On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin spaces

where $x\in {\mathbb{R}}^d$, $t>0$, 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 $\gamma \in {\mathbb{R}}^d$ is the last canonical vector $e_d$. For simplicity, we only consider $k=g=1$.

where $\mathcal{T}=({\mathcal{T}}_k)$, $\mathcal{C}=({\mathcal{C}}_k)$, $\mathcal{S}=({\mathcal{S}}_k)$, $1\le k\le N$ are all operators mapping scalar functions to vector-valued functions and ${\mathcal{C}}_k$ equals a constant multiplication operator whereas ${\mathcal{S}}_k$ 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 [2], they treated the (IMP) in 2D case and obtained the local existence and uniqueness in Hölder space $C^{\delta }$ for $0<\delta <1$ 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 [9]. For the dissipative system related (IPM), in [1], 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 $H^s$, $s>(N-\alpha )/2+1$ and extended to be global under a small condition ${\parallel {\theta }^0\parallel }_{H^s}\le c\nu$, for $s>N/2+1$, 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.[13] 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.

Let $\mathcal{B}=\{\xi \in {\mathbb{R}}^d,|\xi |\le \frac{4}{3}\}$ and $\mathcal{C}=\{\xi \in {\mathbb{R}}^d,\frac{3}{4}\le |\xi |\le \frac{8}{3}\}$. Choose two nonnegative smooth radial functions χ, φ supported, respectively, in ℬ and such that

$\chi (\xi )+\sum _{j\ge 0}\phi \left(2^{-j}\xi \right)=1,\xi \in {\mathbb{R}}^d,\sum _{j\in \mathbb{Z}}\phi \left(2^{-j}\xi \right)=1,\xi \in {\mathbb{R}}^d\mathrm{\setminus }\{0\}.$

and ${\mathcal{Z}}^{\prime }({\mathbb{R}}^d)$ denotes the dual space of $\mathcal{Z}({\mathbb{R}}^d)=\{f\in \mathcal{S}({\mathbb{R}}^d);{\partial }^{\alpha }\hat{f}(0)=0;\mathrm{\forall }\alpha \in {\mathbb{N}}^d\text{ multi-index}\}$ and can be identified by the quotient space of ${\mathcal{S}}^{\prime }/\mathcal{P}$ with the polynomials space .

We refer to [14] for more details.

Lemma 2.1 (Bernstein’s inequality) [15]

Lemma 2.2[14]

Proposition 2.1[10]

Proposition 2.2[10]

Proposition 2.3[13]

where Ω is defined on the unit sphere of ${\mathbb{R}}^N$, ${\mathbb{S}}^{N-1}$, and is integrable with zero average and where $y^{\prime }:=\frac{y}{|y|}\in {\mathcal{S}}^{N-1}$. Clearly, the definition is meaningful for Schwartz functions. Moreover if $\mathrm{\Omega }\in C^1({\mathcal{S}}^{N-1})$, $T_{cz}$ is $L^p$ bounded, $1<p<\mathrm{\infty }$.

The general version (1.4) of the relationship between u and θ is in fact ensured by the following result (see e.g.[16]).

Remark 2.1 Since $-\mathrm{\Delta }v=({\partial }_1{\partial }_N\theta ,\dots ,-{\partial }_{N-1}{\partial }_N\theta ,{\partial }_1^2\theta +\cdots +{\partial }_{N-1}^2\theta )$, the Fourier multiplier of the operator is rather clear. In fact, each component of its multiplier is the linear combination of the term like $\frac{{\xi }_i{\xi }_j}{{|\xi |}^2}$, $i,j\in \{1,2,\dots ,N\}$, which of course belongs to $C^{\mathrm{\infty }}({\mathbb{R}}^N\mathrm{\setminus }\{0\})$ and is homogeneous of degree 0.

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 ${\dot{F}}_{p,q}^s$.

Step 2. Approximate solutions and uniform estimates.

which implies that ${\partial }_t{\theta }^{(n+1)}\in C([0,T_0];F_{p,q}^{s-1}({\mathbb{R}}^d))$. This together with (3.17) gives the uniform estimate of ${\theta }^{(n)}(x,t)$ in n.

Step 3. Existence.

Thus, ${\{{\theta }^{(n)}\}}_{n\in {\mathbb{N}}_0}$ is a Cauchy sequence in $X_{T_1}^{s-1}$. By the standard argument, for $T_1\le min\{T_0,\frac{1}{4C_1}\}$, the limit $\theta \in X_{T_1}^s$ solves (1.1) with the initial data ${\theta }_0$. The fact that $\theta \in Lip([0,T];F_{p,q}^s)$ follows from the uniform estimate (3.18).

Step 4. Uniqueness.

for sufficiently small T. This implies that $\delta \theta \equiv 0$, i.e., $\theta \equiv \tilde{\theta }$.

Blow-up criterion.