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Homogenization of Smoluchowski-type equations with transmission boundary conditions

Bruno Franchi, Silvia Lorenzani

TL;DR

It is proved the existence, positivity and boundedness of solutions to the model equations derived at the microscale, and the convergence of the homogenization process to the solution of a macro-model asymptotically consistent with the microscopic one.

Abstract

In this work, we prove a two-scale homogenization result for a set of diffusion-coagulation Smoluchowski-type equations with transmission boundary conditions. This system is meant to describe the aggregation and diffusion of pathological tau proteins in the cerebral tissue, a process associated with the onset and evolution of a large variety of tauopathies (such as Alzheimer's disease). We prove the existence, positivity and boundedness of solutions to the model equations derived at the microscale (that is the scale of single neurons). Then, we study the convergence of the homogenization process to the solution of a macro-model asymptotically consistent with the microscopic one.

Homogenization of Smoluchowski-type equations with transmission boundary conditions

TL;DR

It is proved the existence, positivity and boundedness of solutions to the model equations derived at the microscale, and the convergence of the homogenization process to the solution of a macro-model asymptotically consistent with the microscopic one.

Abstract

In this work, we prove a two-scale homogenization result for a set of diffusion-coagulation Smoluchowski-type equations with transmission boundary conditions. This system is meant to describe the aggregation and diffusion of pathological tau proteins in the cerebral tissue, a process associated with the onset and evolution of a large variety of tauopathies (such as Alzheimer's disease). We prove the existence, positivity and boundedness of solutions to the model equations derived at the microscale (that is the scale of single neurons). Then, we study the convergence of the homogenization process to the solution of a macro-model asymptotically consistent with the microscopic one.
Paper Structure (7 sections, 16 theorems, 215 equations)

This paper contains 7 sections, 16 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Background of this work
  4. Outline of the paper
  5. Existence of solutions
  6. Positivity and boundedness of solutions
  7. Homogenization

Key Result

Theorem 1.1

Let $v_m^{\epsilon}(t,x,z)$ and $u_m^{\epsilon}(t,x,z)$ ($1 \leq m \leq M$) be a family of weak solutions to the system (2.1a) (see Definition d2.1). The sequences $v_m^{\epsilon}$, $\nabla_x v_m^{\epsilon}$, $u_m^{\epsilon}$, $\epsilon \nabla_x u_m^{\epsilon}$ ($1 \leq m \leq M$), two-scale converg where In (hom2.1) and (hom2.2), $A$ is a matrix with constant coefficients defined by with $\hat{

Theorems & Definitions (28)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 18 more