Wellposedness of Hookean Elastic Fluids in Two Dimensions
DescriptionAs a fundamental model in complex fluids and plasma, the Hookean elastic fluid is thestudy of elastic waves and involves linear elasticity with variation in time. The originalstudy of elastic fluids dates back to experiments by physicists Maxwell, Boltzmann, andKelvin in the nineteenth century. Despite of its importance in physics, the global-in-timewellposedness theories of Hookean elastic fluids remain as challenging openproblems in mathematics, even though a state-of-the-art small perturbation theorynear an equilibrium was successfully formulated during the last decade.In this research proposal we intend to push forward the mathematical understanding ofHookean elastic fluids. The first topic in this proposal focuses on the global-in-timewellposedness of classical solutions of Hookean self-gravitational compressibleelastodynamics in two dimensions with small data. This proposed problem is difficultbecause of the slow decay rate of solutions of two dimensional wave equations, and thisphenomenon produces a hurdle to control the growth of norms of nonlinear terms. Thesuccess of this proposed problem needs a careful analysis of the nonlinear terms, andsheds lights on understanding two dimensional nonlinear wave equations.The second topic in this proposal aims to address the global-in-time existence of weaksolutions to multidimensional incompressible viscoelastic flows. Due to the non-compatibilitybetween the weak continuity and the nonlinear term, the possibleoscillation and concentration phenomena of approximating solutions will be two maindifficulties. To overcome these two difficulties, an improvement on the integrability ofthe deformation gradient is needed and a compactness argument to deal with thequadratic term is helpful.?
|Effective start/end date||1/10/15 → 10/03/20|
- compressible elastodynamics,incompressible viscoelasticity,global existence,two dimensions,