Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global strong solutions to a one-dimensional full non-Newtonian fluid with far field vacuum

Li Fang, Yu Wang, Aibin Zang

Abstract

In this paper, the Cauchy problem for a one-dimensional heat conducting compressible non-Newtonian fluid is considered. The constitute equation of the non-Newtonian fluid is determined by two nonlinear terms $(|u_x|^{q-2}u_x)_x$ and $(|θ_x|^{p-2}θ_x)_x$ with $1<p,q<2.$ When the vacuum occurs at the far field, the local and global existence of strong solutions are established for the Cauchy problem. The results indicate that the non-Newtonian fluid possesses time-dependence boundedness of the energy if the vacuum occurs at the far field and the density decays slowly at the far field as time goes to infinity. This is the key difference from the well-known result developed by Li and Xin (Advances in Mathematics 361(2020), 106923) for the one-dimensional heat conductive compressible Navier-Stokes system.

Global strong solutions to a one-dimensional full non-Newtonian fluid with far field vacuum

Abstract

In this paper, the Cauchy problem for a one-dimensional heat conducting compressible non-Newtonian fluid is considered. The constitute equation of the non-Newtonian fluid is determined by two nonlinear terms and with When the vacuum occurs at the far field, the local and global existence of strong solutions are established for the Cauchy problem. The results indicate that the non-Newtonian fluid possesses time-dependence boundedness of the energy if the vacuum occurs at the far field and the density decays slowly at the far field as time goes to infinity. This is the key difference from the well-known result developed by Li and Xin (Advances in Mathematics 361(2020), 106923) for the one-dimensional heat conductive compressible Navier-Stokes system.
Paper Structure (7 sections, 12 theorems, 191 equations)

This paper contains 7 sections, 12 theorems, 191 equations.

Table of Contents

  1. Introduction and Main results
  2. Introduction
  3. Methodology and main Results
  4. Preliminaries
  5. Local existence in the absence of vacuum
  6. Local existence in the presence of far field vacuum
  7. Global existence in the presence of far field vacuum

Key Result

Theorem 1.1

(Local well-posedness) Let $p,q\in(1,2)$ and $\alpha<\min\{-\frac{q}{2(q-1)},-\frac{4-p}{2-p}\}.$ Assume that the initial data $(J_0,\varrho_0,v_0,\Theta_0)$ satisfies for positive constants $\overline{\varrho},$$\overline {J}$ and $\underline{ J}.$ Then there exists a positive time $T$, depending only on $p,q,\underline J,$$\overline J,$$\overline\varrho,$$\|\varrho_0^{\frac{\alpha}{2}}J_0^{\pri

Theorems & Definitions (21)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Proposition 2.1
  • ...and 11 more