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Logarithmic Sobolev Inequalities for Bounded Domains and Applications to Drift-Diffusion Equations

Elie Abdo, Fizay-Noah Lee

Abstract

We prove logarithmic Sobolev inequalities on higher-dimensional bounded smooth domains based on novel Gagliardo-Nirenberg type interpolation inequalities. Moreover, we use them to address the long-time dynamics of some nonlinear nonlocal drift-diffusion models and prove the exponential decay of their solutions to constant steady states.

Logarithmic Sobolev Inequalities for Bounded Domains and Applications to Drift-Diffusion Equations

Abstract

We prove logarithmic Sobolev inequalities on higher-dimensional bounded smooth domains based on novel Gagliardo-Nirenberg type interpolation inequalities. Moreover, we use them to address the long-time dynamics of some nonlinear nonlocal drift-diffusion models and prove the exponential decay of their solutions to constant steady states.
Paper Structure (4 sections, 6 theorems, 44 equations)

This paper contains 4 sections, 6 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Interpolation Inequalities
  3. Logarithmic Sobolev Inequalities
  4. Application: Long-Time Behavior of Solutions to Electrodiffusion Models

Key Result

Proposition 2.1

Let $d \ge 2$. Let $\Omega \subset \mathbb R^d$ be a bounded, connected domain with smooth boundary. If $d=2$, then let $p \ge 2$ be an integer. If $d\ge 3$, then let $p$ be an integer such that $2\le p\le \frac{2d}{d-2}$. Then the following interpolation inequality holds for any nonnegative scalar function $q \in H^1(\Omega)$. Here $\bar{q}$ denotes the average of $q$ over $\Omega$, and $C$ is a

Theorems & Definitions (13)

  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Proposition 4.1
  • Proposition 4.2
  • proof
  • ...and 3 more