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The fractional porous medium equation on manifolds with conical singularities II

Nikolaos Roidos, Yuanzhen Shao

Abstract

This is the second of a series of two papers which studies the fractional porous medium equation, $\partial_t u +(-Δ)^σ(|u|^{m-1}u )=0 $ with $m>0$ and $σ\in (0,1]$, posed on a Riemannian manifold with isolated conical singularities. The first aim of the article is to derive some useful properties for the Mellin-Sobolev spaces including the Rellich-Kondrachov Theorem and Sobolev-Poincaré, Nash and Super Poincaré type inequalities. The second part of the article is devoted to the study the Markovian extensions of the conical Laplacian operator and its fractional powers. Then based on the obtained results, we establish existence and uniqueness of a global strong solution for $L_\infty-$initial data and all $m>0$. We further investigate a number of properties of the solutions, including comparison principle, $L_p-$contraction and conservation of mass. Our approach is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.

The fractional porous medium equation on manifolds with conical singularities II

Abstract

This is the second of a series of two papers which studies the fractional porous medium equation, with and , posed on a Riemannian manifold with isolated conical singularities. The first aim of the article is to derive some useful properties for the Mellin-Sobolev spaces including the Rellich-Kondrachov Theorem and Sobolev-Poincaré, Nash and Super Poincaré type inequalities. The second part of the article is devoted to the study the Markovian extensions of the conical Laplacian operator and its fractional powers. Then based on the obtained results, we establish existence and uniqueness of a global strong solution for initial data and all . We further investigate a number of properties of the solutions, including comparison principle, contraction and conservation of mass. Our approach is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.

Paper Structure

This paper contains 16 sections, 28 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conical Manifolds
  4. Mellin-Sobolev spaces
  5. The Laplace-Beltrami operator on conical manifolds
  6. Closed extensions of $\Delta_g$
  7. Markovian property of $\Delta_g$
  8. Uniqueness of Markovian extension
  9. Functional inequalities in Mellin-Sobolev spaces
  10. Fractional Powers of the Laplace-Beltrami Operator
  11. Construction of the Fractional Laplace-Beltrami Operators
  12. Properties of the Fractional Laplace-Beltrami Operator
  13. Weak Solutions to the Fractional Porous Medium Equation
  14. Some Other Properties of the Fractional Porous Medium Equation
  15. Strong Solutions
  16. ...and 1 more sections

Key Result

Theorem 1.1

Suppose that $(\mathsf{M},g)$ is an $(n+1)$-dimensional conical manifold. Then for any $u_0\in L_\infty(\mathsf{M})$, S1: FPME with $\sigma\in (0,1)$ and $m>0$ has a unique global strong solution $u$. Moreover, the solution $u$ satisfies the following properties.

Theorems & Definitions (62)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 3.1
  • ...and 52 more