- Open Access
Regularity of Hölder continuous solutions of the supercritical porous media equation
© Yu et al.; licensee Springer. 2013
- Received: 25 July 2013
- Accepted: 27 August 2013
- Published: 7 November 2013
In this paper, we present a regularity result for weak solutions of the N-dimensional () porous media equation with supercritical () dissipation . If a Leray-Hopf weak solution is Hölder continuous with on the time interval , then it is actually a classical solution on .
- porous media equation
- supercritical dissipation
- weak solutions
where 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 is the last canonical vector . For brevity, we only consider .
where is the dissipative coefficient, and the differential operator is given by . Considering the scaling transform for , the system will be divided into three cases: the case is called the critical case, the case is subcritical and the case is supercritical.
where , , , , are all operators mapping scalar functions to vector-valued functions and equals a constant multiplication operator, whereas 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 () 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 () is somewhat a generalization of (QG) equation. Very recently, the system () was introduced and investigated by Córdoba and his group. In , the authors obtained some results on strong solutions, weak solutions and attractors for the dissipative system (). For finite energy, they obtained global existence and uniqueness in the subcritical and critical cases. In the supercritical case, they obtained local results in , and extended to be global under a small condition , for , where c is a small fixed constant. In , they treated the nondissipative () 2D case and obtained the local existence and uniqueness in the Hölder space 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 with on the time interval , then it is actually a classical solution on . 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 , it also belongs to the Besov space for any . By taking p sufficiently large, we have for . The idea is to show that with . Through iteration, we establish that with . 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 to denote the estimate for some C.
then for .
for some . 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).
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