On 2D Viscoelasticity with Small Strain

An exact two-dimensional rotation-strain model describing the motion of Hookean incompressible viscoelastic materials is constructed by the polar decomposition of the deformation tensor. The global existence of classical solutions is proved under the smallness assumptions only on the size of initial strain tensor. The proof of global existence utilizes the weak dissipative mechanism of motion, which is revealed by passing the partial dissipation to the whole system.Comment: Different contributions of strain and rotation of the deformation are studied for viscoelastic fluids of Oldroyd-B type in 2

Asymptotic bahavior for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions

We consider the Cauchy problem for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions. Under the null condition for such systems, the global existence of small amplitude solutions is known. In this paper, we will show that the global solution is asymptotically free in the energy sense, by obtaining the asymptotic pointwise behavior of the derivatives of the solution. Nonetheless we can also show that the pointwise behavior of the solution itself may be quite different from that of the free solution. In connection with the above results, a theorem is also developed to characterize asymptotically free solutions for wave equations in arbitrary space dimensions.Comment: The final version. 30 page

Self-gravitating elastic bodies

Extended objects in GR are often modelled using distributional solutions of the Einstein equations with point-like sources, or as the limit of infinitesimally small "test" objects. In this note, I will consider models of finite self-gravitating extended objects, which make it possible to give a rigorous treatment of the initial value problem for (finite) extended objects.Comment: 16 pages. Based on a talk given at the 2013 WE-Heraeus seminar on "Equations of motion in relativistic gravity

Global Solutions for Incompressible Viscoelastic Fluids

We prove the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in the n-dimensional torus for the incompressible viscoelastic system of Oldroyd-B type in the case of near equilibrium initial data. The results hold in both two and three dimensional spaces. The results and methods presented in this paper are also valid for a wide range of elastic complex fluids, such as magnetohydrodynamics, liquid crystals and mixture problems.Comment: We prove the existence of global smooth solutions to the Cauchy problem for the incompressible viscoelastic system of Oldroyd-B type in the case of near equilibrium initial dat