[Paper Review] The initial-boundary value problem for the compressible viscoelastic fluids

Xianpeng Hu, Dehua Wang|arXiv (Cornell University)|Feb 22, 2011

Navier-Stokes equation solutions23 references3 citations

TL;DR

This paper establishes the global existence of strong solutions to the initial-boundary value problem for three-dimensional compressible viscoelastic fluids near equilibrium in a bounded domain. Using energy estimates and weighted Sobolev embeddings, it proves uniform $W^{1,q}$ bounds for density and deformation gradient with $q>3$, extending to the two-dimensional case as well.

ABSTRACT

The global existence of strong solution to the initial-boundary value problem of the three-dimensional compressible viscoelastic fluids near equilibrium is established in a bounded domain. Uniform estimates in $W^{1,q}$ with $q>3$ on the density and deformation gradient are also obtained. All the results apply to the two-dimensional case.

Motivation & Objective

• To establish the global existence of strong solutions for the initial-boundary value problem of compressible viscoelastic fluids in a bounded domain.
• To derive uniform $W^{1,q}$ estimates for density and deformation gradient with $q>3$ in three dimensions.
• To extend the results to the two-dimensional case.
• To analyze the system near equilibrium, where density $\varrho = 1$, velocity $\mathbf{u} = 0$, and deformation gradient $\mathbf{F} = I$.
• To overcome the analytical challenges posed by the coupling between kinetic energy and internal elastic energy in viscoelastic fluids.

Proposed method

• Transform the original system into a perturbation form around equilibrium using $\rho = \varrho - 1$ and $E = \mathbf{F} - I$.
• Employ energy estimates in $L^q$ spaces with $q > 3$ to control the growth of solutions.
• Use weighted $L^q$ estimates and interpolation techniques to bound $\|\nabla \rho\|_{L^q}$ and $\|\nabla E\|_{L^q}$.
• Apply Poincaré inequality to $\rho$ and $E$ using their zero-average properties derived from initial conditions.
• Use a maximal time interval argument and contradiction to prove that the solution exists globally in time.
• Leverage the structure of the Oldroyd-type system and the strong ellipticity of the viscous operator to close the energy estimates.

Experimental results

Research questions

• RQ1 Can strong solutions to the initial-boundary value problem for compressible viscoelastic fluids exist globally in time near equilibrium?
• RQ2 What uniform bounds can be established for the density and deformation gradient in $W^{1,q}$ with $q > 3$?
• RQ3 How does the coupling between fluid motion and elastic memory affect the long-time behavior of solutions?
• RQ4 Can the results be extended from three to two spatial dimensions?
• RQ5 What role does the initial data size play in ensuring global existence?

Key findings

• The global existence of strong solutions is established for the initial-boundary value problem of 3D compressible viscoelastic fluids near equilibrium in a bounded domain.
• Uniform $W^{1,q}$ estimates are obtained for both density $\rho$ and deformation gradient $E$ with $q > 3$.
• The solution exists for all time $t \in [0, \infty)$, and the $W^{1,q}$ norms of $\rho$ and $E$ remain bounded globally.
• The results are valid in both three and two spatial dimensions.
• The $L^q$ estimates on $\nabla \rho$ and $\nabla E$ are controlled via energy methods and Poincaré inequality.
• The proof relies on a contradiction argument on the maximal time of existence, showing that the solution cannot blow up in finite time.