Global Existence of Weak Solutions and Concentration Phenomena for Multi-Dimensional Compressible Viscoelastic Fluids
DescriptionAs a fundamental model in complex fluids and plasma, the Hookean viscoelastic fluid isthe study of a coupling system between Navier-Stokes equations and a transport equationfor the deformation gradient. The original study of viscoelastic fluids dates back to experi-ments by physicists Maxwell, Boltzmann, and Kelvin in the nineteenth century. Despite ofits importance in physics, the global-in-time wellposedness theories of Hookean viscoelas-tic fluids with large initial data remain as challenging open problems in mathematics,even though a state-of-the-art small perturbation theory near a constant equilibrium wassuccessfully formulated during the last decade.In this research proposal we intend to push forward the mathematical understandingof Hookean viscoelastic fluids. The first topic in this proposal focuses on the global-in-time wellposedness of weak solutions of Hookean compressible viscoelastic fluids inmulti-dimensional spaces Rnwith large initial data as the adiabatic constant. This proposed problem is difficult because of the strong coupling between the fluid (or thepressure) and the elasticity (or the deformation gradient), the lack of a priori estimates,and the non-compatibility between the weak convergence and the nonlinearity. Preciselythe lack of a priori estimates makes the compactness of the pressure and the elasticstress hard to establish and the non-compatibility between the weak convergence andthe nonlinearity implies that the possible oscillation and concentration phenomena ofapproximating solutions will be two main difficulties. To overcome these two difficulties,an improvement on the integrability of both the deformation gradient and the density isneeded and a compactness argument to deal with the quadratic term is required. Thesuccess of this proposed problem needs a careful analysis of concentration and oscillationphenomena, and will in turn shed lights on understanding better the weak convergencemethod.The second topic in this proposal aims to address the concentration phenomena of thekinetic energy of multidimensional Hookean compressible viscoelastic flows as the adiabaticconstant.It is intended to construct a concentration set and study its Hausdorffdimension. The Hausdorff dimension is expected to depend on the adiabatic coefficientand outside the concentration set the weak stability of compressible viscoelastic fluidsholds true. The success of this proposed problem relies on a better understanding of therelation between the maximal function of density and the concentration of the kineticenergy.
|Effective start/end date||1/01/18 → 17/05/22|