Regularity of Hölder continuous solutions of the supercritical porous media equation

In this paper, we present a regularity result for weak solutions of the N-dimensional (N=2\text{ or }3) porous media equation with supercritical () dissipation {\mathrm{\Lambda }}^{\alpha }. If a Leray-Hopf weak solution is Hölder continuous \theta \in C^{\delta }(R^N) with \delta >1-\alpha on the time interval [0,t], then it is actually a classical solution on (0,t].

MSC:76S05, 76D03.

We use Darcy’s law to describe the flow velocity, which reads

where v\in {\mathbb{R}}^N is the liquid discharge, p is the scalar pressure, θ is the liquid temperature, k is the matrix 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}}^N is the last canonical vector e_N. For brevity, we only consider k=g=1.

In this article, we study the system of heat transfer with a fractional diffusion in an incompressible N(2\text{ or }3)-dimensional flow [1]

where \nu >0 is the dissipative coefficient, and the differential operator {\mathrm{\Lambda }}^{\alpha } is given by {\mathrm{\Lambda }}^{\alpha }:={(-\mathrm{\Delta })}^{\frac{\alpha }{2}}. Considering the scaling transform \theta (t,x)\to {\theta }_{\lambda }(t,x):={\lambda }^{\alpha -1}\theta ({\lambda }^{\alpha }t,\lambda x) for \lambda >0, the system will be divided into three cases: the case \alpha =1 is called the critical case, the case \alpha >1 is subcritical and the case is supercritical.

Next, by rewriting Darcy’s law, we obtain the expression of velocity v only in terms of temperature θ [2, 3]. In the 2D case, thanks to the incompressibility, taking the curl operator first and the {\mathrm{\nabla }}^{\mathrm{\perp }}:=(-{\partial }_{x_2},{\partial }_{x_1}) operator second on both sides of Darcy’s law, we have

thus the velocity v can be recovered as

Through integration by parts, we finally get

where the kernel H(\cdot ) is defined by

Similarly, in the 3D case, applying the curl operator twice to Darcy’s law, we get

where {\partial }_i:=\frac{\partial }{\partial x_i}, thus

where

We observe that, in general, each coefficient of v(\cdot ,t) (with t as parameter) is only the linear combination of the Calderón-Zygmund singular integral (for the definition, see the sequel) of θ and θ itself. We write the general version as

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 (2) or (3).

We observe that the system ({\mathit{DPM}}_{\alpha }) is not more than a dissipative transport diffusion equation with non-local divergence-free velocity field (the specific relationship between velocity and temperature as (4) shows). It shares many similarities with another flow model - 2D dissipative quasi-geostrophic (QG) equation, which has been intensively studied by many authors [4–11]. From a mathematical point of view, the system ({\mathit{DPM}}_{\alpha }) is somewhat a generalization of (QG) equation. Very recently, the system ({\mathit{DPM}}_{\alpha }) was introduced and investigated by Córdoba and his group. In [2], the authors obtained some results on strong solutions, weak solutions and attractors for the dissipative system ({\mathit{DPM}}_{\alpha }). 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. In [3], they treated the nondissipative (\nu =0) 2D case and obtained the local existence and uniqueness in the Hölder space C^{\delta } for by the particle-trajectory method and gave some blowup criteria of smooth solutions.

In this paper we present a regularity result of weak solutions of the porous media equation with (the supercritical case). The result asserts that if a Leray-Hopf weak solution θ of (1) is in the Hölder class C^{\delta } with \delta >1-\alpha on the time interval [0,t], then it is actually a classical solution on (0,t]. The proof involves representing the functions in the Hölder space in terms of the Littlewood-Paley decomposition and using Besov space techniques. When θ is in C^{\delta }, it also belongs to the Besov space {\dot{B}}_{p,\mathrm{\infty }}^{\delta (1-2/p)} for any p\ge 2. By taking p sufficiently large, we have \theta \in C^{{\delta }_1}\cap {\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1} for {\delta }_1>1-\alpha. The idea is to show that \theta \in C^{{\delta }_2}\cap {\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_2} with {\delta }_2>{\delta }_1. Through iteration, we establish that \theta \in C^{\gamma } with \gamma >1. Then θ becomes a classical solution.

The rest of this paper is divided into two sections. Section 2 provides the definition of Besov spaces and necessary tools. Section 3 states and proves the main result.

Notation Throughout the paper, C denotes various ‘harmless’ large finite constants, and c denotes various ‘harmless’ small constants. We shall sometimes use X\lesssim Y to denote the estimate X\le CY for some C.

In this preparatory section, we give the definition of Besov spaces based on the Littlewood-Paley decomposition, introduce the Calderón-Zygmund singular integral, and finally we review some important results that will be used in the following.

Let us recall the Littlewood-Paley decomposition. Let \mathcal{S}({\mathbb{R}}^N) be the Schwartz class of rapidly decreasing functions. Given f\in \mathcal{S}({\mathbb{R}}^N), its Fourier transform \mathcal{F}f=\hat{f} is defined by

Choose two nonnegative radial functions \chi ,\phi \in \mathcal{S}({\mathbb{R}}^N), supported respectively in \mathcal{B}=\{\xi \in {\mathbb{R}}^N,|\xi |\le \frac{4}{3}\} and \mathcal{C}=\{\xi \in {\mathbb{R}}^N,\frac{3}{4}\le |\xi |\le \frac{8}{3}\}, such that

Setting {\phi }_j(\xi )=\phi (2^{-j}\xi ). Let h={\mathcal{F}}^{-1}\phi and \tilde{h}={\mathcal{F}}^{-1}\chi, we define the frequency localization operator as follows:

Informally, {\mathrm{\Delta }}_j=S_j-S_{j-1} is a frequency projection to the annulus \{|\xi |\approx 2^j\}, while S_j is a frequency projection to the ball \{|\xi |\lesssim 2^j\}. One easily verifies that with our choice of φ,

Now we give the definitions of Besov spaces.

Definition 2.1 Let s\in \mathbb{R}, 1\le p,q\le \mathrm{\infty }, the homogeneous Besov space {\dot{B}}_{p,q}^s is defined by

Here,

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

The following proposition lists a few simple facts that we will use in the subsequent section. The proof is rather standard and one can refer to [12].

Proposition 2.2 Assume that s\in \mathbb{R} and p,q\in [1,\mathrm{\infty }].

1. (1)

   If 1\le q_1\le q_2\le \mathrm{\infty }, then {\dot{B}}_{p,q_1}^s\subset {\dot{B}}_{p,q_2}^s.

2. (2)

   (Besov embedding) If 1\le p_1\le p_2\le \mathrm{\infty } and s_1=s_2+n(\frac{1}{p_1}-\frac{1}{p_2}), then {\dot{B}}_{p_1,q}^{s_1}({\mathbb{R}}^N)\subset {\dot{B}}_{p_2,q}^{s_2}({\mathbb{R}}^N).

3. (3)

   If , then

   {\dot{B}}_{p,min(p,2)}^s\subset {\dot{W}}^{s,p}\subset {\dot{B}}_{p,max(p,2)}^s,

where {\dot{W}}^{s,p} denotes a standard homogeneous Sobolev space.

We next introduce the classical Bernstein inequality [13].

Lemma 2.3 Let ℬ be a ball, \mathcal{C} be a ring, 0\le a\le b\le \mathrm{\infty }. Then \mathrm{\forall }k\in {\mathbb{Z}}^{+}\cup \{0\}, \mathrm{\forall }\lambda >0, there exists a constant C>0 such that

Similar inequalities hold for the fractional derivative {\mathrm{\Lambda }}^{\beta }.

The following proposition provides a lower bound for an integral that originates from the dissipative term in the process of L^p estimates (see [6, 14]).

Proposition 2.4 Assume either \alpha \ge 0 and p=2 or 0\le \alpha \le 1 and . Let j be an integer and f\in {\mathcal{S}}^{\prime }. Then

for some constant C depending on N, α and p.

The classical Calderón-Zygmund singular integrals are operators of the form

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, .

The general version (4) of the relationship between v and θ is in fact ensured by the following result (see, e.g., [15]).

Lemma 2.5 Let m\in C^{\mathrm{\infty }}({\mathbb{R}}^N\mathrm{\setminus }\{0\}) be a homogeneous function of degree 0, and let T_m be the corresponding multiplier operator defined by {(T_{m}f)}^{\wedge }=m\hat{f}, then there exist a\in \mathbb{C} and \mathrm{\Omega }\in C^{\mathrm{\infty }}({\mathbb{S}}^{N-1}) with zero average such that for any Schwartz function f,

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 \mathcal{T} 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 }}({\mathcal{R}}^N\mathrm{\setminus }\{0\}) and is homogeneous of degree 0.

Theorem 3.1 Let θ be a Leray-Hopf weak solution of (1), namely

Let \delta >1-\alpha and let . If

then \theta \in C^{\mathrm{\infty }}((0,t]\times {\mathbb{R}}^N) for N=2\mathit{\text{or}}3.

Proof First we notice that (5) and (6) imply that

for any p\ge 2 and {\delta }_1=\delta (1-\frac{N}{p}). In fact, for any \tau \in [0,t],

Since \delta >1-\alpha, we have {\delta }_1>1-\alpha when

Next, we show that

implies

for some {\delta }_2>{\delta }_1 to be specified. Let j be an integer. Applying {\mathrm{\Delta }}_j to the first equation of (1), we get

By Bony’s notion of paraproduct,

Multiplying (7) by p{|{\mathrm{\Delta }}_j\theta |}^{p-2}{\mathrm{\Delta }}_j\theta, integrating with respect to x and applying the lower bound

of Proposition 2.4, we obtain

We now estimate I_1. The standard idea is to decompose it into three terms: one with commutator, one that becomes zero due to the divergence-free condition and the rest. That is, we rewrite I_1 as

where we have used the simple fact that {\sum }_{|k-j|\le 1}{\mathrm{\Delta }}_k{\mathrm{\Delta }}_j={\mathrm{\Delta }}_j, and the brackets [] represent the commutator, namely

Since u is divergence free, I_{12} becomes zero. We now bound I_{11} and I_{13}. By Hölder’s inequality,

To bound the commutator, we have, by the definition of {\mathrm{\Delta }}_j,

Using the fact that \theta \in C^{{\delta }_1} and thus

we obtain

Therefore,

The estimate for I_{13} is straightforward. By Hölder’s inequality,

We then bound I_2. By Hölder’s inequality, Bernstein’s inequality and the fact 1-{\delta }_1>0, we obtain

Last, we bound I_3. By Hölder’s inequality and Bernstein’s inequality,

Inserting the estimates for I_1, I_2 and I_3 in (8) and eliminating p{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1} from both sides, we get

Integrating with time t, we have

Multiplying both sides by 2^{(\alpha +2{\delta }_1-1)j} and taking the supremum with respect to j, we get

Here we have used the fact that

Therefore, we conclude that if

then

Since {\delta }_1>1-\alpha, we have 2{\delta }_1+\alpha -1>{\delta }_1 and thus gain regularity. In addition, according to the Besov embedding of Proposition 2.2,

where

We have {\delta }_2>{\delta }_1 when

Noticing that

we conclude that, for p>max\{p_0,p_1\},

for some {\delta }_2>{\delta }_1. The above process can then be iterated with {\delta }_1 replaced by {\delta }_2. A finite number of iterations allow us to obtain that

for some \gamma >1. The regularity in the spatial variable can then be converted into regularity in time. We have thus established that θ is a classical solution to the supercritical porous media equation. Higher regularity can be proved by well-known methods. □

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).

Affiliations

1. College of Electrical and Information Engineering, Hunan University, Changsha, Hunan, 410082, P.R. China

   • Wenxin Yu
   • , Yaonan Tong
   • , Qiwu Luo
   •  & Xianming Wu

2. School of Electrical and Automation Engineering, Hefei University of Technology, Hefei, Anhui, 230009, P.R. China

   • Yigang He

Corresponding author

Correspondence to Wenxin Yu.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

In this paper, WY carried ‘Regularity of Hoder continuous solutions of the supercritical porous media equation’. YH, YT, QL and XW participated in the analysis. All authors read and approved the final manuscript.

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