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A variational approach to the Navier-Stokes equations with shear-dependent viscosity

Christina Lienstromberg, Stefan Schiffer, Richard Schubert

TL;DR

This work develops a variational framework for non-Newtonian Navier–Stokes flows with shear-dependent viscosity on the torus by minimizing stabilised WIDE functionals $I_\eta$ and passing to the limit $\eta\to0$. The approach yields Leray–Hopf solutions and, in the mildly/strongly shear-thickening regimes (depending on the growth exponent $p$ relative to the space dimension $d$), achieves energy equality and strong convergence of the nonlinear viscosity term. A key technical contribution is the solenoidal Lipschitz truncation, extended to an elliptic regularisation setting, which enables passing to the limit in the nonlinear viscosity and identifications via Minty’s trick. The results illuminate how weak vs strong convergence properties reflect rheology: subcritical exponents lead to energy-inequality solutions with potential anomalous dissipation, while supercritical exponents yield energy solutions with full dissipation balance. Overall, the paper provides a rigorous variational route to existence, regularity, and convergence, bridging non-Newtonian rheology with PDE methods and truncation techniques valuable beyond this model.

Abstract

We present a variational approach for the construction of Leray-Hopf solutions to the non-Newtonian Navier-Stokes system. Inspired by the work [42] on the corresponding Newtonian problem, we minimise certain stabilised Weighted Inertia-Dissipation-Energy (WIDE) functionals and pass to the limit of a vanishing parameter in order to recover a Leray-Hopf solution of the non-Newtonian Navier-Stokes equations. The investigation of the non-Newtonian Navier-Stokes system via this variational approach is particularly well suited to gain insights into weak, respectively strong convergence properties of approximating sequences for different flow-behaviour exponents. With this analysis we extend the results of [4] to power-law exponents $\tfrac{2d}{d+2} < p < \tfrac{3d+2}{d+2}$, where weak solutions do not satisfy the energy equality and the involved convergence is genuinely weak. Key of the argument is to pass to the limit in the nonlinear viscosity term in the time-dependent setting. For this we provide an elliptic-parabolic solenoidal Lipschitz truncation that might be of independent interest.

A variational approach to the Navier-Stokes equations with shear-dependent viscosity

TL;DR

This work develops a variational framework for non-Newtonian Navier–Stokes flows with shear-dependent viscosity on the torus by minimizing stabilised WIDE functionals and passing to the limit . The approach yields Leray–Hopf solutions and, in the mildly/strongly shear-thickening regimes (depending on the growth exponent relative to the space dimension ), achieves energy equality and strong convergence of the nonlinear viscosity term. A key technical contribution is the solenoidal Lipschitz truncation, extended to an elliptic regularisation setting, which enables passing to the limit in the nonlinear viscosity and identifications via Minty’s trick. The results illuminate how weak vs strong convergence properties reflect rheology: subcritical exponents lead to energy-inequality solutions with potential anomalous dissipation, while supercritical exponents yield energy solutions with full dissipation balance. Overall, the paper provides a rigorous variational route to existence, regularity, and convergence, bridging non-Newtonian rheology with PDE methods and truncation techniques valuable beyond this model.

Abstract

We present a variational approach for the construction of Leray-Hopf solutions to the non-Newtonian Navier-Stokes system. Inspired by the work [42] on the corresponding Newtonian problem, we minimise certain stabilised Weighted Inertia-Dissipation-Energy (WIDE) functionals and pass to the limit of a vanishing parameter in order to recover a Leray-Hopf solution of the non-Newtonian Navier-Stokes equations. The investigation of the non-Newtonian Navier-Stokes system via this variational approach is particularly well suited to gain insights into weak, respectively strong convergence properties of approximating sequences for different flow-behaviour exponents. With this analysis we extend the results of [4] to power-law exponents , where weak solutions do not satisfy the energy equality and the involved convergence is genuinely weak. Key of the argument is to pass to the limit in the nonlinear viscosity term in the time-dependent setting. For this we provide an elliptic-parabolic solenoidal Lipschitz truncation that might be of independent interest.
Paper Structure (27 sections, 34 theorems, 318 equations)

This paper contains 27 sections, 34 theorems, 318 equations.

Table of Contents

  1. Introduction
  2. Functional setup and statement of the main result
  3. Functional setup and the potential $W$
  4. Leray--Hopf solutions
  5. Definition of the WIDE functional
  6. Main result
  7. Interpolation results
  8. Remarks on the regularity of weak solutions
  9. Properties of solutions to the approximate problem
  10. Existence of minimisers and a weak bound
  11. Euler--Lagrange equations
  12. Energy inequality, a-priori bounds and proof of Theorem \ref{['thm:main']}
  13. Proof of Theorem \ref{['thm:main']} under an additional Lipschitz bound
  14. Solenoidal Lipschitz truncation & the proof of Theorem \ref{['thm:main']} under no additional assumptions.
  15. Weak convergence of $u_\eta$
  16. ...and 12 more sections

Key Result

Theorem A

Let $p > \tfrac{2d}{d+2}$. For each $\eta > 0$ the functional $I_\eta$ possesses a minimiser $u_\eta \in U_\eta$. Moreover, there exists a subsequence $u_\eta$ (not relabeled) that converges weakly to a Leray--Hopf solution of the non-Newtonian Navier--Stokes system

Theorems & Definitions (72)

  • Theorem A: Existence of Leray--Hopf solutions, cf. Theorem \ref{['thm:main']}
  • Theorem B: Strongly shear-thickening fluids -- cf. Theorem \ref{['prop:strong']}
  • Example 2.1
  • Definition 2.2: Leray--Hopf solution
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Theorem 2.7: Main result
  • Theorem 2.8: Bibel
  • ...and 62 more