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1D pressureless gas dynamics systems in a strip

Abhrojyoti Sen

TL;DR

The paper develops an explicit boundary-potential framework to construct global measure-valued entropy solutions for the 1D pressureless gas dynamics system in a strip. By formulating initial and boundary potentials and minimizing them, it builds a pair $(m,u)$ with $dq=udm$ and $dE=\tfrac12 u^2 dm$, proving weak existence and entropy admissibility while capturing boundary mass accumulation as Dirac masses when classical boundary fulfillment fails. It also provides detailed Riemann-type examples that illustrate boundary concentration, interior curve concentration, and rarefaction–shock interactions, highlighting how boundary data govern singularity formation and propagation. The results illuminate boundary effects and singularity formation in measure-valued solutions for hyperbolic systems and point toward extensions to nonhomogeneous terms and discontinuous sources.

Abstract

We construct explicit measure-valued solutions to the one-dimensional pressureless gas dynamics system in a strip-like domain by introducing a new boundary potential. The constructed solutions satisfy an entropy condition, and depending on the boundary data and the behavior of the potentials, mass accumulation can occur at the boundaries. The approach relies on a systematic treatment of boundary potentials and their interactions with the initial data, providing a more precise understanding of the formation and propagation of singularities in measure-valued solutions.

1D pressureless gas dynamics systems in a strip

TL;DR

The paper develops an explicit boundary-potential framework to construct global measure-valued entropy solutions for the 1D pressureless gas dynamics system in a strip. By formulating initial and boundary potentials and minimizing them, it builds a pair with and , proving weak existence and entropy admissibility while capturing boundary mass accumulation as Dirac masses when classical boundary fulfillment fails. It also provides detailed Riemann-type examples that illustrate boundary concentration, interior curve concentration, and rarefaction–shock interactions, highlighting how boundary data govern singularity formation and propagation. The results illuminate boundary effects and singularity formation in measure-valued solutions for hyperbolic systems and point toward extensions to nonhomogeneous terms and discontinuous sources.

Abstract

We construct explicit measure-valued solutions to the one-dimensional pressureless gas dynamics system in a strip-like domain by introducing a new boundary potential. The constructed solutions satisfy an entropy condition, and depending on the boundary data and the behavior of the potentials, mass accumulation can occur at the boundaries. The approach relies on a systematic treatment of boundary potentials and their interactions with the initial data, providing a more precise understanding of the formation and propagation of singularities in measure-valued solutions.

Paper Structure

This paper contains 17 sections, 11 theorems, 161 equations, 9 figures.

Table of Contents

  1. Introduction and main result
  2. State of the art
  3. Main result
  4. Extensions and open problems
  5. Organization of the article
  6. Construction of solution
  7. Characteristic triangles and their properties
  8. Proof of Theorem \ref{['TH1.3']}
  9. Verification of the weak formulation
  10. Boundary condition and entropy admissibility
  11. Boundary condition
  12. Entropy condition
  13. Explicit examples with Riemann data
  14. Concentration of mass on the right boundary
  15. Concentration of mass on the left boundary
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $\rho_0(x), \rho_{b_l}(t)>0, \rho_{b_r}(t)>0,$ and $u_0(x), u_{b_l}(t)>0, u_{b_r}(t)<0,$ are locally bounded measurable functions, then the pair $(m,u)$ given by Definition d3-d2 is a global weak entropy solution to the system e1.1-e1.4 in the sense of Definition intro-defn-weak formulation and

Figures (9)

  • Figure 2.1: Figures of interior characteristic triangles (i)-(v)
  • Figure 2.2: Figure of interior characteristic triangle (vi)
  • Figure 2.3: Figures of boundary characteristic triangles (i)-(ii)
  • Figure 2.4: Figure of boundary characteristic triangle (iii)
  • Figure 2.5: Figure of boundary characteristic triangle (iv)
  • ...and 4 more figures

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem 1.2
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4: Interior characteristic lines
  • Definition 2.5: Boundary characteristic lines
  • Definition 2.6: Interior characteristic triangles
  • ...and 24 more