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On Convolution in Variable Lebesgue Spaces and Applications to Fractional Navier_Stokes Equations

Salah BenMahmoud

Abstract

In this paper, we introduce a new class of convolution-type inequalities in variable exponent Lebesgue spaces and derive several related estimates, including the \(L^{r(\cdot)}\)--\(L^{p(\cdot)}\) smoothing estimate for the fractional heat kernel. We demonstrate the usefulness of these inequalities by establishing the local well-posedness results for mild solutions to the fractional Navier--Stokes equations, and we further extend these results to global-in-time well-posedness for sufficiently small initial data. Our analysis is carried out in a wide range of mixed-norm variable exponent Lebesgue spaces, including the fully variable setting $L^{p(\cdot)}_t L^{q(\cdot)}_x$, highlighting the robustness of the proposed framework under non-constant integrability. Moreover, the proposed framework is expected to serve as a key tool for similar applications in other related variable exponent function spaces.

On Convolution in Variable Lebesgue Spaces and Applications to Fractional Navier_Stokes Equations

Abstract

In this paper, we introduce a new class of convolution-type inequalities in variable exponent Lebesgue spaces and derive several related estimates, including the \(L^{r(\cdot)}\)--\(L^{p(\cdot)}\) smoothing estimate for the fractional heat kernel. We demonstrate the usefulness of these inequalities by establishing the local well-posedness results for mild solutions to the fractional Navier--Stokes equations, and we further extend these results to global-in-time well-posedness for sufficiently small initial data. Our analysis is carried out in a wide range of mixed-norm variable exponent Lebesgue spaces, including the fully variable setting , highlighting the robustness of the proposed framework under non-constant integrability. Moreover, the proposed framework is expected to serve as a key tool for similar applications in other related variable exponent function spaces.
Paper Structure (7 sections, 29 theorems, 175 equations)

This paper contains 7 sections, 29 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On Convolution in Variable Lebesgue Spaces
  4. Applications to Fractional Navier–-Stokes Equations
  5. Estimates for the Fractional Heat Semigroup
  6. Existence of Local Mild Solutions
  7. Global Existence with Small initial Data

Key Result

Lemma 2.1

Let $p \in \mathcal{P}^{\log}(\mathbb{R}^n)$ and let $\psi \in L^1(\mathbb{R}^n)$. Assume that the least bell-shaped majorant $\Psi$ of $\psi$ is integrable. Then for all $f \in L^{p(\cdot)}(\mathbb{R}^n)$, where $\psi_\varepsilon(x):=\varepsilon^{-n}\psi(\varepsilon^{-1}x),\, x\in \mathbb{R}^n$ and $c$ is a constant of depends only on $p$ and $n$.

Theorems & Definitions (54)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6: Banach--Picard principle
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 44 more