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Global solutions of the 2D inhomogeneous incompressible viscoelastic system

Chengfei Ai, Yong Wang, Yunshun Wu

Abstract

In this paper, we investigate the global existence of strong solutions for the inhomogeneous incompressible viscoelastic system with only velocity dissipation on $\mathbb{R}^{2}$. Due to the criticality of the time-weight, the methods for the corresponding problem on $\mathbb{R}^{3}$ cannot be directly applied to the two-dimensional case. To overcome the main difficulties, we first transform the original system into a suitable dissipative system by introducing an effective tensor. Then we develop a new fractional time-weighted energy framework, combined with elegant commutator and bilinear estimates, to prove the global existence of strong solutions without the help of the common ``div-curl" structure on the viscoelastic system.

Global solutions of the 2D inhomogeneous incompressible viscoelastic system

Abstract

In this paper, we investigate the global existence of strong solutions for the inhomogeneous incompressible viscoelastic system with only velocity dissipation on . Due to the criticality of the time-weight, the methods for the corresponding problem on cannot be directly applied to the two-dimensional case. To overcome the main difficulties, we first transform the original system into a suitable dissipative system by introducing an effective tensor. Then we develop a new fractional time-weighted energy framework, combined with elegant commutator and bilinear estimates, to prove the global existence of strong solutions without the help of the common ``div-curl" structure on the viscoelastic system.
Paper Structure (9 sections, 12 theorems, 136 equations)

This paper contains 9 sections, 12 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Energy Framework
  3. Reformulation
  4. Various Refined Energies
  5. Estimates of Assistant Energy $\mathcal{E}_{a}(t)$
  6. Estimates of Basic Energy $\mathcal{E}_{0}(t)$
  7. Estimates of Fractional Time-weighted Energy $\mathcal{E}_{1}(t)$
  8. Proof of Theorem 1.1
  9. Tools

Key Result

Theorem 1.1

Suppose that the initial data $(\tilde{\rho}_{0}, u_{0}, \mathbb{F}_{0})$ with $\mathop{\mathrm{div}}\nolimits u_0=0$ satisfy for some sufficiently small constant $\varepsilon>0$, where $s_{0}, s_{1}$ are any given constants satisfying the bounds Then the Cauchy problem 1.1--1.1' admits a unique global solution $(\tilde{\rho}, u, \mathbb{F})(t)$ such that where $G:=\tilde{\rho}\mathbb{F}\mathbb

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 15 more