Regularity results for a class of widely degenerate parabolic equations
Pasquale Ambrosio, Antonia Passarelli di Napoli
Abstract
Abstract Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>-</m:mo> <m:mrow> <m:mi>div</m:mi> <m:mo></m:mo> <m:mrow> <m:mo maxsize="160%" minsize="160%">(</m:mo> <m:mrow> <m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mo>-</m:mo> <m:mi>ν</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo></m:mo> <m:mfrac> <m:mrow> <m:mi>D</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> </m:mfrac> </m:mrow> <m:mo maxsize="160%" minsize="160%">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> </m:mrow> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mrow> <m:mrow> <m:mtext>in </m:mtext> <m:mo></m:mo> <m:msub> <m:mi mathvariant="normal">Ω</m:mi> <m:mi>T</m:mi> </m:msub> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>×</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> u_{t}-\operatorname{div}\Bigl{(}(\lvert Du\rvert-\nu)_{+}^{p-1}\frac{Du}{% \lvert Du\rvert}\Bigr{)}=f\quad\text{in }\Omega_{T}=\Omega\times(0,T), where Ω is a bounded domain in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {\mathbb{R}^{n}} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {n\geq 2} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {p\geq 2} , ν is a positive constant and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">(</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> </m:msub> </m:math> {(\,\cdot\,)_{+}} stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue–Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>u</m:mi> <m:mi>t</m:mi> </m:msub> </m:math> {u_{