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Non-Newtonian compressible fluids with stochastic right hand side

Pavel Ludvík, Václav Mácha

TL;DR

The paper addresses the existence of solution concepts for compressible barotropic flows of shear-rate dependent non-Newtonian fluids, where classical weak solutions are obstructed by nonlinear elliptic terms. It introduces measure-valued solutions constructed via Young measures and proves an abstract existence theorem in a very general setting. The framework is extended to accommodate stochastic right-hand side forcing, outlining how randomness is integrated within the measure-valued formulation and the corresponding proof strategy. The results provide a rigorous, broadly applicable basis for analyzing complex, stochastic non-Newtonian compressible flows within a generalized Navier–Stokes model, with implications for continuum mechanics and mathematical fluid dynamics.

Abstract

Fluids with shear-rate-dependent viscosity form a special class of non-Newtonian fluids that play a crucial role in continuum dynamics. We consider a compressible barotropic flow of such fluids, governed by a generalized Navier-Stokes system. Due to the nonlinearity in the elliptic term, obtaining the existence of a weak solutions is challenging. To address this, we introduce the concept of a measure-valued solution, whose existence is established in a very general setting using an abstract theorem on the existence of a Young measure. We also discuss the proof of this key result.

Non-Newtonian compressible fluids with stochastic right hand side

TL;DR

The paper addresses the existence of solution concepts for compressible barotropic flows of shear-rate dependent non-Newtonian fluids, where classical weak solutions are obstructed by nonlinear elliptic terms. It introduces measure-valued solutions constructed via Young measures and proves an abstract existence theorem in a very general setting. The framework is extended to accommodate stochastic right-hand side forcing, outlining how randomness is integrated within the measure-valued formulation and the corresponding proof strategy. The results provide a rigorous, broadly applicable basis for analyzing complex, stochastic non-Newtonian compressible flows within a generalized Navier–Stokes model, with implications for continuum mechanics and mathematical fluid dynamics.

Abstract

Fluids with shear-rate-dependent viscosity form a special class of non-Newtonian fluids that play a crucial role in continuum dynamics. We consider a compressible barotropic flow of such fluids, governed by a generalized Navier-Stokes system. Due to the nonlinearity in the elliptic term, obtaining the existence of a weak solutions is challenging. To address this, we introduce the concept of a measure-valued solution, whose existence is established in a very general setting using an abstract theorem on the existence of a Young measure. We also discuss the proof of this key result.
Paper Structure (16 sections, 2 theorems, 4 equations, 3 figures, 2 tables)

This paper contains 16 sections, 2 theorems, 4 equations, 3 figures, 2 tables.

Table of Contents

  1. Part Title
  2. Section Heading
  3. Section Heading
  4. Subsection Heading
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  6. Paragraph Heading
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  8. Section Heading
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  11. Paragraph Heading
  12. Styling of References
  13. Chapter Heading
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  16. ...and 1 more sections

Key Result

theorem 1

Theorem text goes here.

Figures (3)

  • Figure 1: If the width of the figure is less than 7.8 cm use the sidecapion command to flush the caption on the left side of the page. If the figure is positioned at the top of the page, align the sidecaption with the top of the figure -- to achieve this you simply need to use the optional argument [t] with the sidecaption command
  • Figure 2: If the width of the figure is less than 7.8 cm use the sidecapion command to flush the caption on the left side of the page. If the figure is positioned at the top of the page, align the sidecaption with the top of the figure -- to achieve this you simply need to use the optional argument [t] with the sidecaption command
  • Figure 3: Please write your figure caption here

Theorems & Definitions (6)

  • theorem 1
  • definition 1
  • proof
  • theorem 2
  • definition 2
  • proof