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Local-in-time existence of strong solutions to a class of compressible non-Newtonian Navier-Stokes equations

Martin Kalousek, Václav Mácha, Šárka Nečasová

Abstract

The aim of this article is to show a local-in-time existence of a strong solution to the generalized compressible Navier-Stokes equation for arbitrarily large initial data. The goal is reached by $L^p$-theory for linearized equations which are obtained with help of the Weis multiplier theorem and can be seen as generalization of the work of Shibata and Enomoto (devoted to compressible fluids) to compressible non-Newtonian fluids.

Local-in-time existence of strong solutions to a class of compressible non-Newtonian Navier-Stokes equations

Abstract

The aim of this article is to show a local-in-time existence of a strong solution to the generalized compressible Navier-Stokes equation for arbitrarily large initial data. The goal is reached by -theory for linearized equations which are obtained with help of the Weis multiplier theorem and can be seen as generalization of the work of Shibata and Enomoto (devoted to compressible fluids) to compressible non-Newtonian fluids.

Paper Structure

This paper contains 9 sections, 10 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Discussion and main result
  3. Notation
  4. Local-in-time well posedness of the compressible Navier-Stokes
  5. System in Lagrangian coordinates
  6. Linearized system
  7. The fix-point argument
  8. someref regularity of the linearized system
  9. Appendix

Key Result

Theorem 1.1

Let $\mu \in C^3([0,\infty))$ and $\eta \in C^2(\mathbb{R})$ satisfy $\mu(s) + 2\mu'(s) s >0$ for all $s\geq 0$ and $\eta(r) + \eta'(r)r >0$ for all $r\in\mathbb{R}$. Let, moreover, $\pi \in C^2([0,\infty))$, $q>d$ and $p\in \left(\frac{2q}{q-d},\infty\right)$ be given. Then for every $u_0\in W^{2,q with which satisfies SystemClass.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 3.1
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm.non-constant']}
  • ...and 6 more