Global Regularity of Solutions to Complex Fluids in Dimensions Two

Description

As well-known models in complex fluids, the Hookean incompressible viscoelasticity andthe incompressible magnetohydrodynamic flows with zero magnetic diffusivity are activetopics in the current study of complex fluids from both the mathematical viewpoint andthe engineering viewpoint. These models have solid physical backgrounds and engineeringapplications in plasmas, astrophysics, and biology. Despite of their importance in physicsand engineering, fundamental issues, such as the global-in-time regularity of these models,remain as very challenging open problems in mathematics, even though the state-of-the-artglobal-in-time weak solutions with finite energy and the global-in-time classical solutionswith small data were successfully formulated during the past decade.In this research proposal the principal investigator intends to push forward the mathe-matical understanding of complex fluids which include the incompressible viscoelasticityand the incompressible magnetohydrodynamic flows with zero magnetic diffusivity:·Global regularity of solutions to incompressible viscoelasticity in dimensions two;·Global regularity of solutions to incompressible magnetohydrodynamic flows withzero magnetic diffusivity in dimensions two.The first topic in this proposal focuses on the global regularity of solutions to Hookean in-compressible viscoelasticity in dimensions two. This proposed problem is difficult becauseof the hyperbolic property of the deformation gradient, and this hyperbolicity produces ahurdle to estimate the growth of the L¥bound of the deformation gradient. The successof this proposed problem needs a new idea to control the growth of the deformation gra-dient with a careful analysis of the nonlinear terms, and will shed lights on understandingother related problems such as the regularity of compressible Navier-Stokes equations indimensions two.The second topic in this proposal aims to address the global regularity of solutionsto incompressible magnetohydrodynamic flows with zero magnetic diffusivity in dimen-sions two. Similar to Hookean incompressible viscoelasticity, the magnetic field satisfies atransport equation and hence introduces a difficulty to control its L¥norm. Except this,the fact that makes the incompressible magnetohydrodynamic flows with zero magneticdiffusivity much harder than the Hookean incompressible viscoelasticity is the degeneracyof the hyperbolicity of the magneticfield which forces us to find another new idea toovercome the degeneracy.