Electronic Research Archive

Special issue on PDEs arising from nonlinear waves and fluid dynamics

Robin Ming Chen, Department of Mathematics, University of Pittsburgh mingchen@pitt.edu

Runzhang Xu, College of Mathematical Sciences, Harbin Engineering University xurunzh@163.com

Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersionSpecial Issues
Guenbo Hwang and Byungsoo Moon
2020, 28(1): 15-25 doi: 10.3934/era.2020002 +[Abstract](14)+[HTML](8) +[PDF](327.55KB)
Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscositiesSpecial Issues
Yue Cao
2020, 28(1): 27-46 doi: 10.3934/era.2020003 +[Abstract](11)+[HTML](9) +[PDF](346.98KB)

In this paper, the Cauchy problem of the \begin{document}$ 3 $\end{document}D compressible Navier-Stokes equations with degenerate viscosities and far field vacuum is considered. We prove that the \begin{document}$ L^\infty $\end{document} norm of the deformation tensor \begin{document}$ D(u) $\end{document} (\begin{document}$ u $\end{document}: the velocity of fluids) and the \begin{document}$ L^6 $\end{document} norm of \begin{document}$ \nabla \log \rho $\end{document} (\begin{document}$ \rho $\end{document}: the mass density) control the possible blow-up of regular solutions. This conclusion means that if a solution with far field vacuum to the Cauchy problem of the compressible Navier-Stokes equations with degenerate viscosities is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of \begin{document}$ D(u) $\end{document} or \begin{document}$ \nabla \log \rho $\end{document} as the critical time approaches; equivalently, if both \begin{document}$ D(u) $\end{document} and \begin{document}$ \nabla \log \rho $\end{document} remain bounded, a regular solution persists.

The existence of solutions for a shear thinning compressible non-Newtonian modelsSpecial Issues
Yukun Song, Yang Chen, Jun Yan and Shuai Chen
2020, 28(1): 47-66 doi: 10.3934/era.2020004 +[Abstract](10)+[HTML](9) +[PDF](413.61KB)

This paper is concerned with the initial boundary value problem for a shear thinning fluid-particle interaction non-Newtonian model with vacuum. The viscosity term of the fluid and the non-Newtonian gravitational force are fully nonlinear. Under Dirichlet boundary for velocity and the no-flux condition for density of particles, the existence and uniqueness of strong solutions is investigated in one dimensional bounded intervals.

Initial boundary value problem for a inhomogeneous pseudo-parabolic equationSpecial Issues
Jun Zhou
2020, 28(1): 67-90 doi: 10.3934/era.2020005 +[Abstract](11)+[HTML](22) +[PDF](777.08KB)

This paper deals with the global existence and blow-up of solutions to a inhomogeneous pseudo-parabolic equation with initial value \begin{document}$ u_0 $\end{document} in the Sobolev space \begin{document}$ H_0^1( \Omega) $\end{document}, where \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} (\begin{document}$ n\geq1 $\end{document} is an integer) is a bounded domain. By using the mountain-pass level \begin{document}$ d $\end{document} (see (14)), the energy functional \begin{document}$ J $\end{document} (see (12)) and Nehari function \begin{document}$ I $\end{document} (see (13)), we decompose the space \begin{document}$ H_0^1( \Omega) $\end{document} into five parts, and in each part, we show the solutions exist globally or blow up in finite time. Furthermore, we study the decay rates for the global solutions and lifespan (i.e., the upper bound of blow-up time) of the blow-up solutions. Moreover, we give a blow-up result which does not depend on \begin{document}$ d $\end{document}. By using this theorem, we prove the solution can blow up at arbitrary energy level, i.e. for any \begin{document}$ M\in \mathbb{R} $\end{document}, there exists \begin{document}$ u_0\in H_0^1( \Omega) $\end{document} satisfying \begin{document}$ J(u_0) = M $\end{document} such that the corresponding solution blows up in finite time.

Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systemsSpecial Issues
Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu and Yuanran Zhu
2020, 28(1): 91-102 doi: 10.3934/era.2020006 +[Abstract](11)+[HTML](13) +[PDF](299.33KB)

We study the initial boundary value problem of linear homogeneous wave equation with dynamic boundary condition. We aim to prove the finite time blow-up of the solution at critical energy level or high energy level with the nonlinear damping term on boundary in control systems.

2018  Impact Factor: 0.263



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