CPAA
Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity
Xulong Qin Zheng-An Yao
A free boundary problem is investigated for viscous, compressible, heat-conducting, one-dimensional real gas with general large initial data. More precisely, the viscosity is assumed to be $\mu(\rho)=\rho^{\lambda}(\lambda>0)$, where $\rho$ is the density of the gas, and there is nonlinear dependence upon the density and temperature for the equations of state which are different from the linear dependence of perfect gas. It is also proved that no shock wave, vacuum, mass or heat concentration will be developed in a finite time and that the free boundary (interface) separating the gas and vacuum expands at a finite velocity.
keywords: density-dependent viscosity global existence. heat-conducting real gas initial boundary problem Viscous
CPAA
A blow-up criterion for the 3D compressible MHD equations
Ming Lu Yi Du Zheng-An Yao Zujin Zhang
In this paper, we study the 3D compressible magnetohydrodynamic equations. We extend the well-known Serrin's blow-up criterion(see [32]) for the 3D incompressible Navier-Stokes equations to the 3D compressible magnetohydrodynamic equations. In addition, initial vacuum is allowed in our case.
keywords: strong solutions 3D compressible magnetohydrodynamic equations. Blow-up criterion
CPAA
Localization of blow-up points for a nonlinear nonlocal porous medium equation
Lili Du Zheng-An Yao
This paper deals with the porous medium equation with a nonlinear nonlocal source

$u_t=\Delta u^m + au^p\int_\Omega u^q dx,\quad x\in \Omega, t>0$

subject to homogeneous Dirichlet condition. We investigate the influence of the nonlocal source and local term on blow-up properties for this system. It is proved that: (i) when $p\leq 1$, the nonlocal source plays a dominating role, i.e. the system has global blow-up and the blow-up profile which is uniformly away from the boundary either at polynomial scale or logarithmic scale is obtained. (ii) When $p > m$, this system presents single blow-up pattern. In other words, the local term dominates the nonlocal term in the blow-up profile. This extends the work of Li and Xie in Appl. Math. Letter, 16 (2003) 185--192.

keywords: nonlinear nonlocal source blow-up set. Porous medium equation blow-up
CPAA
The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources
Zhong Tan Zheng-An Yao
In this paper we consider the existence, nonexistence and the asymptotic behavior of the global solutions of the quasilinear parabolic equation of the following form:

$u_t-\Delta_pu=|u|^{q-2}u, \quad (x,t)\in\Omega\times (0,T),$

$u(x,t)=0,\quad (x,t)\in\partial\Omega\times (0,T), $

$ u(x,0)=u_0(x), \quad u_0(x)\geq 0, u_0(x)$ ≠ $0, $

where $\Omega$ is a smooth bounded domain in $R^N(N\geq 3)$, $\Delta_pu=$ div$(|\nabla u|^{p-2}\nabla u )$, $\frac{2N}{N+2}$ < $p$ < $N$, $q=p^\star=\frac{pN}{N-p}$ is the critical Sobolev exponent. In particular, we employ the concentration-compactness principle to prove that the global solutions with the initial data in "stable set" converge strongly to zero in $W_0^{1,p}(\Omega)$.

keywords: critical Sobolev exponent p-Laplace evolution equation Finite time blowup. Asymptotic behavior
DCDS
Blow-up phenomena for the 3D compressible MHD equations
Ming Lu Yi Du Zheng-An Yao
In this paper, we study the three-dimensional(3D) compressible magnetohydrodynamic equations. Firstly, we obtain a blow-up criterion for the local strong solutions in terms of the gradient of the velocity, which is similar to the Beal-Kato-Majda criterion(see [1]) for the ideal incompressible flow. Secondly, we extend the well-known Serrin's blow-up criterion for the 3D incompressible Navier-Stokes equations to the 3D compressible magnetohydrodynamic equations. In addition, initial vacuum is allowed in our cases.
keywords: 3D compressible magnetohydrodynamic equations. Blow-up criterion strong solutions
KRM
On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces
Qunyi Bie Qiru Wang Zheng-An Yao
This paper is devoted to the study of the inviscid Boussinesq equations. We establish the local well-posedness and blow-up criteria in Besov-Morrey spaces $N_{p,q,r}^s(\mathbb{R}^n)$ for super critical case $s > 1 + \frac{n}{p}, 1 < q \leq p < \infty, 1 \leq r\leq \infty$, and critical case $s=1+\frac{n}{p}, 1 < q \leq p < \infty, r=1$. Main analysis tools are Littlewood-Paley decomposition and the paradifferential calculus.
keywords: Besov-Morrey space blow-up criteria Littlewood-Paley decomposition. local well-posedness Boussinesq equations
CPAA
Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model
Haibo Cui Lei Yao Zheng-An Yao
In this paper, we consider global existence and optimal time decay rates of global smooth solutions to three-dimensional reduced gravity two and a half layer model. Indeed we show that the upper and middle layer thicknesses and horizontal velocities converge to their equilibrium state at the $L^2$-rate $(1+t)^{-\frac{3}{4}}$ or $L^\infty$-rate $(1+t)^{-\frac{3}{2}}$, respectively. These convergence rates are also shown to be optimal. The proof is based on the detailed analysis of the Green's function to the linearized system and elaborate energy estimates to the nonlinear system.
keywords: global existence Cauchy problem decay estimates. Two and a half layer model
CPAA
Decay of the compressible viscoelastic flows
W. Wei Yin Li Zheng-An Yao
In this paper we study the time decay rates of the solution to the Cauchy problem for the compressible viscoelastic flows via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, and the $\dot{H}^{-s}(0\leq s<\frac{3}{2})$ negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.
keywords: Viscoelastic flows energy method negative Sobolev space.
CPAA
One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries
Xulong Qin Zheng-An Yao Hongxing Zhao
A free-boundary problem is studied for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity that decreases (to zero) with decreasing density, i.e., $\mu=A\rho^\theta$, where $A$ and $\theta$ are positive constants. The existence and uniqueness of the global weak solutions are obtained with $\theta\in (0,1]$, which improves the previous results and no vacuum is developed in the solutions in a finite time provided the initial data does not contain vacuum.
keywords: Compressible Navier-Stokes equations existence uniqueness density-dependent viscosity free boundary.
DCDS
Global existence and optimal decay rates of solutions for compressible Hall-MHD equations
Jincheng Gao Zheng-An Yao
In this paper, we are concerned with global existence and optimal decay rates of solutions for the compressible Hall-MHD equations in dimension three. First, we prove the global existence of strong solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in $H^2$-framework. Second, optimal decay rates of strong solutions in $L^2$-norm are obtained if the initial data belong to $L^1$ additionally. Finally, we apply Fourier splitting method by Schonbek [Arch. Rational Mech. Anal. 88 (1985)] to establish optimal decay rates for higher order spatial derivatives of classical solutions in $H^3$-framework, which improves the work of Fan et al.[Nonlinear Anal. Real World Appl. 22 (2015)].
keywords: global solutions optimal decay rates Compressible Hall-MHD equations Fourier splitting method.

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