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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.

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 $(-\Delta)^{\alpha}$ in the viscous damping with $\alpha\in[0,1]$, we derive optimal decay rates for global (in time) solutions with small data in certain Sobolev spaces. Furthermore, by introducing a threshold $\alpha=1/2$ for the power of the fractional viscous dissipation, we derive an anomalous diffusion profile when $\alpha\in[0,1/2)$ and a diffusion wave profile when $\alpha\in[1/2,1]$ 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., Comment: Blow-up for the inviscid model (or the JMGT equation in the critical case) is added

We study an ocean related system with a small viscosity parameter, which is the linearized version of the modified Primitive Equations. As the parameter goes to zero, a $L^\infty$ convergence result is obtained together with the estimation on the thickness of the boundary layer., Comment: 7 pages

We consider an initial boundary problem for the planar MHD system under the general condition on the heat conductivity $\kappa$ that may depend on both the density $\rho$ and the temperature $\theta$ satisfying $\kappa(\rho,\theta)\geq\kappa_1 \theta^{q}$ for some constants $\kappa_1>0$ 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., Comment: 31 pages. arXiv admin note: text overlap with arXiv:1511.00201

In this paper, we consider an initial-boundary problem for plane magnetohydrodynamics flows under the general condition on the heat conductivity $\kappa$ that may depend on both the density $\rho$ and the temperature $\theta$ and satisfies $$ \kappa(\rho,\theta)\geq\kappa_1(1+\theta^{q}) \quad \hbox{\rm with constants}~ \kappa_1>0 ~\hbox{\rm and}~ q>0. $$ We prove the global existence of strong solutions for large initial data and justify the passage to the limit as the shear viscosity $\mu$ goes to zero. Furthermore, the value $\mu^\alpha$ with any $0<\alpha<1/2$ is established for the boundary layer thickness., Comment: 41 pages