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Abstract integrodifferential equations and applications

Bruno de Andrade, Marcos Gabriel de Santana

TL;DR

This work develops a general framework for initial-value problems in abstract integrodifferential equations on interpolation scales, proving local existence, uniqueness, continuation, and blow-up alternatives for regular mild solutions. Central to the approach are resolvent families associated with sectorial operators and smoothing estimates that link the memory kernel g to spatial regularity via interpolation spaces. The theory is then applied to Navier–Stokes equations with hereditary viscosity and reaction–diffusion equations with memory, obtaining ε-regular mild solution results in Stokes- and Besov/Lebesgue-space settings and detailing how memory parameters ζ_g govern regularity and existence. Overall, the paper unifies memory-driven parabolic-type models under a robust analytic framework, clarifying how material memory shapes well-posedness and regularity across continuous function spaces relevant to fluids and diffusion processes.

Abstract

In this work, we study the initial value problem associated with an abstract integrodifferential equation in interpolation scales. We prove local-in-time existence, uniqueness, continuation, and a blow-up alternative for regular mild solutions to the problem. Additionally, we apply this theory to the Navier-Stokes equations with hereditary viscosity, taking initial data in the scale of fractional power spaces associated with the Stokes operator. We also explore reaction-diffusion problems with memory, considering the effects of super-linear and gradient-type nonlinearities, and initial data in Lebesgue and Besov spaces, respectively.

Abstract integrodifferential equations and applications

TL;DR

This work develops a general framework for initial-value problems in abstract integrodifferential equations on interpolation scales, proving local existence, uniqueness, continuation, and blow-up alternatives for regular mild solutions. Central to the approach are resolvent families associated with sectorial operators and smoothing estimates that link the memory kernel g to spatial regularity via interpolation spaces. The theory is then applied to Navier–Stokes equations with hereditary viscosity and reaction–diffusion equations with memory, obtaining ε-regular mild solution results in Stokes- and Besov/Lebesgue-space settings and detailing how memory parameters ζ_g govern regularity and existence. Overall, the paper unifies memory-driven parabolic-type models under a robust analytic framework, clarifying how material memory shapes well-posedness and regularity across continuous function spaces relevant to fluids and diffusion processes.

Abstract

In this work, we study the initial value problem associated with an abstract integrodifferential equation in interpolation scales. We prove local-in-time existence, uniqueness, continuation, and a blow-up alternative for regular mild solutions to the problem. Additionally, we apply this theory to the Navier-Stokes equations with hereditary viscosity, taking initial data in the scale of fractional power spaces associated with the Stokes operator. We also explore reaction-diffusion problems with memory, considering the effects of super-linear and gradient-type nonlinearities, and initial data in Lebesgue and Besov spaces, respectively.
Paper Structure (12 sections, 24 theorems, 314 equations)

This paper contains 12 sections, 24 theorems, 314 equations.

Table of Contents

  1. Introduction
  2. Resolvent families and elements of abstract interpolation theory
  3. Local well-posedness
  4. The subcritical case
  5. The critical case
  6. Continuation
  7. Blow-up alternative
  8. Applications
  9. Navier-Stokes equations with hereditary viscosity Navier-Stokes
  10. Semilinear reaction-diffusion equations with memory Reaction-diffusion equations
  11. Super-linear nonlinearity and initial data in Lebesgue spaces
  12. Gradient nonlinearity and initial data in Besov spaces

Key Result

Theorem 2.1

Let $A:\mathcal{D}(A)\subset X_0 \to X_0$ such that $-A$ is a sectorial operator of angle $\psi_0 \in [0,\pi/2)$ and $g \in L^1_{loc}([0,\infty);\mathbb{C})$ a non identically zero Laplace transformable function. Suppose, for some $\omega_0 \geq 0$ and $\eta_0 \in (0,\pi/2]$, the following condition Then, there exists the analytic resolvent family $\{S(t)\}_{t \geq 0}$ associated with the pair $(A

Theorems & Definitions (63)

  • Definition 2.1: Resolvent family
  • Definition 2.2: Analytic resolvent family
  • Definition 2.3: Sectorial operator
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Proposition 2.1
  • proof
  • Lemma 2.1
  • proof
  • ...and 53 more