[Paper Review] Existence and maximal $L^{p}$-regularity of solutions for the porous medium equation on manifolds with conical singularities

Nikolaos Roidos, Elmar Schrohe|arXiv (Cornell University)|Apr 20, 2015

Advanced Mathematical Modeling in Engineering31 references4 citations

TL;DR

This paper establishes the existence, uniqueness, and maximal $L^p$-regularity of short-time solutions to the porous medium equation on manifolds with conical singularities. Using bounded imaginary powers and $R$-sectoriality techniques for cone differential operators on Mellin-Sobolev spaces, it proves that the solution exhibits controlled asymptotic behavior near the conical tip under specified weight and integrability conditions on the data and domain.

ABSTRACT

We consider the porous medium equation on manifolds with conical singularities and show existence, uniqueness and maximal $L^{p}$-regularity of a short time solution. In particular, we obtain information on the short time asymptotics of the solution near the conical point. Our method is based on bounded imaginary powers results for cone differential operators on Mellin-Sobolev spaces and $R$-sectoriality perturbation techniques.

Motivation & Objective

• To establish existence and maximal $L^p$-regularity for the porous medium equation on manifolds with conical singularities.
• To analyze the short-time asymptotic behavior of solutions near the conical point.
• To extend maximal regularity theory for the Laplacian to negative-order Mellin-Sobolev spaces with conical singularities.
• To control the trace space (interpolation space) for initial data in the context of cone differential operators.
• To apply the theorem of Clément and Li to the quasilinear parabolic problem via linearized maximal regularity techniques.

Proposed method

• Utilizes Mellin-Sobolev spaces $\mathcal{H}^{s,\gamma}_p(\mathbb{B})$ to model function spaces with weight $\gamma$ at the conical tip.
• Constructs closed extensions $\underline{\Delta}_s$ of the Laplacian with domain $\mathcal{H}^{s+2,\gamma+2}_p(\mathbb{B}) \oplus \mathbb{C}$ to ensure maximal $L^p$-regularity.
• Applies bounded imaginary powers results and $R$-sectoriality perturbation techniques to the linearized problem.
• Employs Neumann series to construct $R$-sectorial resolvents for conically degenerate operators with frozen coefficients.
• Uses interpolation theory to embed the trace space $Y_{1/q,q}$ into a Banach algebra of multipliers on $Y_0$.
• Applies the theorem of Clément and Li to the quasilinear problem by verifying the required conditions on the linearized operator.

Experimental results

Research questions

• RQ1 Can maximal $L^p$-regularity be established for the porous medium equation on manifolds with conical singularities?
• RQ2 How does the conical singularity affect the short-time asymptotic behavior of the solution?
• RQ3 What conditions on the weight $\gamma$, $p$, and $q$ ensure maximal regularity for the Laplacian in negative-order Mellin-Sobolev spaces?
• RQ4 How can the trace space for initial data be characterized in the context of cone differential operators?
• RQ5 Under what conditions does the linearized operator $-u_0^{m-1}\underline{\Delta}$ have maximal $L^q$-regularity?

Key findings

• The Laplacian admits a closed extension $\underline{\Delta}_s$ on $\mathcal{H}^{s+2,\gamma+2}_p(\mathbb{B}) \oplus \mathbb{C}$ with maximal $L^p$-regularity for all $s \in \mathbb{R}$ and suitable $\gamma$.
• For $\dim(\mathbb{B}) \neq 3$, and under conditions on $\gamma$, $p$, and $q$, the operator $-u_0^{m-1}\underline{\Delta}$ has maximal $L^q$-regularity for strictly positive initial data $u_0 \in Y_{1/q,q}$.
• The solution $u$ to the porous medium equation exists uniquely in $L^q(0,T; Y_1) \cap W^{1,q}(0,T; Y_0)$ for some $T > 0$.
• The trace space $Y_{1/q,q}$ embeds into a Banach algebra of multipliers on $Y_0$, ensuring the necessary regularity for the nonlinear term.
• The conical singularity is handled via weighted Mellin-Sobolev spaces and asymptotics spaces, with precise control on the solution's behavior near the tip.
• The results are valid for $\dim(\mathbb{B}) \geq 1$, with explicit conditions on $\gamma$ and $q$ ensuring $R$-sectoriality and maximal regularity.