Yu and He Boundary Value Problems 2014, 2014:95 http://www.boundaryvalueproblems.com/content/2014/1/95 RESEARCH Open Access On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin spaces Wenxin Yu1* and Yigang He2 * Correspondence: slowbird@sohu.com College of Electrical and Information Engineering, Hunan University, Changsha, 410000, P.R China Full list of author information is available at the end of the article Abstract 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 Introduction In this paper, we are concerned with the incompressible porous media equation (IPM) in Rd (d = or ): (IPM) ∂t θ + u · ∇θ = , u = –k(∇p + gγ θ ), θ (x, ) = θ , div u = , (.) where x ∈ Rd , t > , 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 γ ∈ Rd is the last canonical vector ed For simplicity, we only consider k = g = By rewriting Darcy’s law we obtain the expression of velocity u only in terms of temperature θ [, ] In the D case, thanks to the incompressibility, taking the curl operator first and the ∇ ⊥ := (–∂x , ∂x ) operator second on both sides of Darcy’ law, we have – u = ∇ ⊥ (∂x θ ) = –∂x ∂x θ , ∂x θ , thus the velocity u can be recovered as u(x, t) = – π R ln |x – y| – ∂ θ ∂ θ (y, t), (y, t) dy, ∂y ∂y ∂ y x ∈ R ©2014 Yu and He; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Yu and He Boundary Value Problems 2014, 2014:95 http://www.boundaryvalueproblems.com/content/2014/1/95 Page of 11 Through integration by parts we finally get u(x, t) = – , θ (x, t) + PV π H(x – y)θ (y, t) dy, R x ∈ R , (.) where the kernel H(·) is defined by H(x) = x x x – x , |x| |x| Similarly, in D case, applying the curl operator twice to Darcy’s law, we get – u = –∂ ∂ θ , –∂ ∂ θ , ∂ θ + ∂ θ , where ∂i := ∂ , ∂xi u(x, t) = – thus PV , , θ (x, t) + π R K(x – y)θ (y, t) dy, x ∈ R , (.) where K(x) = x x x x x – x – x , , |x| |x| |x| We observe that, in general, each coefficient of u(· , t) (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 u := T (θ) = C (θ) + S (θ), (.) where T = (Tk ), C = (Ck ), S = (Sk ), ≤ k ≤ N are all operators mapping scalar functions to vector-valued functions and Ck equals a constant multiplication operator whereas Sk means a Calderón-Zygmund singular integral operator Especially the corresponding specific forms in D or D are shown as (.) or (.) 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 (.) shows) It shares many similarities with another flow model - the D dissipative quasi-geostrophic (QG) equation, which has been intensively studied by many authors [–] 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 D case and obtained the local existence and uniqueness in Hölder space C δ for < δ < 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 H s , s > (N – α)/ + and extended to be global under a small condition θ H s ≤ cν, for s > N/ + , where c is a small fixed constant Yu and He Boundary Value Problems 2014, 2014:95 http://www.boundaryvalueproblems.com/content/2014/1/95 Page of 11 Recently, Chae studied the local well-posedness and blow-up criterion for the incompressible Euler equations [, ], 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 Lp (Rd ), Sobolev spaces Hps (Rd ), Lipschitz spaces C s (Rd ), and so on In [], the author first used the Littlewood-Paley operator to localize the Euler equation to the frequency annulus {|ξ | ∼ j }, then obtained an integral representation of the frequency-localized solution on the Lagrangian coordinates by introducing a family of particle-trajectory mappings {Xj (α, t)} defined by ∂ X (α, t) = (Sj– v)(Xj (α, t), t), ∂t j (.) Xj (α, ) = α, where v is a divergence-free velocity field and Sj– is a frequency projection to the ball {|ξ | j } (see Section ) He also used the following equivalent relation: jsq j v Xj (α, t) q q ∼ = Lp (·dα) j∈Z jsq j∈Z j v(x) q q = v Lp (·dx) s F˙ p,q (.) 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 (.) 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 X(α, t) independent of j defined by ∂ X(α, t) = v(X(α, t), t), ∂t X(α, ) = α, 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 wellposedness for the incompressible porous media equation (.) 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 . (i) Local-in-time existence Let s > dp + , < p, q < ∞ Assume that θ ∈ s s ) such that (.) has a unique solution θ ∈ Fp,q (Rd ), then there exists T = T( θ Fp,q s d s– d C([, T]; Fp,q (R )) ∩ Lip([, T]; Fp,q (R )) s (ii) Blow-up criterion The local-in-time solution θ ∈ C([, T]; Fp,q ) constructed in (i) ∗ s blows up at T > T in Fp,q , i.e lim sup θ (t) t T∗ s Fp,q = +∞, if and only if T∗ θ (t) F˙ ∞,∞ dt = +∞ T ∗ < ∞, Yu and He Boundary Value Problems 2014, 2014:95 http://www.boundaryvalueproblems.com/content/2014/1/95 Page of 11 Preliminaries Let B = {ξ ∈ Rd , |ξ | ≤ } and C = {ξ ∈ Rd , ≤ |ξ | ≤ } Choose two nonnegative smooth radial functions χ , ϕ supported, respectively, in B and C such that ξ ∈ Rd , ϕ –j ξ = , χ (ξ ) + ξ ∈ Rd \{} ϕ –j ξ = , j≥ j∈Z We denote ϕj (ξ ) = ϕ(–j ξ ), h = F – ϕ and h˜ = F – χ Then the dyadic blocks be defined as follows: jf = ϕ –j D f = jd Sj f = kf Rd j and Sj can h j y f (x – y) dy, = χ –j D f = jd Rd k≤j– h˜ j y f (x – y) dy Formally, j = Sj+ – Sj is a frequency projection to the annulus {|ξ | ∼ j }, and Sj is a frequency projection to the ball {|ξ | j } One easily verifies that with our choice of ϕ j kf ≡ if |j – k| ≥ and j (Sk– f kf ) ≡ if |j – k| ≥ (.) With the introduction of j and Sj , let us recall the definition of the Triebel-Lizorkin s space Let s ∈ R, (p, q) ∈ [, ∞) × [, ∞], the homogeneous Triebel-Lizorkin space F˙ p,q is defined by s F˙ p,q = f ∈ Z Rd ; f s F˙ p,q and (p, q) ∈ [, ∞) × [, ∞], we define the inhomogeneous Triebel-Lizorkin s space Fp,q as follows: s Fp,q = f ∈ S Rd ; f s Fp,q , (p, q) ∈ (, ∞) × (, ∞], or p = q = ∞, then there exists a constant C such that fg s F˙ p,q ≤C f L∞ g s F˙ p,q + g L∞ f s F˙ p,q , fg s Fp,q ≤C f L∞ g s Fp,q + g L∞ f s Fp,q s , then there exists a Proposition . [] Let s > d/ with p, q ∈ [, ∞] Suppose f ∈ Fp,q constant C such that the following inequality holds: f L∞ ≤C + f s F˙ ∞,∞ log+ f s Fp,q + Proposition . [] Let (p, q) ∈ (, ∞) × (, ∞], or p = q = ∞, and f be a solenoidal vector field Then for s > ks [f , k ] · ∇g q (Z) Lp k ] · ∇g q (Z) p ∇f g L∞ s F˙ p,q + ∇g L∞ f , s F˙ p,q (.) or for s > – ks [f , ∇f L∞ g s F˙ p,q + g L∞ ∇f s F˙ p,q (.) The classical Calderón-Zygmund singular integrals are operators of the form Tcz f (x) := PV RN (y ) f (x – y) dy = lim → |y|N |y|> (y ) f (x – y) dy, |y|N where is defined on the unit sphere of RN , SN– , and is integrable with zero average y ∈ S N– Clearly, the definition is meaningful for Schwartz functions and where y := |y| Moreover if ∈ C (S N– ), Tcz is Lp bounded, < p < ∞ The general version (.) of the relationship between u and θ is in fact ensured by the following result (see e.g []) Lemma . Let m ∈ C ∞ (RN \{}) be a homogeneous function of degree , and Tm be the corresponding multiplier operator defined by (Tm f )∧ = mfˆ , then there exist a ∈ C and ∈ C ∞ (SN– ) with zero average such that for any Schwartz function f , Tm f = af + PV (x ) ∗f |x|N Remark . Since – v = (∂ ∂N θ , , –∂N– ∂N θ , ∂ θ + · · · + ∂N– θ ), the Fourier multiplier of the operator T is rather clear In fact, each component of its multiplier is the linear ξξ combination of the term like |ξi |j , i, j ∈ {, , , N}, which of course belongs to C ∞ (RN \{}) and is homogeneous of degree Yu and He Boundary Value Problems 2014, 2014:95 http://www.boundaryvalueproblems.com/content/2014/1/95 Page of 11 Proof of Theorem 1.1 We divide the proof of Theorem . into several steps Step A priori estimates Taking the operation k on both sides of the first equation of (.), we have kθ ∂t +u·∇ kθ =u·∇ – kθ k (u∇θ ) k ] · ∇θ [u, (.) Let X(α, t) be the solution of the following ordinary differential equations: ∂t X(α, t) = u(X(α, t), t), X(α, ) = α (.) Then it follows from (.) that d dt kθ k ] · ∇θ X(α, t), t = [u, X(α, t), t , (.) which implies that kθ t X(α, t), t ≤ k θ (α) + k ] · ∇θ [u, X(α, τ ), τ dτ (.) Multiplying ks , taking the Minkowski inequality ks q q q X(α, t), t kθ (Z) norm on both sides of (.), we get by using the k ≤ ks k θ (α) q q t ks [u, + k k ] · ∇θ X(α, τ ), τ q q dτ (.) k Next, taking the Lp norm with respect to α ∈ Rd on both sides of (.), we get by using the Minkowski inequality that Rd ks kθ X(α, t), t q p q p dα k ≤ θ t s F˙ p,q + ks [u, Rd k ] · ∇θ X(α, τ ), τ q q p p dα dτ (.) k Using the fact that α → X(α, t) is a volume-preserving diffeomorphism due to div u = , we get from (.) that θ (t) s F˙ p,q ≤ θ t s F˙ p,q ks [u, + k ] · ∇θ q (k∈Z) Lp dτ (.) Thanks to Proposition ., the last term on the right side of (.) is dominated by t ∇u L∞ θ s F˙ p,q + ∇θ L∞ u s F˙ p,q dτ , (.) Yu and He Boundary Value Problems 2014, 2014:95 http://www.boundaryvalueproblems.com/content/2014/1/95 Page of 11 and thus θ (t) s F˙ p,q t ≤ θ s F˙ p,q + ∇u L∞ + ∇θ θ L∞ s F˙ p,q dτ , (.) where we used (.) and the boundedness of the Calderón-Zygmund singular integral ops erator on F˙ p,q Now from (.) we have immediately θ (t) Lp = θ (.) Lp for all ≤ p ≤ ∞, since div u = Summing up (.) and (.) yields θ (t) s Fp,q t ≤ θ s Fp,q + ∇u L∞ + ∇θ θ L∞ dτ , (.) dτ (.) s Fp,q which together with the Gronwall inequality gives θ (t) s Fp,q t ≤ θ s Fp,q exp C ∇u L∞ + ∇θ L∞ Step Approximate solutions and uniform estimates We construct the approximate solutions of (.) Define the sequence {θ (n) , u(n) }N =N∪{} by solving the following systems: ⎧ (n+) ∂t θ + u(n) · ∇θ (n+) = , ⎪ ⎪ ⎪ ⎨u(n) = C (θ (n) ) + S (θ (n) ), ⎪ div u(n) = , ⎪ ⎪ ⎩ (n+) θ |t= = Sn+ θ (.) We set (θ () , u() ) = (, ) and let X (n) (α, t) be the solutions of the following ordinary differential equations: ∂t X (n) (α, t) = u(n) (X (n) (α, t), t), X (n) (α, ) = α (.) for each n ∈ N Then, following the same procedure of estimate leading to (.), we obtain θ (n+) (t) s Fp,q ≤ Sn+ θ ≤ θ ≤ θ t s Fp,q + t s Fp,q + s Fp,q + ∇u(n) ∇u(n) s– Fp,q L∞ θ (n+) θ (n+) s Fp,q s Fp,q + ∇θ (n+) + ∇θ (n+) s– Fp,q L∞ u(n) u(n) s Fp,q s Fp,q dτ dτ t θ (n) s Fp,q θ (n+) s Fp,q dτ , (.) Yu and He Boundary Value Problems 2014, 2014:95 http://www.boundaryvalueproblems.com/content/2014/1/95 Page of 11 s– where we used the fact that Sn+ θ ≤ θ , Sobolev embedding theorem Fp,q → ∞ L for s – > d/p, (.) and the boundedness of the Calderón-Zygmund singular ins tegral operator on Fp,q Equation (.) together with the Gronwall inequality implies that θ (n+) (t) s Fp,q t ≤ θ s Fp,q θ (n) exp C s Fp,q dτ , (.) for some C > independent of n Thus, if we choose T = T ( θ T θ s Fp,q ≤ s )> Fp,q such that ln , C we have, for any n ∈ N , θ (n+) (t) sup ≤t≤T s Fp,q ≤ C θ s , Fp,q (.) s (Rd )) Moreover, it folby the standard induction arguments Then, θ (n+) ∈ C([, T ]; Fp,q lows from Proposition . that ∂t θ (n+) (t) s– Fp,q = u(n) · ∇θ (n+) ≤ C u(n) ≤ C θ (n) L∞ s– Fp,q s– Fp,q ∇θ (n+) θ (n+) s– Fp,q + u(n) s– Fp,q ∇θ (n+) L∞ s Fp,q by Sobolev embedding and the boundedness of the Calderón-Zygmund singular integral s operator on Fp,q , and then ∂t θ (n+) (t) sup ≤t≤T s Fp,q ≤ C θ s , Fp,q (.) s– (Rd )) This together with (.) gives the uniwhich implies that ∂t θ (n+) ∈ C([, T ]; Fp,q (n) form estimate of θ (x, t) in n Step Existence We will show that there exists a positive time T (≤ T ) independent of n such that θ (n) s– and u(n) are Cauchy sequences in XTs– C ([, T ]; Fp,q ) For this purpose, we set δθ (n+) = θ (n+) – θ (n) , δu(n+) = u(n+) – u(n) Then, it follows that δθ (n+) satisfies the equations ∂t δθ (n+) + u(n) · ∇δθ (n+) = –δu(n) · ∇θ (n) , δθ (n+) |t= = n+ θ Applying ∂t k k δθ (.) to the first equation of (.), we get (n+) + u(n) · ∇ k δθ (n+) = u(n) , k · ∇δθ (n+) – k δu(n) · ∇θ (n) (.) Yu and He Boundary Value Problems 2014, 2014:95 http://www.boundaryvalueproblems.com/content/2014/1/95 Page of 11 Exactly as in the proof of (.), we get δθ (n+) s– F˙ p,q t ≤C s– n+ θ F˙ p,q t + k(s–) u(n) , + δu(n) · ∇θ (n) (τ ) t s– n+ θ F˙ p,q δu(n) ∇u(n) + ∇θ (n) L∞ δθ (n+) L∞ t + q (Z) p dτ dτ s– F˙ p,q ≤C · ∇δθ (n+) (τ ) k + δu(n) s– F˙ p,q s– F˙ p,q + δθ (n+) ∇θ (n) s– F˙ p,q L∞ ∇u(n) s– F˙ p,q dτ dτ L∞ t ≤C s– n+ θ F˙ p,q + s– n+ θ F˙ p,q + u(n) s Fp,q δθ (n+) s Fp,q + δu(n) θ (n) s– Fp,q s Fp,q dτ t ≤C θ (n) s Fp,q δθ (n+) s Fp,q + δθ (n) θ (n) s– Fp,q s Fp,q dτ , (.) s– → L∞ , and the where we used Proposition ., Proposition ., the embedding Fp,q s boundedness of the Calderón-Zygmund singular integral operator on Fp,q Thanks to the Fourier support of n+ θ , we have s– n+ θ F˙ p,q ≤ C–(n+) θ s F˙ p,q (.) Now, we estimate the Lp norm of δθ (n+) Multiplying |δθ (n+) |p– δθ (n+) on both sides of the first equation of (.), and integrating the resulting equations over Rd , we obtain δθ (n+) (t) Lp ≤ t δu(n) · ∇θ (n) (τ ) + n+ θ Lp t ≤ –(n+) z+ s F˙ p,q +C ≤ –(n+) z+ s F˙ p,q +C Lp dτ δu(n) Lp ∇θ (n) δθ (n) Lp θ (n) L∞ dτ t s Fp,q dτ , which together with (.) and (.) gives δθ (n+) s– Fp,q ≤ C–(n+) θ + δθ (n) t s Fp,q s– Fp,q ≤ C–(n+) θ θ (n) s Fp,q s Fp,q t∈[,T] s Fp,q δθ (n+) s– Fp,q dτ s Fp,q + CT sup θ (n) + CT sup θ (n) t∈[,T] θ (n) +C δθ (n) s– Fp,q s Fp,q δθ (n+) s– Fp,q (.) Equation (.) together with (.) yields δθ (n+) XTs– ≤ C –(n+) + C T δθ (n+) + C T δθ (n) XTs– , XTs– (.) Yu and He Boundary Value Problems 2014, 2014:95 http://www.boundaryvalueproblems.com/content/2014/1/95 where C = C ( θ δθ (n+) XTs– Page 10 of 11 Thus, if C T ≤ , then s ) Fp,q ≤ C –n + C T δθ (n) XTs– This implies that δθ (n+) XTs– ≤ C –(n+) Thus, {θ (n) }n∈N is a Cauchy sequence in XTs– By the standard argument, for T ≤ min{T , C }, the limit θ ∈ XTs solves (.) with the initial data θ The fact that θ ∈ s ) follows from the uniform estimate (.) Lip([, T]; Fp,q Step Uniqueness s )) is another solution to (.) with the same initial data Let Consider θ˜ ∈ C([, T ]; (Fp,q ˜ Then δθ satisfies the following equation: δθ = θ – θ˜ and δu = u – u ∂t δθ + u · ∇δθ = –δu · ∇ θ˜ , δθ|t= = In the same way as the derivation in (.), we obtain δθ XTs– ≤ C T δθ XTs– for sufficiently small T This implies that δθ ≡ , i.e., θ ≡ θ˜ Blow-up criterion For the a priori estimate (.), we only need to dominate ∇u L∞ and ∇θ L∞ From into Proposition . and the boundedness of the Calderón-Zygmund operator from F˙ ∞,∞ itself, we have ∇u L∞ ≤ C + ∇u F˙ ∞,∞ ≤C + θ F˙ ∞,∞ ≤C + θ F˙ ∞,∞ log+ ∇u s– Fp,q + log+ θ s Fp,q + log+ θ s Fp,q + Similarly, ∇θ L∞ Thus, the a priori estimate (.) gives θ s Fp,q ≤ C θ t s Fp,q θ (τ ) exp C F˙ ∞,∞ log+ θ (τ ) s Fp,q + dτ By the Gronwall inequality θ s Fp,q ≤ C θ Therefore, if lim supt t s Fp,q θ (τ ) exp C exp T∗ θ(t) s Fp,q = ∞, then F˙ ∞,∞ T∗ dτ θ(t) F˙ ∞,∞ dt = ∞ Yu and He Boundary Value Problems 2014, 2014:95 http://www.boundaryvalueproblems.com/content/2014/1/95 Page 11 of 11 s On the other hand, it follows from the Sobolev embedding Fp,q → W ,∞ → F˙ ∞,∞ for s > d/p + that T∗ θ (t) F˙ ∞,∞ dt ≤ T ∗ sup ≤τ ≤T ∗ ∇θ (τ ) ≤ T ∗ sup θ (τ ) ≤τ ≤T ∗ T∗ θ(t) F˙ ∞,∞ F˙ ∞,∞ ≤ T ∗ sup ≤τ ≤T ∗ Then θ (τ ) L∞ s Fp,q dt = ∞ implies lim supt T∗ θ(t) s Fp,q = ∞ Competing interests The authors declare that they have no competing interests Authors’ contributions All authors contributed equally to the writing of this paper All authors read and approved the final manuscript Author details College of Electrical and Information Engineering, Hunan University, Changsha, 410000, P.R China School of Electrical and Automation Engineering, Hefei University of Technology, Anhui Pro., Hefei, 230009, P.R China Acknowledgements 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) Received: 22 January 2014 Accepted: 24 March 2014 Published: 06 May 2014 References Castro, A, Córdoba, D, Gancedo, F, Orive, R: Incompressible flow in porous media with fractional diffusion Nonlinearity 22, 1791-1815 (2009) Córdoba, D, Gancedo, F, Orive, R: Analytical behaviour of the two-dimensional incompressible flow in porous media J Math Phys 48, 1-19 (2007) Abidi, H, Hmidi, T: On the global well posedness of the critical quasi-geostrophic equation SIAM J Math Anal 40, 167-185 (2008) Caffarelli, L, Vasseur, V: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equations Ann Math 171, 1903-1930 (2010) Chen, Q, Miao, C, Zhang, Z: A new Bernstein’s inequality and the 2D dissipative quasigeostrophic equation Commun Math Phys 271, 821-838 (2007) Córdoba, A, Córdoba, D: A maximum principle applied to the quasi-geostrophic equations Commun Math Phys 249, 511-528 (2004) Kislev, A, Nazarov, F, Volberg, A: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation Invent Math 167, 445-453 (2007) Wu, J: Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces SIAM J Math Anal 36, 1014-1030 (2004) Córdoba, D, Faraco, D, Gancedo, F: Lack of uniqueness for weak solutions of the incompressible porous media equation Arch Ration Mech Anal 200, 725-746 (2011) doi:10.1007/s00205-010-0365-z 10 Chae, D: On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces Commun Pure Appl Math 55, 654-678 (2002) 11 Chae, D: On the Euler equations in the critical Triebel-Lizorkin spaces Arch Ration Mech Anal 170, 185-210 (2003) 12 Chae, D: The quasi-geostrophic equations in the Triebel-Lizorkin spaces Nonlinearity 16, 479-495 (2003) 13 Chen, Q, Miao, C, Zhang, Z: On the well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces Arch Ration Mech Anal 195, 561-578 (2010) doi:10.1007/s00205-008-0213-6 14 Triebel, H: Theory of Function Spaces Monograph in Mathematics, vol 78 Birkhäuser Verlag, Basel (1983) 15 Chemin, J-Y: Perfect Incompressible Fluids Oxford University Press, New York (1998) 16 Duoandikoetxea, J: Fourier Analysis GSM, vol 29 Am Math Soc., Providence (2001) (Translated and revised by D Cruz-Uribe) 10.1186/1687-2770-2014-95 Cite this article as: Yu and He: On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin spaces Boundary Value Problems 2014, 2014:95 ... The quasi-geostrophic equations in the Triebel- Lizorkin spaces Nonlinearity 16, 479-495 (2003) 13 Chen, Q, Miao, C, Zhang, Z: On the well- posedness of the ideal MHD equations in the Triebel- Lizorkin. .. to establish the local wellposedness for the incompressible porous media equation (.) and to obtain a blow-up criterion of smooth solutions in the framework of Triebel- Lizorkin spaces Now we... criterion for the incompressible Euler equations [, ], and quasi-geostrophic equations [] in Triebel- Lizorkin spaces As is well known, Triebel- Lizorkin spaces are the unification of several