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Local-in-time existence of strong solutions to a class of compressible Power-Law flows

Fang Li, Chang Mengge

TL;DR

The paper studies local-in-time existence of strong solutions for compressible non-Newtonian fluids with a spatially/time-varying power-law exponent $p(t,x)$ on a 3D periodic domain, using a $p(t,x)$-potential framework to define the stress. A Faedo–Galerkin scheme combined with transport-coupled density, together with sharp a priori estimates built from the $\\mathcal{I}_{\\Phi}$ and $\\mathcal{J}_{\\Phi}$ functionals, yields local strong solutions for $ rac{7}{5} < p^- \,\le\, p(t,x) \le 2$. The work also proves an improved blow-up criterion based on the $L^ abla(0,T;L^3(\\Omega))$ norm of the velocity gradient and density, and provides a detailed passage to the limit to handle the variable-exponent nonlinearity. These results extend the theory of compressible power-law fluids to dynamically varying exponents and elucidate conditions under which strong solutions break down or can be continued.

Abstract

We consider a model of the compressible non-Newtonian fluids for power-law flow fulfilling a periodic domain in ${\mathbb R}^3,$ in which the extra stress tensor is induced by a potential with $p(t,x)$-structure. The local-in-time existence of strong solution is proved for all $\frac{7}{5} < \inf p(t,x) \leqslant \sup p(t,x) \leqslant 2.$ Further, an improved blow-up criterion for strong solutions is given in terms of the $L^\infty(0,T;L^3(Ω))$-norm of the gradient of the velocity.

Local-in-time existence of strong solutions to a class of compressible Power-Law flows

TL;DR

The paper studies local-in-time existence of strong solutions for compressible non-Newtonian fluids with a spatially/time-varying power-law exponent on a 3D periodic domain, using a -potential framework to define the stress. A Faedo–Galerkin scheme combined with transport-coupled density, together with sharp a priori estimates built from the and functionals, yields local strong solutions for . The work also proves an improved blow-up criterion based on the norm of the velocity gradient and density, and provides a detailed passage to the limit to handle the variable-exponent nonlinearity. These results extend the theory of compressible power-law fluids to dynamically varying exponents and elucidate conditions under which strong solutions break down or can be continued.

Abstract

We consider a model of the compressible non-Newtonian fluids for power-law flow fulfilling a periodic domain in in which the extra stress tensor is induced by a potential with -structure. The local-in-time existence of strong solution is proved for all Further, an improved blow-up criterion for strong solutions is given in terms of the -norm of the gradient of the velocity.

Paper Structure

This paper contains 8 sections, 15 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A priori estimates
  4. Proof of Theorem 1
  5. The Faedo-Galerkin approximations
  6. Estimates independent of $N$ and $\delta$
  7. Passage to the limit
  8. Proof of Theorem 2

Key Result

Theorem 1

Let $T>0,$$\Omega = \mathbb{T} ^3$ and $\frac{7}{5}<p^-\leqslant p(t,x) \leqslant p^+\leqslant 2$ with $p(t,x)\in W^{1,\infty}((0,T)\times\Omega).$ Assume that the initial data $(\rho_0,{\mathbf u}_0)$ and the external force ${\mathbf f}$ satisfy the following regularity and the compatibility condition for some ${\bf g}\in L^2(\Omega).$ Then, there exist a small time $T^{* }\in ( 0, T)$ and a st

Theorems & Definitions (32)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • ...and 22 more