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Functional Inequalities involving Nonlocal Operators on Complete Riemannian Manifolds and Their Applications to The Fractional Porous Medium Equation

Nikolaos Roidos, Yuanzhen Shao

Abstract

The objective of this paper is twofold. First, we conduct a careful study of various functional inequalities involving the fractional Laplacian operators, including nonlocal Sobolev-Poincaré, Nash, Super Poincaré and logarithmic Sobolev type inequalities, on complete Riemannian manifolds satisfying some mild geometric assumptions. Second, based on the derived nonlocal functional inequalities, we analyze the asymptotic behavior of the solution to the fractional porous medium equation, $\partial_t u +(-Δ)^σ(|u|^{m-1}u )=0 $ with $m>0$ and $σ\in (0,1)$. In addition, we establish the global well-posedness of the equation on an arbitrary complete Riemannian manifold.

Functional Inequalities involving Nonlocal Operators on Complete Riemannian Manifolds and Their Applications to The Fractional Porous Medium Equation

Abstract

The objective of this paper is twofold. First, we conduct a careful study of various functional inequalities involving the fractional Laplacian operators, including nonlocal Sobolev-Poincaré, Nash, Super Poincaré and logarithmic Sobolev type inequalities, on complete Riemannian manifolds satisfying some mild geometric assumptions. Second, based on the derived nonlocal functional inequalities, we analyze the asymptotic behavior of the solution to the fractional porous medium equation, with and . In addition, we establish the global well-posedness of the equation on an arbitrary complete Riemannian manifold.

Paper Structure

This paper contains 21 sections, 17 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Main Results and Geometric Assumptions
  3. Subordinated Semigroups and the Fractional Laplacian
  4. Functional Inequalities via Subordination
  5. A Logarithmic Sobolev Inequality on Complete Non-compact Manifolds
  6. A Logarithmic Sobolev Inequality on Closed Manifolds
  7. Other Functional Inequalities, Heat Kernel and Semigroup Estimates via Subordination
  8. Solutions to the Fractional Porous Medium Equation
  9. Definition of Solutions
  10. Existence of Weak Solution
  11. Existence and Uniqueness of Strong Solution
  12. Proof of Theorem \ref{['S4: global weak sol']}
  13. Asymptotic Behavior: Complete and Non-compact Manifolds
  14. Asymptotic behavior for $u_0\in L_1(\mathsf{M}) \cap L_\infty(\mathsf{M})$
  15. Proof of Theorem \ref{['S5: Large time behavior']}
  16. ...and 6 more sections

Key Result

Theorem 2.1

Suppose that $(\mathsf{M},g)$ is a Riemannian manifold satisfying (A1) with $n>2$. Then the followings hold true.

Theorems & Definitions (34)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 4.1: Theorem II.5.2 in VarCosteCoul92
  • proof
  • ...and 24 more