Regularity of Hölder continuous solutions of the supercritical porous media equation

Abstract

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 ($\alpha <1$) 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.

1 Introduction

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

$$v=-k(\mathrm{\nabla }p+g\gamma \theta ),$$

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]

$${(\mathit{DPM})}_{\alpha }\{\begin{array}{c}{\partial }_t\theta +v\cdot \mathrm{\nabla }\theta +\nu {\mathrm{\Lambda }}^{\alpha }\theta =0,\hfill \\ v=-(\mathrm{\nabla }p+\gamma \theta ),divv=0,\hfill \\ \theta (0,x)={\theta }^0(x),\hfill \end{array}$$

(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 $\alpha <1$ 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

$$-\mathrm{\Delta }v={\mathrm{\nabla }}^{\mathrm{\perp }}({\partial }_{x_1}\theta )=(-{\partial }_{x_1}{\partial }_{x_2}\theta ,{\partial }_{x_1}^2\theta ),$$

thus the velocity v can be recovered as

$$v(t,x)=-\frac{1}{2\pi }{\int }_{{\mathbb{R}}^2}ln|x-y|(-\frac{{\partial }^2\theta }{\partial y_2\partial y_1}(t,y),\frac{{\partial }^2\theta }{{\partial }^2{y}_1}(t,y))dy,x\in {\mathbb{R}}^2.$$

Through integration by parts, we finally get

$$v(t,x)=-\frac{1}{2}(0,\theta (t,x))+\frac{1}{2\pi }\mathit{PV}{\int }_{{\mathbb{R}}^2}H(x-y)\theta (t,y)dy,x\in {\mathbb{R}}^2,$$

(2)

where the kernel $H(\cdot )$ is defined by

$$H(x)=(\frac{2x_1{x}_2}{{|x|}^4},\frac{x_2^2-x_1^2}{{|x|}^4}).$$

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

$$-\mathrm{\Delta }v=(-{\partial }_1{\partial }_3\theta ,-{\partial }_2{\partial }_3\theta ,{\partial }_1^2\theta +{\partial }_2^2\theta ),$$

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

$$v(t,x)=-\frac{2}{3}(0,0,\theta (t,x))+\frac{1}{4\pi }\mathit{PV}{\int }_{{\mathbb{R}}^3}K(x-y)\theta (t,y)dy,x\in {\mathbb{R}}^3,$$

(3)

where

$$K(x)=(\frac{3x_1{x}_3}{{|x|}^5},\frac{3x_2{x}_3}{{|x|}^5},\frac{2x_3^2-x_1^2-x_2^2}{{|x|}^5}).$$

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

$$v:=\mathcal{T}(\theta )=\mathcal{C}(\theta )+\mathcal{S}(\theta ),$$

(4)

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 $0<\delta <1$ 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 $\alpha <1$ (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.

2 Besov spaces and related tools

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

$$\hat{f}(\xi )={(2\pi )}^{-\frac{N}{2}}{\int }_{{\mathbb{R}}^N}{e}^{-ix\cdot \xi }f(x)dx.$$

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

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

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:

$$\begin{array}{c}{\mathrm{\Delta }}_{j}f=\phi \left(2^{-j}D\right)f=2^{jN}{\int }_{R^N}h\left(2^{j}y\right)f(x-y)dy,\hfill \\ S_{j}f=\sum _{k\le j-1}{\mathrm{\Delta }}_{k}f=\chi \left(2^{-j}D\right)f=2^{jN}{\int }_{R^N}\tilde{h}\left(2^{j}y\right)f(x-y)dy.\hfill \end{array}$$

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,

$${\mathrm{\Delta }}_j{\mathrm{\Delta }}_{k}f\equiv 0\text{if }|j-k|\ge 2\text{and}{\mathrm{\Delta }}_j(S_{k-1}f{\mathrm{\Delta }}_{k}f)\equiv 0\text{if }|j-k|\ge 5.$$

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

$${\dot{B}}_{p,q}^s=\{f\in {\mathcal{Z}}^{\prime }\left({\mathbb{R}}^N\right);{\parallel f\parallel }_{{\dot{B}}_{p,q}^s}<\mathrm{\infty }\}.$$

Here,

$${\parallel f\parallel }_{{\dot{B}}_{p,q}^s}=\{\begin{array}{cc}{({\sum }_{j\in \mathbb{Z}}{2}^{jsq}{\parallel {\mathrm{\Delta }}_{j}f\parallel }_p^q)}^{\frac{1}{q}}\hfill & \text{for }q<\mathrm{\infty }\hfill \\ {sup}_{j\in \mathbb{Z}}{\parallel {\mathrm{\Delta }}_{j}f\parallel }_p\hfill & \text{for }q=\mathrm{\infty },\hfill \end{array}$$

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) 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) (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) If $1<p<\mathrm{\infty }$, 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

$$\begin{array}{c}\underset{|\alpha |=k}{sup}{\parallel {\partial }^{\alpha }f\parallel }_{L^b}\le C{\lambda }^{k+N(\frac{1}{a}-\frac{1}{b})}{\parallel f\parallel }_{L^a}\mathit{\text{if}}supp\mathcal{F}f\subset \lambda \mathcal{B},\hfill \\ C^{-1}{\lambda }^k{\parallel f\parallel }_{L^a}\le \underset{|\alpha |=k}{sup}{\parallel {\partial }^{\alpha }f\parallel }_{L^a}\le C{\lambda }^k{\parallel f\parallel }_{L^a}\mathit{\text{if}}supp\mathcal{F}f\subset \lambda \mathcal{C}.\hfill \end{array}$$

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 $2<p<\mathrm{\infty }$. Let j be an integer and $f\in {\mathcal{S}}^{\prime }$. Then

$${\int }_{{\mathbb{R}}^N}{|{\mathrm{\Delta }}_{j}f|}^{p-2}{\mathrm{\Delta }}_{j}f{\mathrm{\Lambda }}^{2\alpha }{\mathrm{\Delta }}_{j}fdx\ge C{2}^{2\alpha j}{\parallel {\mathrm{\Delta }}_{j}f\parallel }_{L^p}^p$$

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

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

$$T_{cz}f(x):=\mathit{PV}{\int }_{{\mathbb{R}}^N}\frac{\mathrm{\Omega }(y^{\prime })}{{|y|}^N}f(x-y)dy=\underset{ϵ\to 0}{lim}{\int }_{|y|>ϵ}\frac{\mathrm{\Omega }(y^{\prime })}{{|y|}^N}f(x-y)dy,$$

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 (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,

$$T_{m}f=af+\mathit{PV}\frac{\mathrm{\Omega }(x^{\prime })}{{|x|}^N}\ast f.$$

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.

3 The main theorem and its proof

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

$$\theta \in L^{\mathrm{\infty }}([0,\mathrm{\infty });L^2\left({\mathbb{R}}^N\right))\cap L^2([0,\mathrm{\infty });{\dot{H}}^{\alpha /2}\left({\mathbb{R}}^N\right)).$$

(5)

Let $\delta >1-\alpha$ and let $0<t<\mathrm{\infty }$. If

$$\theta \in L^{\mathrm{\infty }}([0,t];C^{\delta }\left({\mathbb{R}}^N\right)),$$

(6)

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

$$\theta \in L^{\mathrm{\infty }}([0,t];{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}\left({\mathbb{R}}^N\right))$$

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

$${\parallel \theta (\cdot ,\tau )\parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}=\underset{j}{sup}{2}^{{\delta }_{1}j}{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}\le \underset{j}{sup}{2}^{{\delta }_{1}j}{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^{\mathrm{\infty }}}^{1-\frac{2}{p}}{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^2}^{\frac{2}{p}}\le {\parallel \theta (\cdot ,\tau )\parallel }_{C^{\delta }}^{1-\frac{2}{p}}{\parallel \theta (\cdot ,\tau )\parallel }_{L^2}^{\frac{2}{p}}.$$

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

$$p>p_0\equiv \frac{N\delta }{\delta -(1-\alpha )}.$$

Next, we show that

$$\theta \in L^{\mathrm{\infty }}([0,t];{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}\cap C^{{\delta }_1})$$

implies

$$\theta (\cdot ,t)\in {\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_2}\cap C^{{\delta }_2}$$

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

$${\partial }_t{\mathrm{\Delta }}_j\theta +\nu {\mathrm{\Lambda }}^{\alpha }{\mathrm{\Delta }}_j\theta =-{\mathrm{\Delta }}_j(v\cdot \mathrm{\nabla }\theta ).$$

(7)

By Bony’s notion of paraproduct,

$${\mathrm{\Delta }}_j(v\cdot \mathrm{\nabla }\theta )=\sum _{|k-j|\le 4}{\mathrm{\Delta }}_j(S_{k-1}v\cdot \mathrm{\nabla }{\mathrm{\Delta }}_k\theta )+\sum _{|k-j|\le 4}{\mathrm{\Delta }}_j({\mathrm{\Delta }}_{k}v\cdot \mathrm{\nabla }{S}_{k-1}\theta )\underset{|k-l|\le 1}{\sum _{k\ge j-2}}{\mathrm{\Delta }}_j({\mathrm{\Delta }}_{k}v\cdot \mathrm{\nabla }{\mathrm{\Delta }}_l\theta ).$$

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

$${\int }_{{\mathbb{R}}^N}{|{\mathrm{\Delta }}_{j}f|}^{p-2}{\mathrm{\Delta }}_{j}f{\mathrm{\Lambda }}^{2\alpha }{\mathrm{\Delta }}_{j}fdx\ge C{2}^{2\alpha j}{\parallel {\mathrm{\Delta }}_{j}f\parallel }_{L^p}^p$$

of Proposition 2.4, we obtain

$$\begin{array}{rcl}\frac{d}{dt}{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^p+C\nu 2^{\alpha j}{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^p& \le & -p\sum _{|k-j|\le 4}{\int }_{{\mathbb{R}}^N}{|{\mathrm{\Delta }}_j\theta |}^{p-2}{\mathrm{\Delta }}_j\theta {\mathrm{\Delta }}_j(S_{k-1}v\cdot \mathrm{\nabla }{\mathrm{\Delta }}_k\theta )dx\\ -p\sum _{|k-j|\le 4}{\int }_{{\mathbb{R}}^N}{|{\mathrm{\Delta }}_j\theta |}^{p-2}{\mathrm{\Delta }}_j\theta {\mathrm{\Delta }}_j({\mathrm{\Delta }}_{k}v\cdot \mathrm{\nabla }{S}_{k-1}\theta )dx\\ -p\sum _{\begin{array}{c}k\ge j-2\\ |k-l|\le 1\end{array}}{\int }_{{\mathbb{R}}^N}{|{\mathrm{\Delta }}_j\theta |}^{p-2}{\mathrm{\Delta }}_j\theta {\mathrm{\Delta }}_j({\mathrm{\Delta }}_{l}v\cdot \mathrm{\nabla }{\mathrm{\Delta }}_k\theta )dx\\ =& I_1+I_2+I_3.\end{array}$$

(8)

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

$$\begin{array}{rcl}{I}_1& =& -p\sum _{|k-j|\le 4}{\int }_{{\mathbb{R}}^N}{|{\mathrm{\Delta }}_j\theta |}^{p-2}{\mathrm{\Delta }}_j\theta \cdot [{\mathrm{\Delta }}_j,S_{k-1}v\cdot \mathrm{\nabla }]{\mathrm{\Delta }}_k\theta dx\\ -p{\int }_{{\mathbb{R}}^N}{|{\mathrm{\Delta }}_j\theta |}^{p-2}{\mathrm{\Delta }}_j\theta \cdot (S_{j}v\cdot \mathrm{\nabla }{\mathrm{\Delta }}_j\theta )dx\\ -p\sum _{|k-j|\le 1}{\int }_{{\mathbb{R}}^N}{|{\mathrm{\Delta }}_j\theta |}^{p-2}{\mathrm{\Delta }}_j\theta \cdot (S_{k-1}v-S_{j}v)\cdot \mathrm{\nabla }{\mathrm{\Delta }}_j{\mathrm{\Delta }}_k\theta dx\\ =& I_{11}+I_{12}+I_{13},\end{array}$$

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

$$[{\mathrm{\Delta }}_j,S_{k-1}v\cdot \mathrm{\nabla }]{\mathrm{\Delta }}_k\theta ={\mathrm{\Delta }}_j(S_{k-1}v\cdot \mathrm{\nabla }{\mathrm{\Delta }}_k\theta )-S_{k-1}v\cdot \mathrm{\nabla }{\mathrm{\Delta }}_j{\mathrm{\Delta }}_k\theta .$$

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

$$I_{11}\le p{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}\sum _{|k-j|\le 4}{\parallel [{\mathrm{\Delta }}_j,S_{k-1}v\cdot \mathrm{\nabla }]{\mathrm{\Delta }}_k\theta \parallel }_{L^p}.$$

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

$$[{\mathrm{\Delta }}_j,S_{k-1}v\cdot \mathrm{\nabla }]{\mathrm{\Delta }}_k\theta =2^{kN}{\int }_{{\mathbb{R}}^N}h(2^k(x-y))(S_{k-1}(v)(x)-S_{k-1}(v)(y))\cdot \mathrm{\nabla }{\mathrm{\Delta }}_k\theta dy.$$

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

$${\parallel S_{k-1}(v)(x)-S_{k-1}(v)(y)\parallel }_{L^{\mathrm{\infty }}}\le {\parallel v\parallel }_{C^{{\delta }_1}}{|x-y|}^{{\delta }_1},$$

we obtain

$${\parallel [{\mathrm{\Delta }}_j,S_{k-1}v\cdot \mathrm{\nabla }]{\mathrm{\Delta }}_k\theta \parallel }_{L^p}\le 2^{-{\delta }_{1}j}{\parallel v\parallel }_{C^{{\delta }_1}}{2}^k{\parallel {\mathrm{\Delta }}_k\theta \parallel }_{L^p}.$$

Therefore,

$$\begin{array}{rcl}|I_{11}|& \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{2}^{-{\delta }_{1}j}{\parallel v\parallel }_{C^{{\delta }_1}}\sum _{|k-j|\le 4}{2}^k{\parallel {\mathrm{\Delta }}_k\theta \parallel }_{L^p}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{2}^{(1-2{\delta }_1)j}{\parallel v\parallel }_{C^{{\delta }_1}}\sum _{|k-j|\le 4}{2}^{{\delta }_{1}k}{\parallel {\mathrm{\Delta }}_k\theta \parallel }_{L^p}{2}^{(k-j)(1-{\delta }_1)}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{2}^{(1-2{\delta }_1)j}{\parallel v\parallel }_{C^{{\delta }_1}}{\parallel \theta \parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}.\end{array}$$

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

$$\begin{array}{rcl}|I_{13}|& \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}\sum _{|k-j|\le 4}{\parallel S_{k-1}v-S_{j}v\parallel }_{L^p}{\parallel \mathrm{\nabla }{\mathrm{\Delta }}_j\theta \parallel }_{L^{\mathrm{\infty }}}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{2}^{(1-{\delta }_1)j}{\parallel \theta \parallel }_{C^{{\delta }_1}}\sum _{|k-j|\le 4}{\parallel {\mathrm{\Delta }}_{k}v\parallel }_{L^p}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{2}^{(1-2{\delta }_1)j}{\parallel \theta \parallel }_{C^{{\delta }_1}}\sum _{|k-j|\le 4}{2}^{{\delta }_{1}k}{\parallel {\mathrm{\Delta }}_{k}v\parallel }_{L^p}{2}^{(j-k){\delta }_{1}k}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{2}^{(1-2{\delta }_1)j}{\parallel \theta \parallel }_{C^{{\delta }_1}}{\parallel v\parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}.\end{array}$$

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

$$\begin{array}{rcl}{I}_2& \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}\sum _{|k-j|\le 4}{\parallel {\mathrm{\Delta }}_{k}v\parallel }_{L^p}{\parallel \mathrm{\nabla }{S}_{k-1}\theta \parallel }_{L^{\mathrm{\infty }}}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}\sum _{|k-j|\le 4}{\parallel {\mathrm{\Delta }}_{k}v\parallel }_{L^p}\sum _{k^{\prime }\le k-2}{2}^{k^{\prime }}{\parallel {\mathrm{\Delta }}_{k^{\prime }}\theta \parallel }_{L^{\mathrm{\infty }}}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}\sum _{|k-j|\le 4}{\parallel {\mathrm{\Delta }}_{k}v\parallel }_{L^p}{2}^{(1-{\delta }_1)k}\sum _{k^{\prime }\le k-2}{2}^{(k^{\prime }-k)(1-{\delta }_1)}{2}^{k^{\prime }{\delta }_1}{\parallel {\mathrm{\Delta }}_{k^{\prime }}\theta \parallel }_{L^{\mathrm{\infty }}}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{\parallel \theta \parallel }_{C^{{\delta }_1}}\sum _{|k-j|\le 4}{\parallel {\mathrm{\Delta }}_{k}v\parallel }_{L^p}{2}^{(1-{\delta }_1)k}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{\parallel \theta \parallel }_{C^{{\delta }_1}}{2}^{(1-2{\delta }_1)j}\sum _{|k-j|\le 4}{2}^{{\delta }_{1}k}{\parallel {\mathrm{\Delta }}_{k}v\parallel }_{L^p}{2}^{(k-j)(1-2{\delta }_1)}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{\parallel \theta \parallel }_{C^{{\delta }_1}}{2}^{(1-2{\delta }_1)j}{\parallel v\parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}.\end{array}$$

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

$$\begin{array}{rcl}{I}_3& \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{\parallel {\mathrm{\Delta }}_j\mathrm{\nabla }\cdot \left(\sum _{\begin{array}{c}k\ge j-2\\ |k-l|\le 1\end{array}}{\mathrm{\Delta }}_{l}v{\mathrm{\Delta }}_k\theta \right)\parallel }_{L^p}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{2}^j{\parallel v\parallel }_{C^{{\delta }_1}}\sum _{k\ge j-2}{2}^{-{\delta }_{1}k}{\parallel {\mathrm{\Delta }}_k\theta \parallel }_{L^p}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{2}^{(1-2{\delta }_1)j}{\parallel v\parallel }_{C^{{\delta }_1}}\sum _{k\ge j-2}{2}^{-2{\delta }_1(k-j)}{2}^{{\delta }_{1}k}{\parallel {\mathrm{\Delta }}_k\theta \parallel }_{L^p}\\ \le & Cp{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}^{p-1}{2}^{(1-2{\delta }_1)j}{\parallel v\parallel }_{C^{{\delta }_1}}{\parallel \theta \parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}.\end{array}$$

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

$$\begin{array}{rcl}\frac{d}{dt}{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}+C\nu 2^{\alpha j}{\parallel {\mathrm{\Delta }}_j\theta \parallel }_{L^p}& \le & 2^{(1-2{\delta }_1)j}{\parallel v\parallel }_{C^{{\delta }_1}}{\parallel \theta \parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}+2^{(1-2{\delta }_1)j}{\parallel \theta \parallel }_{C^{{\delta }_1}}{\parallel v\parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}\\ +2^{(1-2{\delta }_1)j}{\parallel \theta \parallel }_{C^{{\delta }_1}}{\parallel v\parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}+2^{(1-2{\delta }_1)j}{\parallel v\parallel }_{C^{{\delta }_1}}{\parallel \theta \parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}.\end{array}$$

Integrating with time t, we have

$$\begin{array}{rcl}{\parallel {\mathrm{\Delta }}_j\theta (t)\parallel }_{L^p}& \le & e^{-C\nu 2^{\alpha j}t}{\parallel {\mathrm{\Delta }}_j\theta (0)\parallel }_{L^p}\\ +C{\int }_0^t{e}^{-C\nu 2^{\alpha j}(t-\tau )}{2}^{(1-2{\delta }_1)j}({\parallel v\parallel }_{C^{{\delta }_1}}{\parallel \theta \parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}+{\parallel \theta \parallel }_{C^{{\delta }_1}}{\parallel v\parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}})d\tau .\end{array}$$

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

$$\begin{array}{rcl}{\parallel \theta (t)\parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{\alpha +2{\delta }_1-1}}& \le & \underset{j}{sup}\left\{e^{-C\nu 2^{\alpha j}t}{2}^{(\alpha +{\delta }_1-1)j}\right\}{\parallel \theta (0)\parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}\\ +C{\nu }^{-1}\underset{j}{sup}\left\{(1-e^{-C\nu 2^{\alpha j}t})\right\}\underset{\tau \in [0,t]}{max}{\parallel \theta (\tau )\parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}{\parallel \theta (\tau )\parallel }_{C^{{\delta }_1}}.\end{array}$$

Here we have used the fact that

$${\parallel v\parallel }_{C^{{\delta }_1}}\le {\parallel \theta \parallel }_{C^{{\delta }_1}}\text{and}{\parallel v\parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}\le {\parallel \theta \parallel }_{{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}}.$$

Therefore, we conclude that if

$$\theta \in L^{\mathrm{\infty }}([0,t];{\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_1}\cap C^{{\delta }_1}),$$

then

$$\theta (\cdot ,t)\in {\dot{B}}_{p,\mathrm{\infty }}^{\alpha +2{\delta }_1-1}.$$

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,

$${\dot{B}}_{p,\mathrm{\infty }}^{\alpha +2{\delta }_1-1}\subset {\dot{B}}_{\mathrm{\infty },\mathrm{\infty }}^{{\delta }_2},$$

where

$${\delta }_2=2{\delta }_1+\alpha -1-\frac{N}{p}={\delta }_1+({\delta }_1-(1-\alpha +\frac{N}{p})).$$

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

$$p>p_1\equiv \frac{N}{{\delta }_1-(1-\alpha )}.$$

Noticing that

$${\dot{B}}_{\mathrm{\infty },\mathrm{\infty }}^{{\delta }_2}\cap L^{\mathrm{\infty }}=C^{{\delta }_2},$$

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

$$\theta (\cdot ,t)\in {\dot{B}}_{p,\mathrm{\infty }}^{{\delta }_2}\cap C^{{\delta }_2}$$

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

$$\theta (\cdot ,t)\in C^{\gamma }$$

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. □