Existence and Incompressible Limit of Compressible Complex Flows with Discontinuous Data

Project: Research

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It is both physically significant and mathematically challenging for understanding, in compressible complex flows, which quantities are smoothed out in the flows, which are not. It is also well-known that different physical phenomena in classical mechanics appear as the physical parameters tend to their limits. Examples include the shear 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 for classical solutions around an equilibrium was successfully formulated during the last two decades, the global well posedness theories of compressible complex flows with discontinuous initial data remain as challenging open problems in the mathematical community, especially when the degeneracy is present. Moreover as the bulk viscosity approaches infinity, the rigorous mathematical verification of the limit from compressible models to incompressible models is classical and challenging, in particularly when the regularity of solutions is weak.The first topic in this proposal focuses on the existence of weak solutions of incompressible/compressible magnetohydrodynamic flows (MHD) in multidimensional spatial spaces with small data around the constant equilibrium when the magnetic diffusivity is zero. This proposed problem is interesting and challenging because of the degeneracy of dissipation mechanism of the magnetic field, and the non-compatibility between the weak convergence and the nonlinearity. Precisely due to the degeneracy of dissipation mechanism of the magnetic field, the oscillation of density and the magnetic field would appear and it makes the compactness of the pressure and the lorentz force hard to be established. The “effective viscous flux” will be applied to obtain the L∞ estimate of both density and the magnetic field. The success of this proposed problem needs a careful analysis which guarantees appropriate pointwise bounds on the magnetic field B and the density ρ in the absence of regularity of ∇u, and will in turn shed lights on understanding better the weak convergence method in compressible flows with degeneracy.The second topic in this proposal aims to address the incompressible limit of compressible complex flows, including both the viscoelastic and magnetohydrodynamic flows with zero magnetic diffusivity, near an equilibrium with discontinuous initial data when the bulk viscosity approaches infinity. It is expected that the width of the initial layer has a uniform lower bound with respect to the bulk viscosity; while the density is expected to converge to a positive constant as the bulk viscosity approaches infinity. The decay mechanisms in the linearised hyperbolic-parabolic coupling system provide the necessary uniform control on the growth of norms of solutions. The success of this proposed problem relies on a better understanding of the smoothing effect of velocity and the damping effect of density with the help of “effective viscous fluxes”, and will pave the way to the justification of parameter limits of other compressible models with discontinuous initial data in the physical applications. 


Project number9043376
Grant typeGRF
StatusNot started
Effective start/end date1/01/23 → …