Wellposedness and Low Mach Number Limit in Compressible Complex Fluids

Project: Research

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Description

It is well-known that both the classical Newtonian fluids, such as compressible Navier- Stokes/Euler equations, and the models from complex fluids, such as compressible viscoelastic fluids and compressible magnetohydrodynamic (MHD) fluids, involve several physical parameters. Examples include the shear viscosity, the bulk viscosity, the Mach number, the Reynold number, and so on. Similar as Hilbert's sixth problem, the mathematical understanding for these parameter limits is analytically challenging and is physically interesting, in particular for numerical simulations and engineering applications. Despite of their importance in physics and engineering, even though a state-of-the-art small perturbation theory around an equilibrium was successfully formulated during the last two decades as the shear viscosity is nonnegative, the global wellposedness theories of Hookean elastic fluids with initial data of arbitrary size remain as challenging open problems in the mathematical community when the shear viscosity is large. Moreover as the Mach number approaches zero, the rigorous mathematical verification of the limit from inviscid compressible models to inviscid incompressible models is classical and challenging, especially in low spatial dimensions.The first topic in this proposal focuses on the wellposedness of the global-in-time classical solutions of Hookean compressible viscoelasticity with sufficiently large shear viscosity when the initial data is of arbitrary size. Comparing with the classical global wellposedness theories near an equilibrium, one of the advantages of the current project allows solutions to be of arbitrary size. This proposed problem is interesting and difficult because we lose the control for the compressible components and the ellipticity of elasticity is degenerated. For instance, the equilibrium provides an elliptic structure for the deformation gradient as the solutions is a small perturbation of non-zero equilibrium, while for initial data of arbitrary size, one only has the transport structure for the density and the deformation gradient. The success of this proposed problem needs a careful application of logarithmic Sobolev embeddings and effective viscous fluxes, and sheds lights on understanding the behavior of solutions when the deformation gradient is large.The second topic in this proposal aims to address the low Mach number limits of both compressible elastodynamics in dimensions two and ideal compressible magnetohydrodynamic (MHD) fluid flows in multidimensional spaces. On one hand the low Mach number limit of compressible elastodynamics in dimensions two is challenging due to the weaker dispersive estimates and the strong coupling among waves with different wave speeds. On the other hand the ideal compressible MHD is an important model in astrophysics, and is formally related to compressible elastodynamics. However the analysis of MHD is more challenging because of the degeneracy for the magnetic field. This degeneracy is compensated with a more careful analysis of the associated linearized system. The success of this proposed problem also needs a detailed analysis of compressible components which would disappear in the limit, and will provide insights for studying other compressible models in complex fluids. 

Detail(s)

Project number9043012
Grant typeGRF
StatusActive
Effective start/end date1/01/21 → …