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Xulong Qin

Publications and source records attributed to Xulong Qin.

9 recordsLinked to original sources
arXiv2025

$L^p-L^q$ estimates for the dissipative and conservative Moore-Gibson-Thompson equations

This paper studies some $L^p-L^q$ estimates for the dissipative or conservative Moore-Gibson-Thompson (MGT) equations in the whole space $\mathbb{R}^n$. Our contributions are twofold. By applying the Fourier analysis associated with the modified Bessel function in the dissipative case, we derive some $L^p-L^q$ estimates of solutions. Then, introducing a good unknown related to the free wave equation in the conservative case, some $L^p-L^q$ estimates of solutions with the admissible closed triangle range of exponents are deduced. These results show some essential influences of dissipation from the MGT equations in the $L^q$ framework.

math.AP↗
arXiv2023

Asymptotic behavior of solutions for the thermoviscous acoustic systems

We study some asymptotic properties of solutions for the acoustic coupled systems in thermoviscous fluids which was proposed by [Karlsen-Bruus, \emph{Phys. Rev. E} (2015)]. Basing on the WKB analysis and the Fourier analysis, we derive optimal estimates and large time asymptotic profiles of the energy term via diagonalization procedure, and of the velocity potential via reduction methodology. We found that the wave effect has a dominant influence for lower dimensions comparing with thermal-viscous effects. Moreover, by employing suitable energy methods, we rigorously demonstrate global (in time) inviscid limits as the momentum diffusion coefficient vanishes, whose limit model can be regarded as the thermoelastic acoustic systems in isotropic solids. These results explain some influence of the momentum diffusion on asymptotic behavior of solutions.

math.AP↗
arXiv2023

The influence of viscous dissipations on the nonlinear acoustic wave equation with second sound

We study the effect of a viscous dissipation on the Cauchy problem for a Cattaneo-type model in nonlinear acoustics, established by applying the Lighthill approximation for the viscous or inviscid fluid model. The contribution of this paper is twofold. For the nonlinear viscous Cattaneo-type model involving a fractional Laplacian $(-Δ)^α$ in the viscous damping with $α\in[0,1]$, we derive optimal decay rates for global (in time) solutions with small data in certain Sobolev spaces. Furthermore, by introducing a threshold $α=1/2$ for the power of the fractional viscous dissipation, we derive an anomalous diffusion profile when $α\in[0,1/2)$ and a diffusion wave profile when $α\in[1/2,1]$ for large-time. Whereas, for the nonlinear inviscid Cattaneo-type model (or the Jordan-Moore-Gibson-Thompson equation in the critical case), we obtain the blow-up of the energy solutions in finite time under suitable assumptions for the initial data. Thus, the presence of a viscous dissipation in the nonlinear Cattaneo-type model is a criterion for the global (in time) existence and blow-up of solutions.

math.AP↗
arXiv2018

Vanishing Shear Viscosity Limit and Boundary Layer Study on the Planar MHD system

We consider an initial boundary problem for the planar MHD system under the general condition on the heat conductivity $κ$ that may depend on both the density $ρ$ and the temperature $θ$ satisfying $κ(ρ,θ)\geqκ_1 θ^{q}$ for some constants $κ_1>0$ and $q>0.$ Firstly, the global existence of strong solution for large initial data is obtained, and then the limit of the vanishing shear viscosity is justified. In addition, the $L^2$ convergence rate is obtained together with the estimation on the thickness of the boundary layer.

math.AP↗
arXiv2015

Vanishing shear viscosity and boundary layers for plane magnetohydrodynamics flows

In this paper, we consider an initial-boundary problem for plane magnetohydrodynamics flows under the general condition on the heat conductivity $κ$ that may depend on both the density $ρ$ and the temperature $θ$ and satisfies $$ κ(ρ,θ)\geqκ_1(1+θ^{q}) \quad \hbox{\rm with constants}~ κ_1>0 ~\hbox{\rm and}~ q>0. $$ We prove the global existence of strong solutions for large initial data and justify the passage to the limit as the shear viscosity $μ$ goes to zero. Furthermore, the value $μ^α$ with any $0<α<1/2$ is established for the boundary layer thickness.

math.AP↗
arXiv2013

Global Solvability and Vanishing Shear Viscosity Limit of Planar Magnetohydrodynamic Equations with Large Initial Data

By observing a new relation between the magnetic pressure and the hydrodynamic pressure, global existence of classical solution to the full perfect MHD equations with large data is established, in particular including the case when all the viscosity, heat conductivity and diffusivity coefficients are constant. This can be viewed as an analog of the classical work by Kazhikhov-Shelukhin for the Navier-Stokes equations to the MHD equations. In addition, the vanishing shear viscosity limit is proved.

math.AP↗