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Weak solutions and weak-strong uniqueness for a compressible power-law-Oldroyd--B fluid model

Yong Lu, Milan Pokorny

TL;DR

The paper analyzes a compressible viscoelastic fluid model of Oldroyd–B type with power-law rheology and linear stress diffusion in three dimensions, proving the existence of renormalized weak finite-energy solutions for $r \ge \frac{5}{2}$ and establishing a weak-strong uniqueness principle for data admitting a sufficiently regular strong solution. The authors develop a layered approximation scheme (involving mollifications, stress regularization, a Galerkin projection, and sequential limit passages in $\delta$, $\sigma$, $n$, $\Theta$, and $\alpha$) and derive robust a priori estimates anchored by the energy balance, the renormalized continuity equation, and entropy-type bounds. A key technical feature is ensuring the positivity and regularity of the stress tensor ${\mathbb T}$ through logarithmic entropic terms and convex-function truncations, which enables compactness and strong convergence necessary to pass to the limit in the nonlinear terms. The weak-strong uniqueness result uses a relative entropy method to compare any weak solution with a suitably regular strong solution, showing they must coincide when the strong solution satisfies precise boundedness assumptions. Collectively, the results advance well-posedness for a 3D compressible Oldroyd–B model with power-law viscosity, under physically relevant linear stress diffusion, and provide a rigorous bridge between weak and strong solution theories in this setting.

Abstract

We consider a model of a viscoelastic compressible flow in $R^{3}$ which is additionally shear thickening (the stress tensor corresponds to the power law model, however, the divergence of the velocity is due to the model bounded). We prove existence of a weak solution to this model provided the growth in the power law model is larger or equal than $\frac 52$. We also prove that any sufficiently smooth solution of this model is unique in the class of weak solution, provided extra integrability of the initial value for the extra stress tensor is assumed.

Weak solutions and weak-strong uniqueness for a compressible power-law-Oldroyd--B fluid model

TL;DR

The paper analyzes a compressible viscoelastic fluid model of Oldroyd–B type with power-law rheology and linear stress diffusion in three dimensions, proving the existence of renormalized weak finite-energy solutions for and establishing a weak-strong uniqueness principle for data admitting a sufficiently regular strong solution. The authors develop a layered approximation scheme (involving mollifications, stress regularization, a Galerkin projection, and sequential limit passages in , , , , and ) and derive robust a priori estimates anchored by the energy balance, the renormalized continuity equation, and entropy-type bounds. A key technical feature is ensuring the positivity and regularity of the stress tensor through logarithmic entropic terms and convex-function truncations, which enables compactness and strong convergence necessary to pass to the limit in the nonlinear terms. The weak-strong uniqueness result uses a relative entropy method to compare any weak solution with a suitably regular strong solution, showing they must coincide when the strong solution satisfies precise boundedness assumptions. Collectively, the results advance well-posedness for a 3D compressible Oldroyd–B model with power-law viscosity, under physically relevant linear stress diffusion, and provide a rigorous bridge between weak and strong solution theories in this setting.

Abstract

We consider a model of a viscoelastic compressible flow in which is additionally shear thickening (the stress tensor corresponds to the power law model, however, the divergence of the velocity is due to the model bounded). We prove existence of a weak solution to this model provided the growth in the power law model is larger or equal than . We also prove that any sufficiently smooth solution of this model is unique in the class of weak solution, provided extra integrability of the initial value for the extra stress tensor is assumed.

Paper Structure

This paper contains 15 sections, 6 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Weak solution, main results
  3. Weak formulation
  4. Main results
  5. A priori estimates
  6. Preliminaries
  7. Formal a priori estimates
  8. Construction of a weak solution
  9. Construction of approximate solution
  10. Estimates independent of $\delta$, limit passage $\delta \to 0^+$
  11. Estimates independent of $\sigma$, limit passage $\sigma \to 0^+$
  12. Estimates independent of $n$, limit passage $n \to \infty$
  13. Estimates independent of $\Theta$, limit passage $\Theta \to 0^+$
  14. Estimates independent of $\alpha$, limit passage $\alpha \to 0^+$
  15. Weak-strong uniqueness

Key Result

Theorem 2.2

Let $\Omega$ be a bounded domain of the class $C^{2,\beta}$ for some $0<\beta \leq 1$. Let $\varrho_0 \in L^\infty(\Omega)$, $0<C_0 \leq \varrho_0$ a.e. in $\Omega$, $\frac{{\bf m}_0}{\varrho_0} \in L^2(\Omega;{\mathbb R}^3)$, ${\mathbb T}_0 \in L^2(\Omega;{\mathbb R}^{3\times 3})$ be symmetric posi

Theorems & Definitions (12)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • Lemma 5.1
  • ...and 2 more