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Discrete & Continuous Dynamical Systems - A

1996 , Volume 2 , Issue 4

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Some computational aspects of approximate inertial manifolds and finite differences
Rolf Bronstering
1996, 2(4): 417-454 doi: 10.3934/dcds.1996.2.417 +[Abstract](131) +[PDF](4861.3KB)
An approach to the concept of approximate inertial manifolds for dissipative evolutionary equations in combination with finite difference semidiscretizations is presented. We introduce general frequency decompositions of the underlying finite dimensional solution space and consider the inertial form corresponding to this decomposition. It turns out that, under certain restrictions, all terms in the inertial form can be explicitly expanded as functions of the new coefficients. The calculations are carried out for reaction diffusion equations in 1D, 2D and 3D and for the Kuramoto-Sivashinsky equation in 1D, and numerical results are presented.
Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions
Boling Guo and Bixiang Wang
1996, 2(4): 455-466 doi: 10.3934/dcds.1996.2.455 +[Abstract](114) +[PDF](182.8KB)
In the present paper , we show the Gevrey class regularity of solutions for the generalized Ginzburg-Landau equation in two spatial dimensions. We also introduce an approximate inertial manifold for this system.
Nonlinear Galerkin approximation of the two dimensional exterior Navier-Stokes problem
Yinnian He and Kaitai Li
1996, 2(4): 467-482 doi: 10.3934/dcds.1996.2.467 +[Abstract](86) +[PDF](212.0KB)
In this paper, we present an Oseen coupling problem to approximate the two dimensional exterior unsteady Navier-Stokes problem with the nonhomogeneous boundary conditions. The Oseen coupling problem consists of the Navier-Stokes equations in a bounded region and the Oseen equations in an unbounded region. Then we derive the reduced Oseen coupling problem by use of the integral representations of the solution of the Oseen equations in an unbounded region. Moreover, we present the Galerkin approximation and the nonlinear Galerkin approximation for the reduced Oseen coupling problem. By analysing their convergence rates, we find that the nonlinear Galerkin approximation provides the same convergence order as the classical Galerkin approximation if we choose the space discrete parameter $H=O(h^{1/2})$. However, in this approximation, the nonlinearity is treated on the coarse grid finite element space and only the linear problem needs to be solved on the fine grid finite element space.
The asymptotic behavior of solutions of a semilinear parabolic equation
Minkyu Kwak and Kyong Yu
1996, 2(4): 483-496 doi: 10.3934/dcds.1996.2.483 +[Abstract](123) +[PDF](197.8KB)
We study the long-time behavior of solutions of the Cauchy problem

$u_t=\Delta u - (u^q)_y- u^p, \quad p, q >1,$

defined in the domain $Q=\{ (x, t): x=(x, y) \in \mathbf{R}^{N-1} \times \mathbf{R}, t >0 \}$ with nonnegative initial data in $L^1( \mathbf{R}^N)$. We completely classify the asymptotic profiles of solutions as $t \to \infty$ according to the parameters $p$ and $q$. We use rescaling transformations and a priori estimates.

Fourier nonlinear Galerkin method for Navier-Stokes equations
Kaitai Li and Yanren Hou
1996, 2(4): 497-524 doi: 10.3934/dcds.1996.2.497 +[Abstract](82) +[PDF](580.8KB)
Nonlinear Galerkin Methods (NGMs) are numerical schemes for evolutionary partial differential equations based on the theory of Inertial Manifolds (IMs) [1] and Approximate Inertial Manifolds (AIMs) [2]. In this paper, we focus our attention on the 2-D Navier- Stokes equations with periodic boundary conditions, and use Fourier methods to study its nonlinear Galerkin approximation which we call Fourier Nonlinear Galerkin Methods (FNGMs) here. The first part is contributed to the semidiscrete case. In this part, we derive the well-posedness of the nonlinear Galerkin form and the distance between the nonlinear Galerkin approximation and the genuine solution in Sobolev spaces of any orders. The second part is concerned about the full discrete case, in which, for a given numerical scheme based on NGMs, we investigate the stability and error estimate respectively. We derive the stability conditions for the scheme in any fractional Sobolev spaces. Finally, we give its error estimate in $H^r$ for any $r\geq 0$.
On the structure of the permanence region for competing species models with general diffusivities and transport effects
Julián López-Gómez
1996, 2(4): 525-542 doi: 10.3934/dcds.1996.2.525 +[Abstract](94) +[PDF](926.8KB)
In this work we study the problem of the coexistence of two competing species in an inhabited region by analyzing the shape of the region where the species exhibit permanence. To make this analysis we first obtain a singular perturbation result for an elliptic boundary value problem associated to a logistic equation with a general differential operator. Then, we analyze how varies the principal eigenvalue of the operator at its singular limit in the case when the operator admits a reduction to the selfadjoint case. These results are new and of a great interest by themselves. Finally, we shall apply them to the problem of the permanence.
Some new generalizations of inertial manifolds
A. Debussche and R. Temam
1996, 2(4): 543-558 doi: 10.3934/dcds.1996.2.543 +[Abstract](196) +[PDF](225.7KB)
In this article we introduce two new generalizations of inertial manifolds. The first one is the concept of Inertial Manifold with Delay which may be physically more suitable than the usual concept of inertial manifold since it allows a certain delay time for the adjustment of the small scales to the large ones. We also introduce an invariant manifold which is the graph of a multivalued function and which exists under general conditions. We conjecture that, like an inertial manifold, this manifold attracts all orbits at an exponential rate.
Large optimal truncated low-dimensional dynamical systems
Chui-Jie Wu
1996, 2(4): 559-583 doi: 10.3934/dcds.1996.2.559 +[Abstract](79) +[PDF](1117.8KB)
A new theory of constructing optimal truncated Low-Dimensional Dynamical Systems (LDDS), either based on known databases or directly from partial differential equations, is presented. Applying the new theory to four examples, i.e., the one--dimensional linear heat transfer equation, the nonlinear Burgers' equation and the two-dimensional Navier-Stokes equations with closed or open domains, it is shown that the optimal truncated LDDS in which the projecting errors have already been reduced to the minimum, can be constructed. Depending upon different optimal conditions, different LDDS can be found. Within the framework of optimal truncated LDDS, the initial bases chosen for iterations are the crucial factors in the optimization processes. The nonlinear Galerkin method can significantly improve the results, after the optimal bases have been gotten.
Some physical considerations attendant to the approximate inertial manifolds for Navier-Stokes equations
Oscar P. Manley
1996, 2(4): 585-593 doi: 10.3934/dcds.1996.2.585 +[Abstract](104) +[PDF](149.3KB)
In contrast with the other papers presented at the 1995 Xian Workshop, this article explores some physical ideas evoked by the concept of approximate inertial manifolds. Notably, while there is no clear separation of scales, such as needed for the existence of an inertial manifold, in turbulent flows there is a separation of physical phenomena. Thus the behavior of the spectrum ranging from the energy containing modes, through the inertial range is unaffected by dissipation, while the low energy content modes in the dissipation range are dominated by dissipation. An iterative process is developed yielding an effective viscosity useful for large eddy simulation. Some conjectures about the spectrum resulting from such a simulation are explored.

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