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Vanishing viscosity solution to a 2 x 2 system of conservation laws with linear damping

Kayyunnapara Divya Joseph

TL;DR

The paper addresses a $2\times 2$ conservation-law system with linear damping that modelizes Eulerian droplet dynamics, focusing on general initial data beyond Riemann-type. It regularizes the system with a parabolic term $\epsilon e^{-\alpha t}$, and derives explicit weakly regularized solutions via a Hopf-Cole reduction, enabling a rigorous vanishing-viscosity analysis. By employing the weak asymptotic method, Volpert product, and Colombeau algebra, it constructs weak, distributional, and Colombeau solutions, proves the vanishing-viscosity limit yields a distributional solution, and establishes large-time behavior of the viscous system. The results provide a coherent framework for handling non-conservative products and measure-valued components (delta-type shocks) in a general-initial-data setting, with implications for the stability and physical interpretation of the Eulerian droplet model.

Abstract

Systems of the first order partial differential equations with singular solutions appear in many multiphysics problems and the weak formulation of solutions involve in many cases product of distributions. In this paper we study such a system derived from Eulerian droplet model for air particle flow. This is a 2 x 2 non - strictly hyperbolic system of conservation laws with linear damping. We first study a regularized viscous system with variable viscosity term and obtain a weak asymptotic solution with general initial data and also get solution in the Colombeau algebra. We also study the vanishing viscosity limit and show that this limit is a distribution solution. Further we study the large time asymptotic behaviour of the viscous system. This important system, is not very well studied due to complexities in the analysis. As far as we know the only work done on this system is for Riemann type of initial data. The significance of this paper is that we work on the system having general initial data and not just initial data of the Riemann type.

Vanishing viscosity solution to a 2 x 2 system of conservation laws with linear damping

TL;DR

The paper addresses a conservation-law system with linear damping that modelizes Eulerian droplet dynamics, focusing on general initial data beyond Riemann-type. It regularizes the system with a parabolic term , and derives explicit weakly regularized solutions via a Hopf-Cole reduction, enabling a rigorous vanishing-viscosity analysis. By employing the weak asymptotic method, Volpert product, and Colombeau algebra, it constructs weak, distributional, and Colombeau solutions, proves the vanishing-viscosity limit yields a distributional solution, and establishes large-time behavior of the viscous system. The results provide a coherent framework for handling non-conservative products and measure-valued components (delta-type shocks) in a general-initial-data setting, with implications for the stability and physical interpretation of the Eulerian droplet model.

Abstract

Systems of the first order partial differential equations with singular solutions appear in many multiphysics problems and the weak formulation of solutions involve in many cases product of distributions. In this paper we study such a system derived from Eulerian droplet model for air particle flow. This is a 2 x 2 non - strictly hyperbolic system of conservation laws with linear damping. We first study a regularized viscous system with variable viscosity term and obtain a weak asymptotic solution with general initial data and also get solution in the Colombeau algebra. We also study the vanishing viscosity limit and show that this limit is a distribution solution. Further we study the large time asymptotic behaviour of the viscous system. This important system, is not very well studied due to complexities in the analysis. As far as we know the only work done on this system is for Riemann type of initial data. The significance of this paper is that we work on the system having general initial data and not just initial data of the Riemann type.
Paper Structure (13 sections, 6 theorems, 105 equations)

This paper contains 13 sections, 6 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Weak formulation of solutions
  3. Weak asymptotic method
  4. Functions of bounded variation and Volpert product
  5. Solution in the sense of Colombeau
  6. Construction of regularized approximate solution.
  7. Explicit formula for the approximate solution
  8. Large time asymptotic behavior of approximate solution
  9. Weak solution to the Cauchy problem \ref{['e1.1']}-\ref{['e1.2']}.
  10. Weak asymptotic solution
  11. Volpert product and distributional solution
  12. Weak solution in the sense of Colombeau
  13. Conclusion

Key Result

Theorem 2.1

Let $u \in BV( \Omega,R^p)$ and $g: R^p \rightarrow R^p$ be a locally bounded Borel function. Then, there exists a unique family of real valued bounded Borel measures $\mu_j$ on $\Omega, j=1,...n$ characterized by the three properties:

Theorems & Definitions (11)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • ...and 1 more