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>

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.

THE BOLTZMANN EQUATION IN THE SPACE $L^2\cap L^\infty_\beta$: GLOBAL AND TIME-PERIODIC SOLUTIONS

2006 ◽  
Vol 04 (03) ◽  
pp. 263-310 ◽  
Author(s):  
SEIJI UKAI ◽  
TONG YANG
Keyword(s):  
Boltzmann Equation ◽  
Source Term ◽  
Decay Rates ◽  
Regularity Conditions ◽  
Unique Existence ◽  
The Boltzmann Equation ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
Derivatives Of ◽  
Time Periodic

We present a function space in which the Cauchy problem for the Boltzmann equation is well-posed globally in time near an absolute Maxwellian in a mild sense without any regularity conditions. The asymptotic stability of the absolute Maxwellian is also established in this space and, moreover, it is shown that the higher order spatial derivatives of the solutions vanish in time faster than the lower order derivatives. No smallness assumptions are imposed on the derivatives of the initial data, and the optimal decay rates are derived. Furthermore, the Boltzmann equation with a time-periodic source term is solved in the same space on the unique existence and stability of a time-periodic solution which has the same period as the source term. The proof is based on the spectral analysis of the linearized Boltzmann operator.

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>

scholarly journals Optimal Convergence Rates for the Strong Solutions to the Compressible MHD Equations with Potential Force

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Miao Ouyang
Keyword(s):  
Convergence Rates ◽  
Strong Solutions ◽  
Large Time Behavior ◽  
Decay Rates ◽  
Mhd Equations ◽  
Time Behavior ◽  
Potential Force ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Compressible Mhd Equations

In this paper, the large-time behavior of solutions to the Cauchy problem for the 3D compressible MHD equations is considered with the effect of external force. We construct the global unique solution with the small initial data near the stationary profile. The optimal Lp-L2(1≤p≤2) time decay rates of the solution to the system are built in multifrequency decompositions.

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>

scholarly journals The Cauchy problem and decay rates for strong solutions of a Boussinesq system

2005 ◽  
Vol 2005 (7) ◽  
pp. 757-766 ◽  
Author(s):  
Francisco Guillén González ◽  
Márcio Santos da Rocha ◽  
Marko Rojas Medar
Keyword(s):  
Heat Transfer ◽  
Cauchy Problem ◽  
Viscous Fluid ◽  
Strong Solutions ◽  
Boussinesq Equations ◽  
Decay Rates ◽  
Boussinesq System ◽  
Three Dimension ◽  
The Cauchy Problem ◽  
Derivatives Of

The Boussinesq equations describe the motion of an incompressible viscous fluid subject to convective heat transfer. Decay rates of derivatives of solutions of the three-dimension-al Cauchy problem for a Boussinesq system are studied in this work.

scholarly journals Well-posedness and optimal decay rates for the viscoelastic Kirchhoff equation

2017 ◽  
Vol 35 (3) ◽  
pp. 203 ◽  
Author(s):  
Aissa Guesmia
Keyword(s):  
Decay Rate ◽  
Bounded Domain ◽  
Decay Rates ◽  
Kirchhoff Equation ◽  
Well Posedness ◽  
Kirchhoff Type ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
Optimal Decay Rate ◽  
Finite Memory

In this paper, we investigate the well-posedness as well as optimal decay rate estimates of the energy associated with a Kirchhoff-Carrier problem in n-dimensional bounded domain under an internal finite memory. The considered class of memory kernels is very wide and allows us to derive new and optimal decay rate estimates then those ones considered previously in the literature for Kirchhoff-type models.

scholarly journals A new stability result for a thermoelastic Bresse system with viscoelastic damping

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Soh Edwin Mukiawa ◽  
Cyril Dennis Enyi ◽  
Tijani Abdulaziz Apalara
Keyword(s):  
Stability Result ◽  
Bending Moment ◽  
Decay Rates ◽  
Viscoelastic Damping ◽  
Physical Parameters ◽  
Bresse System ◽  
Solution Energy ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
Weaker Conditions

AbstractWe investigate a thermoelastic Bresse system with viscoelastic damping acting on the shear force and heat conduction acting on the bending moment. We show that with weaker conditions on the relaxation function and physical parameters, the solution energy has general and optimal decay rates. Some examples are given to illustrate the findings.

Sign in / Sign up

Export Citation Format

Share Document