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On the existence and uniqueness of classical solution for an initial-boundary value problem for a discrete Boltzmann system in two space dimensions

Koudzo Togbévi Selom Sobah, Amah Séna d'Almeida

TL;DR

The paper studies the initial-boundary value problem for a two-dimensional, four-velocity Broadwell discrete Boltzmann system in a rectangle. By reformulating the problem via a transformed mixed system and employing a nonnegative Schauder-fixed-point framework, the authors prove the existence and uniqueness of a global classical solution that is nonnegative and has bounded first-order derivatives, under the data condition $pq\le\tfrac{1}{4}$, where $p$ and $q$ depend on system parameters and boundary data. The method combines a meticulous fixed-point construction with Arzelà–Ascoli–type compactness arguments and derives explicit a priori bounds on the solution, $\|N\|$, and its derivatives. This provides a rigorous foundation for reliability of 2D discrete-velocity Boltzmann simulations in rectangular geometries and highlights the role of fixed-point techniques in ensuring well-posedness for nonlinear kinetic systems.

Abstract

The initial-boundary value problem for the two-dimensional regular four-velocity discrete Boltzmann system is analyzed in a rectangle. The existence and uniqueness of a classical global positive solution, bounded with its first partial derivatives are proved for a range of bounded data by the use of fixed points tools. A bound for the solution and its partial derivatives is provided.

On the existence and uniqueness of classical solution for an initial-boundary value problem for a discrete Boltzmann system in two space dimensions

TL;DR

The paper studies the initial-boundary value problem for a two-dimensional, four-velocity Broadwell discrete Boltzmann system in a rectangle. By reformulating the problem via a transformed mixed system and employing a nonnegative Schauder-fixed-point framework, the authors prove the existence and uniqueness of a global classical solution that is nonnegative and has bounded first-order derivatives, under the data condition , where and depend on system parameters and boundary data. The method combines a meticulous fixed-point construction with Arzelà–Ascoli–type compactness arguments and derives explicit a priori bounds on the solution, , and its derivatives. This provides a rigorous foundation for reliability of 2D discrete-velocity Boltzmann simulations in rectangular geometries and highlights the role of fixed-point techniques in ensuring well-posedness for nonlinear kinetic systems.

Abstract

The initial-boundary value problem for the two-dimensional regular four-velocity discrete Boltzmann system is analyzed in a rectangle. The existence and uniqueness of a classical global positive solution, bounded with its first partial derivatives are proved for a range of bounded data by the use of fixed points tools. A bound for the solution and its partial derivatives is provided.
Paper Structure (16 sections, 12 theorems, 119 equations, 1 figure)

This paper contains 16 sections, 12 theorems, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. The discrete velocity model
  4. Initial-boundary value problem
  5. Main theorem
  6. Non-negativity of the solution
  7. Non-negative operator.
  8. Non-negativity theorem.
  9. Existence and uniqueness of bounded solution
  10. Fixed point problem
  11. Continuity of the operator of the fixed point problem
  12. Convex set on which the operator is compact.
  13. Stable convex set under the operator.
  14. Existence theorem
  15. Proof of the main theorem \ref{['thm:Suppose-.-Then']}
  16. ...and 1 more sections

Key Result

Theorem 2.1

Suppose $pq\leq\dfrac{1}{4}$. Then the system $\Sigma^{0}$eq:erter-eq:lsoo has an unique non-negative solution such that $\dfrac{\partial N_{i}}{\partial t},\dfrac{\partial N_{i}}{\partial x},\dfrac{\partial N_{i}}{\partial y}$ are defined in $\left]0;T\right[\times\left]a_{1},b_{1}\right[\times\left]a_{2},b_{2}\right[$ except possibly on a finite number of planes including the four planes with

Figures (1)

  • Figure 2.1: The model $B_{\theta}$

Theorems & Definitions (23)

  • Theorem 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Proposition 4.1
  • proof
  • ...and 13 more