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Global-in-time existence and uniqueness of classical solutions to the unsteady initial-boundary value problem for the four-velocity planar Broadwell model in a rectangular domain

Koudzo Togbévi Selom Sobah, Amah Séna D'Almeida

Abstract

We establish the global-in-time existence and uniqueness of classical solutions to the unsteady initial-boundary value problem for the four-velocity planar Broadwell model in a rectangular domain. The result is obtained for sufficiently small initial and boundary data within the framework of discrete kinetic theory.

Global-in-time existence and uniqueness of classical solutions to the unsteady initial-boundary value problem for the four-velocity planar Broadwell model in a rectangular domain

Abstract

We establish the global-in-time existence and uniqueness of classical solutions to the unsteady initial-boundary value problem for the four-velocity planar Broadwell model in a rectangular domain. The result is obtained for sufficiently small initial and boundary data within the framework of discrete kinetic theory.
Paper Structure (12 sections, 12 theorems, 116 equations)

This paper contains 12 sections, 12 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Two initial-boundary value problems
  3. The discrete Boltzmann equations
  4. Two initial-boundary value problems
  5. Reformulation of one initial-boundary value problem as fixed-point problems
  6. Continuity of the operators of the fixed-point problems
  7. Uniqueness theorem
  8. Convex set on which the operators are compact
  9. Convex set stable under the operators
  10. Existence theorem for the bounded time interval
  11. Existence theorem for the unbounded time interval
  12. Conclusion

Key Result

Theorem 2.1

The solution to the problem $\Sigma_{\tau,\tau'}$(respectively $\Sigma$) when it exists, is unique.

Theorems & Definitions (24)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Proposition 4.1
  • proof
  • proof
  • Proposition 6.1
  • ...and 14 more