EXPONENTIAL STABILITY OF STATIONARY SOLUTIONS TO THE DISCRETE BOLTZMANN EQUATION IN A BOUNDED DOMAIN

1992 ◽  
Vol 02 (02) ◽  
pp. 239-248 ◽  
Author(s):  
SHUICHI KAWASHIMA
Keyword(s):  
Boltzmann Equation ◽  
Exponential Stability ◽  
Bounded Domain ◽  
Large Time ◽  
Stationary Solutions ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Crucial Point ◽  
Boundary Estimates ◽  
Discrete Boltzmann Equation

Large-time behavior of solutions of the discrete Boltzmann equation in a bounded domain is studied. The boundary conditions considered are pure diffuse relection and general reflection. Under suitable assumptions it is proved that a unique solution exists globally in time and converges to the corresponding unique stationary solution exponentially as time goes to infinity. The crucial point of the proof is in the derivation of desired boundary estimates of the solution subordinate to the general reflection.

scholarly journals Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Rong Zhang ◽  
Liangchen Wang
Keyword(s):  
Boundary Conditions ◽  
Bounded Domain ◽  
Large Time ◽  
Neumann Boundary ◽  
Rates Of Convergence ◽  
Large Time Behavior ◽  
Global Dynamics ◽  
Time Behavior ◽  
Chemical Signalling ◽  
Chemotaxis System

<p style='text-indent:20px;'>This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi_1\nabla\cdot(u\nabla v)+\mu_1 u(1-u-a_1w),\, x\in \Omega,\, t&gt;0,\\ 0 = \Delta v-v+w,\,x\in\Omega,\, t&gt;0,\\ w_t = \Delta w-\chi_2\nabla\cdot(w\nabla z)-\chi_3\nabla\cdot(w\nabla v)+\mu_2 w(1-w-a_2u), \,x\in \Omega,\,t&gt;0,\\ 0 = \Delta z-z+u, \,x\in\Omega,\, t&gt;0, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ n\geq1 $\end{document}</tex-math></inline-formula>, where the parameters <inline-formula><tex-math id="M3">\begin{document}$ a_1,a_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \chi_1, \chi_2, \chi_3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu_1, \mu_2 $\end{document}</tex-math></inline-formula> are positive constants. We first showed some conditions between <inline-formula><tex-math id="M6">\begin{document}$ \frac{\chi_1}{\mu_1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \frac{\chi_2}{\mu_2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \frac{\chi_3}{\mu_2} $\end{document}</tex-math></inline-formula> and other ingredients to guarantee boundedness. Moreover, the large time behavior and rates of convergence have also been investigated under some explicit conditions.</p>

scholarly journals Large Time Behavior of the Vlasov-Poisson-Boltzmann System

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Li Li ◽  
Shuilin Jin ◽  
Li Yang
Keyword(s):  
Boltzmann Equation ◽  
Large Time ◽  
Charged Particles ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Time Stability ◽  
Poisson Boltzmann

The motion of dilute charged particles can be modeled by Vlasov-Poisson-Boltzmann system. We study the large time stability of the VPB system. To be precise, we prove that when time goes to infinity, the solution of VPB system tends to global Maxwellian state in a rateOt−∞, by using a method developed for Boltzmann equation without force in the work of Desvillettes and Villani (2005). The improvement of the present paper is the removal of condition on parameterλas in the work of Li (2008).

GLOBAL SOLUTIONS TO THE BOLTZMANN EQUATION WITH EXTERNAL FORCES

2005 ◽  
Vol 03 (02) ◽  
pp. 157-193 ◽  
Author(s):  
SEIJI UKAI ◽  
TONG YANG ◽  
HUIJIANG ZHAO
Keyword(s):  
Boltzmann Equation ◽  
Type System ◽  
Global Solutions ◽  
Stationary Solutions ◽  
Large Time Behavior ◽  
Classical Solutions ◽  
Time Behavior ◽  
Potential Force ◽  
Global Classical Solutions ◽  
The Boltzmann Equation

For the Boltzmann equation with an external potential force depending only on the space variables, there is a family of stationary solutions, which are local Maxwellians with space dependent density, zero velocity and constant temperature. In this paper, we will study the nonlinear stability of these stationary solutions by using the energy method. The analysis combines the analytic techniques used for the conservation laws using the fluid-type system derived from the Boltzmann equation (cf. [14]) and the dissipative effects on the fluid and non-fluid components of the Boltzmann equation through the celebrated H-theorem. To our knowledge, this is the first result on the global classical solutions to the Boltzmann equation with external force and non-trivial large time behavior in the whole space.

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