scholarly journals Optimal time-decay rates of the compressible Navier–Stokes–Poisson system in $ \mathbb R^3 $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guochun Wu ◽  
Han Wang ◽  
Yinghui Zhang
Keyword(s):  
Electric Field ◽  
Decay Rates ◽  
Navier Stokes ◽  
Optimal Time ◽  
Poisson System ◽  
Navier Stokes Equation ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
The Cauchy Problem ◽  
Time Decay Rates

<p style='text-indent:20px;'>We are concerned with the Cauchy problem of the 3D compressible Navier–Stokes–Poisson system. Compared to the previous related works, the main purpose of this paper is two–fold: First, we prove the optimal decay rates of the higher spatial derivatives of the solution. Second, we investigate the influences of the electric field on the qualitative behaviors of solution. More precisely, we show that the density and high frequency part of the momentum of the compressible Navier–Stokes–Poisson system have the same <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> decay rates as the compressible Navier–Stokes equation and heat equation, but the <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> decay rate of the momentum is slower due to the effect of the electric field.</p>

Global well-posedness and decay estimates for three-dimensional compressible Navier–Stokes–Allen–Cahn systems

2021 ◽  
pp. 1-32
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Three Dimensional ◽  
Decay Rates ◽  
Navier Stokes ◽  
Small Data ◽  
Well Posedness ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
The Cauchy Problem ◽  
Time Decay Rates ◽  
Derivatives Of

We study the small data global well-posedness and time-decay rates of solutions to the Cauchy problem for three-dimensional compressible Navier–Stokes–Allen–Cahn equations via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, the $\dot {H}^{-s}$ ( $0\leq s<\frac {3}{2}$ ) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

Mathematical modeling and qualitative analysis of viscoelastic conductive fluids

2020 ◽  
Vol 18 (06) ◽  
pp. 1077-1117
Author(s):  
Zhong Tan ◽  
Yong Wang ◽  
Wenpei Wu
Keyword(s):  
Variational Approach ◽  
Three Dimensional ◽  
Decay Rates ◽  
Navier Stokes ◽  
Optimal Time ◽  
Poisson Equations ◽  
The Cauchy Problem ◽  
Time Decay Rates ◽  
Spatial Derivatives ◽  
Unique Smooth Solution

We use an energetic variational approach to model the transport of compressible viscoelastic conductive fluids. Such a model can be called the three-dimensional compressible viscoelastic Navier–Stokes–Poisson equations. The global unique smooth solution to the Cauchy problem is obtained. In particular, we obtain the optimal time-decay rates of the solution and its higher-order spatial derivatives by using a pure energy method.

scholarly journals On global well-posedness and decay of 3D Ericksen-Leslie system

2021 ◽  
Vol 6 (11) ◽  
pp. 12660-12679
Author(s):  
Xiufang Zhao ◽  
◽  
Ning Duan ◽  
Keyword(s):  
Strong Solutions ◽  
Decay Rates ◽  
Well Posedness ◽  
Small Initial Data ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
Liquid Crystal System ◽  
The Cauchy Problem ◽  
Derivatives Of ◽  
Leslie System

<abstract><p>In this paper, the small initial data global well-posedness and time decay estimates of strong solutions to the Cauchy problem of 3D incompressible liquid crystal system with general Leslie stress tensor are studied. First, assuming that $ \|u_0\|_{\dot{H}^{\frac12+\varepsilon}}+\|d_0-d_*\|_{\dot{H}^{\frac32+\varepsilon}} $ ($ \varepsilon &gt; 0) $ is sufficiently small, we obtain the global well-posedness of strong solutions. Moreover, the $ L^p $–$ L^2 $ ($ \frac32\leq p\leq2 $) type optimal decay rates of the higher-order spatial derivatives of solutions are also obtained. The $ \dot{H}^{-s} $ ($ 0\leq s &lt; \frac12 $) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.</p></abstract>

scholarly journals A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ruy Coimbra Charão ◽  
Alessandra Piske ◽  
Ryo Ikehata
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Decay Rates ◽  
Plate Equation ◽  
New Model ◽  
Asymptotic Profile ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
The Cauchy Problem ◽  
Logarithmic Type

<p style='text-indent:20px;'>We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in <inline-formula><tex-math id="M1">\begin{document}$ {{\bf R}}^{n} $\end{document}</tex-math></inline-formula>, and study the asymptotic profile and optimal decay rates of solutions as <inline-formula><tex-math id="M2">\begin{document}$ t \to \infty $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M3">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula>-sense. The operator <inline-formula><tex-math id="M4">\begin{document}$ L $\end{document}</tex-math></inline-formula> considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charão-Ikehata [<xref ref-type="bibr" rid="b7">7</xref>]. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the property of the solutions into three parts from the viewpoint of regularity of the initial data, that is, diffusion-like, wave-like, and both of them.</p>

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