论文标题
关于麦克斯韦流体的结构稳定性：抗弹性动力学的收敛
About the structural stability of Maxwell fluids: convergence toward elastodynamics
论文作者
论文摘要
麦克斯韦（Maxwell）的粘弹性流量模型以其在连续力学的透视上统一固体弹性运动的潜力而闻名。但是缺乏严格的证据。目前的说明是对明确定义的粘弹性流的贡献，被证明涵盖了固体和（液体）流体状态。在第一部分中，我们考虑了特定粘弹性流的结构稳定性：1D剪切波解决阻尼波动方程的溶液。我们展示了纯粹的弹性1D剪切波的收敛性，将解决方案提供给标准波方程，因为放松时间$λ$和粘度$μ$生长$λ$ $ $ $ \ equiv $ $ $ $ $ $ $ $ $ $ g $ \ $ \ rightArrow $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ \具有速度U的粘弹性流体。在第二部分中，我们考虑了一般多维粘弹性流的结构稳定性。为此，我们将Maxwell的本构方程嵌入了对称的PDE的纯纤维系统中，我们在先前的出版物[ESAIM：M2AN 55（2021）807-831]中提出，以确定多维粘弹性流。接下来，我们使用C. M. Dafermos之后针对对称的hyperbolic Systems开发的相对渗透工具，显示了多维粘弹性流对$λ$ $ \ equiv $ $ $/ g的连续依赖性。它意味着在[ESAIM：M2AN 55（2021）807-831]中定义的粘弹性流的收敛，当$λ$ $ $ $ \ rightarrow $ $ $ \ iffty $时。
Maxwell's models for viscoelastic flows are famous for their potential to unify elastic motions of solids with viscous motions of liquids in the continuum mechanics perspective. But rigorous proofs are lacking. The present note is a contribution toward well-defined viscoelastic flows proved to encompass both solid and (liquid) fluid regimes. In a first part, we consider the structural stability of particular viscoelastic flows: 1D shear waves solutions to damped wave equations. We show the convergence toward purely elastic 1D shear waves solutions to standard wave equations, as the relaxation time $λ$ and the viscosity $μ$ grow unboundedly $λ$ $\not\equiv$ $μ$/G $\rightarrow$ $\infty$ in Maxwell's constitutive equation $λ$ $τ$ +$τ$ = 2 $μ$D(u) for the stress $τ$ of viscoelastic fluids with velocity u. In a second part, we consider the structural stability of general multi-dimensional viscoelastic flows. To that aim, we embed Maxwell's constitutive equation in a symmetric-hyperbolic system of PDEs which we proposed in our previous publication [ESAIM:M2AN 55 (2021) 807-831]so as to define multi-dimensional viscoelastic flows unequivocally. Next, we show the continuous dependence of multi-dimensional viscoelastic flows on $λ$ $\not\equiv$ $μ$/ G using the relative-entropy tool developped for symmetric-hyperbolic systems after C. M. Dafermos. It implies convergence of the viscoelastic flows defined in [ESAIM:M2AN 55 (2021) 807-831] toward compressible neo-Hookean elastodynamics when $λ$ $\rightarrow$ $\infty$.
