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Dissipative Solutions to a Compressible Non-Newtonian Korteweg System with Density-Dependent Viscous Stress Tensor

Didier Bresch, Christophe Lacave, Maja Szlenk

TL;DR

The paper addresses the existence of dissipative solutions for a compressible, non-Newtonian Navier–Stokes–Korteweg system with density-dependent viscosity and capillarity in a periodic setting. It employs a relative entropy framework to quantify the distance to a smooth reference state and derives a dissipative inequality that yields weak-strong uniqueness. The main contribution is extending the dissipative-solution theory from Newtonian to non-Newtonian viscosities under specified growth and monotonicity conditions, using a Galerkin-based approximation and monotonicity methods to pass to the limit. This advances the understanding of well-posedness for compressible non-Newtonian fluids with capillarity and provides a robust tool for analyzing stability against smooth solutions in applied settings.

Abstract

The main objective of this paper is to prove that if capillarity effect is taken into account then there exist dissipative solutions to a system describing viscoplastic compressible flows with density dependent viscosities in a periodic domain $\T^d$ with $d=2,3$. We calculate the relative entropy inequality and in consequence show existence of dissipative solutions and the weak-strong uniqueness for this system. Our result extends the recent result concerning the link between Euler--Korteweg and Navier--Stokes--Korteweg systems for Newtonian flows (when the viscosity depends on the density) [See D.~Bresch, M. Gisclon, I. Lacroix-Violet, {\it Arch. Rational Mech. Anal.} (2019)] to non-Newtonian flows.

Dissipative Solutions to a Compressible Non-Newtonian Korteweg System with Density-Dependent Viscous Stress Tensor

TL;DR

The paper addresses the existence of dissipative solutions for a compressible, non-Newtonian Navier–Stokes–Korteweg system with density-dependent viscosity and capillarity in a periodic setting. It employs a relative entropy framework to quantify the distance to a smooth reference state and derives a dissipative inequality that yields weak-strong uniqueness. The main contribution is extending the dissipative-solution theory from Newtonian to non-Newtonian viscosities under specified growth and monotonicity conditions, using a Galerkin-based approximation and monotonicity methods to pass to the limit. This advances the understanding of well-posedness for compressible non-Newtonian fluids with capillarity and provides a robust tool for analyzing stability against smooth solutions in applied settings.

Abstract

The main objective of this paper is to prove that if capillarity effect is taken into account then there exist dissipative solutions to a system describing viscoplastic compressible flows with density dependent viscosities in a periodic domain with . We calculate the relative entropy inequality and in consequence show existence of dissipative solutions and the weak-strong uniqueness for this system. Our result extends the recent result concerning the link between Euler--Korteweg and Navier--Stokes--Korteweg systems for Newtonian flows (when the viscosity depends on the density) [See D.~Bresch, M. Gisclon, I. Lacroix-Violet, {\it Arch. Rational Mech. Anal.} (2019)] to non-Newtonian flows.
Paper Structure (19 sections, 14 theorems, 185 equations)

This paper contains 19 sections, 14 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. State of the art and the main result
  3. Density-dependent viscosities.
  4. Dissipative solutions.
  5. Non-Newtonian fluids.
  6. Main result.
  7. Convention.
  8. Dissipative solutions - motivation and relative entropy inequality.
  9. Construction of solutions - approximation scheme
  10. Relative entropy inequality and limit passage with $\nu,\varepsilon\to 0$
  11. Limit passage with $\nu,\varepsilon\to 0$.
  12. Acknowledgements.
  13. Auxiliary lemmas
  14. Poncaré inequality.
  15. Monotonicity property.
  16. ...and 4 more sections

Key Result

Theorem 1.2

Under the assumptions S1-S2 and P1-P2, there exists a global-in-time dissipative solution to NSK in the sense of Definition def:dissipative with the initial conditions $(\varrho_0,u_0)$ satisfying initial.

Theorems & Definitions (30)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 2.4
  • ...and 20 more