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1078-0947
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Discrete & Continuous Dynamical Systems - A
October 1996 , Volume 2 , Issue 4
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1996, 2(4): 417-454
doi: 10.3934/dcds.1996.2.417
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Abstract:
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.
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.
1996, 2(4): 455-466
doi: 10.3934/dcds.1996.2.455
+[Abstract](2412)
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Abstract:
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.
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.
1996, 2(4): 467-482
doi: 10.3934/dcds.1996.2.467
+[Abstract](2162)
+[PDF](212.0KB)
Abstract:
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.
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.
1996, 2(4): 483-496
doi: 10.3934/dcds.1996.2.483
+[Abstract](2205)
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Abstract:
We study the long-time behavior of solutions of the Cauchy problem
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.
1996, 2(4): 497-524
doi: 10.3934/dcds.1996.2.497
+[Abstract](2274)
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Abstract:
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 $r\geq 0$.
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 $r\geq 0$.
1996, 2(4): 525-542
doi: 10.3934/dcds.1996.2.525
+[Abstract](2011)
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Abstract:
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.
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.
1996, 2(4): 543-558
doi: 10.3934/dcds.1996.2.543
+[Abstract](2036)
+[PDF](225.7KB)
Abstract:
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.
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.
1996, 2(4): 559-583
doi: 10.3934/dcds.1996.2.559
+[Abstract](1625)
+[PDF](1117.8KB)
Abstract:
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.
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.
1996, 2(4): 585-593
doi: 10.3934/dcds.1996.2.585
+[Abstract](1955)
+[PDF](149.3KB)
Abstract:
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.
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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