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Global solutions of the 3D inhomogeneous incompressible viscoelastic system without structure assumptions

Chengfei Ai, Mengxing Bei, Yong Wang

Abstract

In this paper, we prove the global existence of strong solutions for the inhomogeneous incompressible viscoelastic system without any additional structure assumptions on $\mathbb{R}^{3}$. Unlike the time weighted energy method presented by Ai and Wang (Nonlinear Anal. 254 (2025), 113747.), by replacing $H^{-1}$ conditions with certain $L^{1}$ conditions on initial data, we need to develop some new transformation techniques for the system (1.1) and make use of elegant spectral analysis method to capture an enhanced time-decay rate of the velocity field $u$, which is essential to establish the uniform bounds of the density and deformation tensor.

Global solutions of the 3D inhomogeneous incompressible viscoelastic system without structure assumptions

Abstract

In this paper, we prove the global existence of strong solutions for the inhomogeneous incompressible viscoelastic system without any additional structure assumptions on . Unlike the time weighted energy method presented by Ai and Wang (Nonlinear Anal. 254 (2025), 113747.), by replacing conditions with certain conditions on initial data, we need to develop some new transformation techniques for the system (1.1) and make use of elegant spectral analysis method to capture an enhanced time-decay rate of the velocity field , which is essential to establish the uniform bounds of the density and deformation tensor.
Paper Structure (5 sections, 13 theorems, 88 equations)

This paper contains 5 sections, 13 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Derivation of the dissipative system
  3. Reformulation
  4. Some useful lemmas
  5. Proof of Theorem 1.1

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 $\delta>0$, and Then the Cauchy problem 1.1--1.1' admits a unique global solution $(\tilde{\rho}, u, \mathbb{F})(t)$ such that where $G:=\tilde{\rho}\mathbb{F}\mathbb{F}^{T}-\mathbb{I}, \sigma\in(1,\sigma_{0})$ and $1<\sigma_{0}<\fr

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 17 more