Regularity of Hölder continuous solutions of the supercritical porous media equation

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 $k=g=1$.

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.

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.

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.

Now we give the definitions of Besov spaces.

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].

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

We next introduce the classical Bernstein inequality [13].

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]).

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

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]).

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.

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

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. □