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Global solutions to the $n$-dimensional incompressible Oldroyd-B model without damping mechanism

Xiaoping Zhai

Abstract

The present work is dedicated to the global solutions to the incompressible Oldroyd-B model without damping on the stress tensor in $\mathbb{R}^n(n=2,3)$. This result allows to construct global solutions for a class of highly oscillating initial velocity. The proof uses the special structure of the system. Moreover, our theorem extends the previous result by Zhu [19] and covers the recent result by Chen and Hao [4].

Global solutions to the $n$-dimensional incompressible Oldroyd-B model without damping mechanism

Abstract

The present work is dedicated to the global solutions to the incompressible Oldroyd-B model without damping on the stress tensor in . This result allows to construct global solutions for a class of highly oscillating initial velocity. The proof uses the special structure of the system. Moreover, our theorem extends the previous result by Zhu [19] and covers the recent result by Chen and Hao [4].

Paper Structure

This paper contains 3 sections, 10 theorems, 110 equations.

Table of Contents

  1. The low frequency estimates of the solutions
  2. The high frequency estimates of the solutions
  3. Complete the proof of our main Theorem \ref{['zhuyaodingli']}

Key Result

Theorem \oldthetheorem

(see zhuyi) Let $n=3$, $\mu$, $K_1$, $K_2 >0$. Suppose that $\hbox{\rm div}\, u_0 = 0, (\tau_0)_{ij} =( \tau_0)_{ji}$ and initial data $\varepsilon$ such that system m admits a unique global classical solution provided that where $\Lambda = (-\Delta)^\frac{1}{2}$.

Theorems & Definitions (19)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 9 more