-578. 10.1007/s00205-008-0213-6MathSciNetView ArticleGoogle Scholar
- Triebel H Monograph in Mathematics 78. In Theory of Function Spaces. Birkhäuser Verlag, Basel; 1983.View ArticleGoogle Scholar
- Chemin J-Y: Perfect Incompressible Fluids. Oxford University Press, New York; 1998.Google Scholar
- Duoandikoetxea J GSM 29. In Fourier Analysis. Am. Math. Soc., Providence; 2001. (Translated and revised by D. Cruz-Uribe)Google Scholar
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