arXiv subjects

Qin Duan

Publications and source records attributed to Qin Duan.

3 recordsLinked to original sources

arXiv2024

Well-posedness of regular solutions for 3-D full compressible Navier-Stokes equations with degenerate viscosities and heat conductivity

For the degenerate viscous and heat conductive compressible fluids, the momentum equations and the energy equation are degenerate both in the time evolution and spatial dissipation when vacuum appears, and then the physical entropy S behaves singularly, which make it challenging to study the corresponding well-posedness of regular solutions with high order regularities of S near the vacuum. In this paper, for the physically important case that the coefficients of viscosities and heat conductivity depend on the absolute temperature θin a power law of Chapman-Enskog, we identify a class of initial data admitting a local-in-time regular solution with far field vacuum to the Cauchy problem of the 3-D full CNS, and such a solution possesses the uniformly high order regularities for S near the vacuum. The key idea here is to study the vacuum problem in terms of the mass density ρ, velocity u and S instead of (ρ, u,θ), which makes it possible to compare the orders of the degeneracy of the time evolution and the spatial dissipations near the vacuum in terms of the powers of ρ. However, for heat conductive fluids, both a degenerate spatial dissipation and a source term related to \triangle ρ^{γ-1}, will appear in the time evolution equation for S, which makes it formidable to study the propagation of regularities of S. Fortunately, based on some elaborate analysis of the intrinsic degenerate-singular structures of the 3-D full CNS, we can choose proper weights to control the behaviors of (ρ, u,S) by introducing an enlarged reformulated system, which includes a singular parabolic system for u, and one degenerate-singular parabolic equation for S. Then one can carry out a series of weighted energy estimates carefully designed for this reformulated system, which provides an effective propagation mechanism for S's high order regularities near the vacuum.

math.AP↗

arXiv2022

On regular solutions for three-dimensional full compressible Navier-Stokes equations with degenerate viscosities and far field vacuum

In this paper, the Cauchy problem for the three-dimensional (3-D) full compressible Navier-Stokes equations (CNS) with zero thermal conductivity is considered. First, when shear and bulk viscosity coefficients both depend on the absolute temperature $θ$ in a power law ($θ^ν$ with $ν>0$) of Chapman-Enskog, based on some elaborate analysis of this system's intrinsic singular structures, we identify one class of initial data admitting a local-in-time regular solution with far field vacuum in terms of the mass density $ρ$, velocity $u$ and entropy $S$. Furthermore, it is shown that within its life span of such a regular solution, the velocity stays in an inhomogeneous Sobolev space, i.e., $u\in H^3(\mathbb{R}^3)$, $S$ has uniformly finite lower and upper bounds in the whole space, and the laws of conservation of total mass, momentum and total energy are all satisfied. Note that due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, and the physical entropy for polytropic gases behaves singularly, which make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the full CNS to the possible singularities of some special source terms related with $S$, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.

math.AP↗

arXiv2020

On the vanishing dissipation limit for the incompressible MHD equations on bounded domains

In this paper, we investigate the solvability, regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magneto-hydrodynamic (MHD) equations in bounded domains. On the boundary, the velocity field fulfills a Navier-slip condition, while the magnetic field satisfies the insulating condition. It is shown that the initial-boundary problem has a global weak solution for a general smooth domain. More importantly, for a flat domain, we establish the uniform local well-posedness of the strong solution with higher order uniform regularity and the asymptotic convergence with a rate to the solution of the ideal MHD as the dissipation tends to zero.

math.AP↗