Local bounds of the gradient of weak solutions to the porous medium equation

Ugo Gianazza () and Juhana Siljander ()
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Ugo Gianazza: Università di Pavia
Juhana Siljander: Imperial College London

Partial Differential Equations and Applications, 2023, vol. 4, issue 2, 1-35

Abstract: Abstract Let u be a nonnegative, local, weak solution to the porous medium equation $$\begin{aligned} \partial _t u-\Delta u^m=0 \end{aligned}$$ ∂ t u - Δ u m = 0 for $$m\ge 2$$ m ≥ 2 in a space-time cylinder $$\Omega _T=\Omega \times (0,T]$$ Ω T = Ω × ( 0 , T ] . Fix a point $$(x_{o},t_{o})\in \Omega _T$$ ( x o , t o ) ∈ Ω T : if the average then the quantity $$|\nabla u^{m-1}|$$ | ∇ u m - 1 | is locally bounded in a proper cylinder, whose center lies at time $$t_o+a^{1-m}r^2$$ t o + a 1 - m r 2 . This implies that in the same cylinder the solution u is Hölder continuous with exponent $$\alpha =\frac{1}{m-1}$$ α = 1 m - 1 , which is known to be optimal. Moreover, u presents a sort of instantaneous regularization, which we discuss.

Keywords: Degenerate parabolic; Porous medium equation; Gradient boundedness; Optimal Hölder continuity; Primary 35K65; Secondary 35B65 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s42985-022-00217-9

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