Higher integrability for the singular porous medium system

Abstract

In this paper we establish in the fast diffusion range the higher integrability of the spatial gradient of weak solutions to porous medium systems. The result comes along with an explicit reverse Hölder inequality for the gradient. The novel feature in the proof is a suitable intrinsic scaling for space-time cylinders combined with reverse Hölder inequalities and a Vitali covering argument within this geometry. The main result holds for the natural range of parameters suggested by other regularity results. Our result applies to general fast diffusion systems and includes both, nonnegative and signed solutions in the case of equations. The methods of proof are purely vectorial in their structure.

Higher integrability for obstacle problem related to the singular porous medium equation

“…The method of intrinsic scaling is also applied to the study of regularity problems of porous medium type equations of the form ∂ t uu m = 0, m > 0, where u = u(x, t) and (x, t) ∈ R n+1 . There are two different approaches to the study of higher integrability of weak solutions to porous medium equations, one is the approach developed by Gianazza and Schwarzacher [9,10], the other is the approach developed by Bögelein et al [1,2]. In fact, the second method can also be used to study the vector-valued weak solutions of porous medium system.…”

“…This idea was also used in [14] to study the higher integrability of singular parabolic systems with non-standard growth. In [2] the authors considered the intrinsic cylinders of the form…”

“…Du m 2 dx dt, (1.4) which are compatible with (1.3). On the other hand, a key ingredient in the approach [1,2] is an application of the properties of the boundary term b u m , a m = m m + 1 |a| m+1 -|u| m+1u a mu m , (1.5) which was first introduced by Bögelein et al [3]. In summary, there are some differences between these two approaches and the second approach has more advantages.…”

“…In summary, there are some differences between these two approaches and the second approach has more advantages. Recently, Cho and Scheven [5,6] proved the higher integrability of weak solutions to obstacle problems related to the porous medium equation and their proofs followed the approach in [1,2]. In [6] the authors used the boundary term (1.5) to establish an energy estimate and a gluing lemma for weak solutions of the obstacle problem.…”

“…Moreover, the intrinsic cylinders in [6] takes the form (1.4). Admittedly, the approach in [2] is effective in treating the higher integrability of obstacle problem for the singular porous medium equation, but it is natural to try to use the old idea in [10] to study the same problem. To this end, the present work is intended as an attempt to follow the approach in [10] to establish a self-improving result for the obstacle problem.…”