Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global solutions of the 3D incompressible inhomogeneous viscoelastic system

Chengfei Ai, Yong Wang

Abstract

In this paper, we prove the global existence of strong solutions for the 3D incompressible inhomogeneous viscoelastic system. We do not assume the "initial state" assumption and the "div-curl" structure inspired by the works [59,61]. It is a key to transform the original system into a suitable dissipative system by introducing a new effective tensor, which is useful to establish a series of energy estimates with appropriate time weights.

Global solutions of the 3D incompressible inhomogeneous viscoelastic system

Abstract

In this paper, we prove the global existence of strong solutions for the 3D incompressible inhomogeneous viscoelastic system. We do not assume the "initial state" assumption and the "div-curl" structure inspired by the works [59,61]. It is a key to transform the original system into a suitable dissipative system by introducing a new effective tensor, which is useful to establish a series of energy estimates with appropriate time weights.
Paper Structure (9 sections, 14 theorems, 110 equations)

This paper contains 9 sections, 14 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Energy estimate
  3. Transformation and analysis for (\ref{['1.1']})
  4. The estimate of assistant energy $\mathcal{E}_{a}(t)$
  5. The estimate of basic energy $\mathcal{E}(t)$
  6. The estimate of strongly dissipative energy $\mathcal{E}_{s}(t)$
  7. The proof of Theorem 1.1
  8. Derivation of models
  9. Tools

Key Result

Theorem 1.1

Suppose that the initial data $(\tilde{\rho}_{0}, u_{0}, \mathbb{F}_{0})$ with $\mathop{\mathrm{div}}\nolimits u_0=0$ satisfies for some sufficiently small constant $\varepsilon>0$, where $|\nabla|=(-\Delta)^{\frac{1}{2}}$. Then the Cauchy problem 1.1--1.1' admits a unique global solution $(\tilde{\rho}, u, \mathbb{F})(t)$ such that for some $\gamma_{0}\in(0,\frac{1}{2})$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 17 more