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Relative entropy technique in terms of position and momentum and its application to Euler-Poisson system

Jan Giesselmann, Kiwoong Kwon, Min-Gi Lee

Abstract

This paper presents a systematic study of the relative entropy technique for compressible motions of continuum bodies described as Hamiltonian flows. While the description for the classical mechanics of $N$ particles involves a Hamiltonian in terms of position and momentum vectors, that for the continuum fluid involves a Hamiltonian in terms of density and momentum. For space dimension $d\ge 2$, the Hamiltonian functional has a non-convex dependency on the deformation gradient or placement map due to material frame indifference. Because of this, the applicability of the relative entropy technique with respect to the deformation gradient or the placement map is inherently limited. Despite these limitations, we delineate the feasible applications and limitations of the technique by pushing it to its available extent. Specifically, we derive the relative Hamiltonian identity, where the Hamiltonian takes the position and momentum field as its primary and conjugate state variables, all within the context of the referential coordinate system that describes the motion. This approach, when applicable, turns out to yield rather strong stability statements. As instances, we consider Euler-Poisson systems in one space dimension. For a specific pressureless model, we verify non-increasing $L^2$ state differences before the formation of $δ$-shock. In addition, weak-strong uniqueness, stability of rarefaction waves, and convergence to the gradient flow in the singular limit of large friction are shown. Depending on the presence or absence of pressure, assumptions are made to suitably accommodate phenomena such as $δ$-shocks, vacuums, and shock discontinuities in the weak solutions.

Relative entropy technique in terms of position and momentum and its application to Euler-Poisson system

Abstract

This paper presents a systematic study of the relative entropy technique for compressible motions of continuum bodies described as Hamiltonian flows. While the description for the classical mechanics of particles involves a Hamiltonian in terms of position and momentum vectors, that for the continuum fluid involves a Hamiltonian in terms of density and momentum. For space dimension , the Hamiltonian functional has a non-convex dependency on the deformation gradient or placement map due to material frame indifference. Because of this, the applicability of the relative entropy technique with respect to the deformation gradient or the placement map is inherently limited. Despite these limitations, we delineate the feasible applications and limitations of the technique by pushing it to its available extent. Specifically, we derive the relative Hamiltonian identity, where the Hamiltonian takes the position and momentum field as its primary and conjugate state variables, all within the context of the referential coordinate system that describes the motion. This approach, when applicable, turns out to yield rather strong stability statements. As instances, we consider Euler-Poisson systems in one space dimension. For a specific pressureless model, we verify non-increasing state differences before the formation of -shock. In addition, weak-strong uniqueness, stability of rarefaction waves, and convergence to the gradient flow in the singular limit of large friction are shown. Depending on the presence or absence of pressure, assumptions are made to suitably accommodate phenomena such as -shocks, vacuums, and shock discontinuities in the weak solutions.
Paper Structure (26 sections, 15 theorems, 167 equations)

This paper contains 26 sections, 15 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Formal relative entropy identity of Hamiltonian flow
  3. Euler-Poisson system as a Hamiltonian flow
  4. Formal relative Hamiltonian identity
  5. Interacting systems of Euler-Poisson type
  6. Background and previous results
  7. Placement and motion maps
  8. Formulation (A) for Section \ref{['sec:EP']}
  9. Formulation (B) for Section \ref{['specialEP']}
  10. Convexity of kernels for the formulation (A)
  11. Relative Hamiltonian computation for the Euler-Poisson system in one space dimension
  12. Dissipative weak solution and strong solution
  13. Convexity and lower bound of relative Hamiltonian
  14. Relative Hamiltonian inequality
  15. Applications of relative Hamiltonian inequality
  16. ...and 11 more sections

Key Result

Proposition 4.1.1

Suppose ${\mathcal{E}}$ is given by the formula H and $\eta\in \hat{\textsf{S}}(X)$ is admissible. Then the functional $\mathbf{f}[\eta] : \phi \mapsto -\frac{d}{ds} {\mathcal{E}}[\eta + s\phi] |_{s=0}$ is continuous on and has the representation

Theorems & Definitions (36)

  • Proposition 4.1.1
  • proof
  • Definition 4.1.2
  • Remark 4.1.3
  • Remark 4.1.4
  • Definition 4.1.5
  • Proposition 4.2.1: Finiteness of relative energies
  • proof
  • Theorem 1
  • proof
  • ...and 26 more