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A Hamilton-Jacobi Framework in a Field-Road System with Unidirectional Advection under Wentzell-Type Boundary Condition

Xinye Xiao, Haomin Huang

TL;DR

The paper develops a Hamilton-Jacobi framework for a field-road propagation model with unidirectional advection under Wentzell-type boundary conditions, deriving a variational inequality as the singular limit of a reaction-diffusion system. It yields an explicit viscosity-solution representation $v(x,y,t) = \max\{0, \varphi^*(x,y,t) - t\}$ with the fundamental solution $\varphi^*(x,y,t) = \min_{s\ge0} \{ ((-x + b s + c t)^2)/(4(t+ a s)) + ((y+s)^2)/(4 t) \}$, and identifies a geometrical transition boundary that separates rectilinear and road-assisted propagation. The analysis combines viscosity-solution theory, optimal control, and variational methods, proves convergence from the singular limit, and extends the framework to conical domains with interacting roads, accompanied by numerical simulations that reveal how advection and road diffusion shape invasion patterns. The results provide a robust variational toolkit for understanding anisotropic propagation in fragmented landscapes and offer insights into how linear infrastructures modulate front speeds. The methodology is positioned to address non-order-preserving systems and complex geometries beyond the canonical field-road setting.

Abstract

This paper develops a comprehensive Hamilton-Jacobi framework to analyze asymptotic propagation dynamics in a field-road system featuring unidirectional advection and Wentzell-type boundary conditions. We rigorously derive a Hamilton-Jacobi variational inequality as the singular limit of a reaction-diffusion system in the upper half-plane, where the road is modeled as a degenerate one-dimensional medium with enhanced diffusion and tangential drift. By synthesizing viscosity solution theory, optimal control formulation, and variational analysis, we establish the existence, uniqueness, and explicit variational representation of the viscosity solution. The solution is characterized by a fundamental solution constructed via optimal paths, revealing a critical transition in propagation behavior governed by a geometrically derived curve that separates rectilinear and road-assisted regimes. Our framework extends to non-order-preserving systems where classical comparison methods fail, and we provide a detailed asymptotic derivation of the Wentzell boundary condition from flux continuity principles. Furthermore, we generalize the approach to conical domains with intersecting roads, demonstrating the robustness of our variational methodology. Numerical simulations illustrate how advection and diffusion parameters shape the invaded region, highlighting the interplay between field and road dynamics in determining propagation patterns.

A Hamilton-Jacobi Framework in a Field-Road System with Unidirectional Advection under Wentzell-Type Boundary Condition

TL;DR

The paper develops a Hamilton-Jacobi framework for a field-road propagation model with unidirectional advection under Wentzell-type boundary conditions, deriving a variational inequality as the singular limit of a reaction-diffusion system. It yields an explicit viscosity-solution representation with the fundamental solution , and identifies a geometrical transition boundary that separates rectilinear and road-assisted propagation. The analysis combines viscosity-solution theory, optimal control, and variational methods, proves convergence from the singular limit, and extends the framework to conical domains with interacting roads, accompanied by numerical simulations that reveal how advection and road diffusion shape invasion patterns. The results provide a robust variational toolkit for understanding anisotropic propagation in fragmented landscapes and offer insights into how linear infrastructures modulate front speeds. The methodology is positioned to address non-order-preserving systems and complex geometries beyond the canonical field-road setting.

Abstract

This paper develops a comprehensive Hamilton-Jacobi framework to analyze asymptotic propagation dynamics in a field-road system featuring unidirectional advection and Wentzell-type boundary conditions. We rigorously derive a Hamilton-Jacobi variational inequality as the singular limit of a reaction-diffusion system in the upper half-plane, where the road is modeled as a degenerate one-dimensional medium with enhanced diffusion and tangential drift. By synthesizing viscosity solution theory, optimal control formulation, and variational analysis, we establish the existence, uniqueness, and explicit variational representation of the viscosity solution. The solution is characterized by a fundamental solution constructed via optimal paths, revealing a critical transition in propagation behavior governed by a geometrically derived curve that separates rectilinear and road-assisted regimes. Our framework extends to non-order-preserving systems where classical comparison methods fail, and we provide a detailed asymptotic derivation of the Wentzell boundary condition from flux continuity principles. Furthermore, we generalize the approach to conical domains with intersecting roads, demonstrating the robustness of our variational methodology. Numerical simulations illustrate how advection and diffusion parameters shape the invaded region, highlighting the interplay between field and road dynamics in determining propagation patterns.

Paper Structure

This paper contains 15 sections, 12 theorems, 200 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. A VARIATIONAL FORMULA AND VISCOSITY SOLUTIONS
  3. ESTIMATES AND LIMITS OF THE PHASE FUNCTION
  4. Convergence of the Phase Function
  5. Formulation for the Asymptotic Behaviour of the Phase Function
  6. CONICAL DOMAINS
  7. NUMERICAL SIMULATION
  8. Asymptotic Derivation of the Wentzell-Type Boundary Condition
  9. Model setup and notation
  10. Interface conditions
  11. Road averaging and integrated equation
  12. Singular limit and scaling assumptions
  13. Limit and the Wentzell boundary condition
  14. Strict Convexity of $\Omega$
  15. Rotational Monotonicity of $J$

Key Result

Proposition 1

Let $\underline{v}$ and $\bar{v}$ be a viscosity sub-solution and a super-solution of (2.1) respectively. Assume that $\underline{v}$ is bounded above, $\bar{v}$ is bounded below, and $\underline{v}(x,y,0) \leq \bar{v}(x,y,0)$ on $\overline{\mathbb{R}_+^2}$, then $\underline{v} \leq \bar{v}$ on $\ov

Figures (2)

  • Figure 1: Optimal path topology for the fundamental solution $\varphi^*$. Panel (I): rectilinear propagation with $s^*=0$; Panel (II): field-road transition with $s^*>0$ where the optimal trajectory first reaches the road then travels along it. Dashed curve marks the critical transition boundary $y=\frac{a}{2t}(x-ct)^2 + b(x-ct)$, which separates the rectilinear and field--road regimes.
  • Figure 2: Level sets of $\varphi^*(x,y,1)<1$ for different parameter sweeps. The panel shows (left) $a=2,\;b=2$ with $c$ varying from $1$ to $10$ and (right) $a=2,\;c=2$ with $b$ varying from $1$ to $10$.

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Proposition 1: Comparison principle for viscosity solutions
  • Theorem 1
  • proof
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • proof
  • Remark 1
  • ...and 16 more