Solutions behavior of some dissipative systems
某些具有耗散效應系統解的性態
Student thesis: Doctoral Thesis
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Award date  4 Oct 2010 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(d81a3757314d49da8e3829c7141b8994).html 

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Abstract
We investigate the solutions behavior of some dissipative systems. Precisely,
we study the following problems: 1. The NavierStokes equations with a time
periodic external force in Rn. We show that a time periodic solution exists when
the space dimension n ≥ 5 under some suitably small assumption. The main
idea is to combine the energy method and the spectral analysis for the optimal
decay estimates on the linearized solution operator. With the optimal decay estimates,
we prove the existence and uniqueness of time periodic solution in some
suitable function space by the contraction mapping theorem. In addition, we
also study the time asymptotic stability of the time periodic solution. 2. Compressible
EulerPoisson system with nonlinear damping added to the momentum
equations. Under some mild conditions, the solutions of the Cauchy problem for
the system globally exist and converge to the nonlinear diffusion waves, which
are the corresponding solutions of nonlinear parabolic equations given by Darcy's
law with specified initial data. The optimal convergence rates are obtained by
Green function method when the initial perturbation is in L1space. 3. 2 x 2
linear damped psystem with boundary effect. By a heuristic analysis, we realize
that the best asymptotic profile for the original solution is the parabolic solution
of the IBVP for the corresponding porous media equation with a specified initial
data. In particular, we further show the convergence rates of the original solution
to its best asymptotic profile, which are much better than the rates obtained in previous works. The approach adopted in this part is the elementary weighted energy method together with Green function one.
 Numerical solutions, Mathematical models, Differential equations, Partial, Fluid dynamics