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Well posedness of linear parabolic partial differential equations posed on a star-shaped network with local time Kirchhoff's boundary condition at the vertex

Miguel Martinez, Isaac Ohavi

TL;DR

The paper addresses the well-posedness of a linear parabolic system defined on a star-shaped network with a dynamic local-time boundary transmission at the junction, driven by an external variable $l$. It develops a parabolic discretization in $l$ and employs an elliptic-approximation scheme to obtain uniform bounds and compactness, enabling existence and uniqueness in Hölder-type spaces. A comparison principle is established, and a refined analysis yields interior regularity and derivative bounds, connecting to Walsh spider diffusions. The results extend classical network-parabolic theory to a setting with time-dependent vertex transmission and lay groundwork for stochastic scattering interpretations and diffusion on graphs with local-time boundary control.

Abstract

The main purpose of this work is to provide an existence and uniqueness result for the solution of a linear parabolic system posed on a star-shaped network, which presents a new type of Kirchhoff's boundary transmission condition at the junction. This new type of Kirchhoff's condition-that we decide to call here local-time Kirchhoff 's condition-induces a dynamical behavior with respect to an external variable that may be interpreted as a local time parameter, designed to drive the system only at the singular point of the network. The seeds of this study point towards a forthcoming theoretical inquiry of a particular generalization of Walsh's random spider motions, whose spinning measures would select the available directions according to the local time of the motion at the junction of the network.

Well posedness of linear parabolic partial differential equations posed on a star-shaped network with local time Kirchhoff's boundary condition at the vertex

TL;DR

The paper addresses the well-posedness of a linear parabolic system defined on a star-shaped network with a dynamic local-time boundary transmission at the junction, driven by an external variable . It develops a parabolic discretization in and employs an elliptic-approximation scheme to obtain uniform bounds and compactness, enabling existence and uniqueness in Hölder-type spaces. A comparison principle is established, and a refined analysis yields interior regularity and derivative bounds, connecting to Walsh spider diffusions. The results extend classical network-parabolic theory to a setting with time-dependent vertex transmission and lay groundwork for stochastic scattering interpretations and diffusion on graphs with local-time boundary control.

Abstract

The main purpose of this work is to provide an existence and uniqueness result for the solution of a linear parabolic system posed on a star-shaped network, which presents a new type of Kirchhoff's boundary transmission condition at the junction. This new type of Kirchhoff's condition-that we decide to call here local-time Kirchhoff 's condition-induces a dynamical behavior with respect to an external variable that may be interpreted as a local time parameter, designed to drive the system only at the singular point of the network. The seeds of this study point towards a forthcoming theoretical inquiry of a particular generalization of Walsh's random spider motions, whose spinning measures would select the available directions according to the local time of the motion at the junction of the network.
Paper Structure (19 sections, 15 theorems, 282 equations)

This paper contains 19 sections, 15 theorems, 282 equations.

Table of Contents

  1. Introduction
  2. Introduction and Main results
  3. Notations and preliminary results
  4. Assumptions and main results
  5. Existence and uniqueness for a Parabolic PDE with Kirchhoff's local time condition
  6. The main result obtained by Von Below in Von Below revisited.
  7. Uniform bound for the solution
  8. Uniform bound for the approximated time derivative
  9. Refined estimates for the approximated time derivative at the junction point.
  10. Global gradient estimate
  11. The main result of Von Below Von Below revisited
  12. Proof of the main result
  13. Uniform bound for $|u^p|_\infty$
  14. Uniform bound for the Lipschitz constant $|u^p(\cdot,0)|_{\lfloor W^{1,\infty}([0,T])\rfloor}$ at the junction point.
  15. Uniform bound for the gradient $|\partial_x u^p|_\infty$
  16. ...and 4 more sections

Key Result

Lemma 2.2

Fix $\underline{K}>0$. Assume that $u:=u(t,x,l) \in \mathcal{C}^{0,1,0}([0,T]\times[0,R]\times [0,\underline{K}])$ satisfies for some given constants $\nu_1, \nu_2, \nu_3\in \mathbb R^+$ and $\alpha, \beta, \gamma \in (0,1)$. Then

Theorems & Definitions (30)

  • Remark 1.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Proposition 3.1
  • ...and 20 more