Global Classical Solutions to the Compressible Viscoelastic System with Zero Shear Viscosity

具有零切向粘性系數的可壓縮粘彈性流體方程組的全局經典解

Student thesis: Doctoral Thesis

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Award date14 May 2019

Abstract

In this thesis, the compressible viscoelastic systems are studied. When considering elastic materials, the stress due to deformation of the materials plays an important role in its dynamical behavior. This, along with some dissipation effect, gives rise to the compressible viscoelastic systems.

In the first part, some basic properties of the viscoelastic systems are presented. We formulate the system in terms of the density, velocity and deformation matrix of the material. Then the constraints among these variables are investigated.

In the second part of the thesis, we focus on the linear theory. We will consider a more general form of first order linear hyperbolic systems with dissipation. Sideris-Thomases' techniques in can be applied to derive a local energy decay result. This will provide a quick method to derive some decay for the wave-like solutions in the following parts. And the Green's function to a more specific hyperbolic-parabolic system is studied. This provides more accurate decay estimates when working on the two dimensional case.

In the third part, we prove the first main result of the thesis, the global existence of solutions to the 3D compressible viscoelastic system with zero shear viscosity. The basic tool is the local energy decay result and we will prove an enhanced decay estimate for the compressible parts by exploiting the structure of the system.

In the last part, we study the 2D compressible viscoelastic system with zero shear viscosity. By reformulating the system in the conservation form, the interaction among the incompressible parts satisfies the strong null condition which is essential to derive the global existence. To close the energy estimates, the Green's function method will be applied to derive the decay rates for the compressible parts.

    Research areas

  • compressible, zero shear viscosity, elastic system