Zero Shear Viscosity for Compressible Complex Fluids

Project: Research

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As fundamental models in complex fluids and plasma, the Hookean compressible elastic fluid and the compressible MHD fluid are the studies of fluid behaviors in elastic materials or in conducting magnetic flows and in particular the magnetic flows usually involve the elastic elect as shown in “frozen-in law”. The original study of complex fluids dates back to experiments by physicists Maxwell, Boltzmann, and Kelvin in the nineteenth century. Due to its wide applications in engineering, the study on complex fluids has attracts a lot of attention not only in modelling and numerical simulations, but also is a hot topic in the mathematical analysis. Despite of its importance in physics, even though a state-of-the-art small perturbation theory near an equilibrium was successfully formulated during the last decades as the shear viscosity is fixed and strictly positive, the global-in-time wellposedness theories of compressible complex fluids remain as challenging open problems in mathematics when the viscosity is smaller and smaller. It is worthy of mentioning that in incompressible flows, the shear viscosity is the reciprocal of Reynold's number, and hence the smaller the shear viscosity, the closer the fluids is to ideal fluids, and the harder the analysis due to the turbulence phenomenon.In this research proposal we intend to push forward the mathematical understanding of compressible complex fluids as the shear viscosity vanishes but the volume viscosity is strictly positive. The first topic in this proposal focuses on questions, such as the low Mach number limit and the global solution of exterior problems, while the global-in-time wellposedness of Cauchy problem of Hookean compressible viscoelasticity with zero shear viscosity has been studied in [32, 34]. The proposed problems are interesting and difficult because the linearized system takes the form of wave equations with partial dissipation only for the compressible component, and hence the lower the dimension is, the weaker the decay of solutions will be. For instance, the local energy would decay in the exponential rate for three dimensional exterior problems of wave equations, but that decay reduces to a polynomial decay in dimensions two. The success of this proposed problem needs a careful analysis of the nonlinear terms, and sheds lights on understanding the behaviour of solutions when both shear and volume viscosities vanishes.The second topic in this proposal aims to address the global-in-time existence of classical solutions to multidimensional compressible MHD flows with zero magnetic diffusivity and zero shear viscosity near the equilibrium. Compressible MHD with zero magnetic diffusivity is formally related to compressible viscoelasticity via the \frozen-in-law", but is more challenging because of the degeneracy from the magnetic field. This degeneracy is compensated with a more careful analysis of the associated linearized system and the \frozen-in-law". The success of this proposed problem also needs a detailed analysis of nonlinear terms and will provide insights for understanding the inviscid MHD and the vanishing viscosity limit problem for compressible MHD, which are important issues in astrophysics. 


Project number9042855
Grant typeGRF
Effective start/end date1/01/20 → …