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Discrete & Continuous Dynamical Systems - B

2008 , Volume 10 , Issue 4

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Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems
Marie-Odile Bristeau and Jacques Sainte-Marie
2008, 10(4): 733-759 doi: 10.3934/dcdsb.2008.10.733 +[Abstract](96) +[PDF](295.5KB)
From the free surface Navier-Stokes system, we derive the non-hydrostatic Saint-Venant system for the shallow waters including friction and viscosity. Based on an asymptotic analysis, the derivation leads to two formulations of growing complexity depending on the level of approximation chosen for the fluid pressure. The obtained models rely on formal estimates and are compared with the Boussinesq models.
Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations
Tomás Caraballo, Peter E. Kloeden and José Real
2008, 10(4): 761-781 doi: 10.3934/dcdsb.2008.10.761 +[Abstract](85) +[PDF](244.3KB)
We obtain regularity results for solutions of the three dimensional system of globally modified Navier-Stokes equations, and we investigate the relationship between global attractors, invariant measures, time-average measures and statistical solutions of these system in the case of temporally independent forcing.
Kawahara equation in a bounded domain
Gleb G. Doronin and Nikolai A. Larkin
2008, 10(4): 783-799 doi: 10.3934/dcdsb.2008.10.783 +[Abstract](88) +[PDF](224.8KB)
This paper deals with an initial-boundary value problem for the nonlinear one-dimensional Kawahara equation posed on a bounded interval. For reasonable initial and boundary conditions we prove the existence and uniqueness of a global regular solution subject to dispersive smoothing. We also show the exponential decay of the obtained solution as $t\to\infty$ and its asymptotics while the coefficient of the higher derivative approaches zero.
The stochastic primitive equations in two space dimensions with multiplicative noise
Nathan Glatt-Holtz and Mohammed Ziane
2008, 10(4): 801-822 doi: 10.3934/dcdsb.2008.10.801 +[Abstract](97) +[PDF](265.8KB)
We study the two dimensional primitive equations in the presence of multiplicative stochastic forcing. We prove the existence and uniqueness of solutions in a fixed probability space. The proof is based on finite dimensional approximations, anisotropic Sobolev estimates, and weak convergence methods.
Iterative method for mass diffusion model with density dependent viscosity
Francisco Guillén-González and Mamadou Sy
2008, 10(4): 823-841 doi: 10.3934/dcdsb.2008.10.823 +[Abstract](78) +[PDF](220.7KB)
The aim of this work is to study the existence of strong solutions for $3D$ fluids models with mass diffusion (also called Kazhikhov-Smagulov type system) assuming density dependent viscosity. The considered system represents a pollutant model.
    We use an iterative method to approach regular solutions. Moreover, some convergence rates are obtained, depending on weak, strong and more regular norms. This work extend to [1], where this technique has been used for the model with constant viscosity.
    The model has a diffusive operator $-\lambda$div$(\rho (\nabla v +\nabla v^t))$ with $v$ the velocity field, which not allows us to use direct Stokes regularity (as has been done in [1]. Thus, it becomes more difficult to obtain the $H^2\times H^1$ and $H^3\times H^2$ regularity for the velocity-pressure pair $(v,p)$. The key is to use a new regularity result for a Stokes type problem with $\rho\Delta v$ as diffusion term.
Asymptotic behavior of linearized viscoelastic flow problem
Yinnian He and Yi Li
2008, 10(4): 843-856 doi: 10.3934/dcdsb.2008.10.843 +[Abstract](76) +[PDF](203.6KB)
In this article, we provide some asymptotic behaviors of linearized viscoelastic flows in a general two-dimensional domain with certain parameters small and the time variable large.
Global dynamics of a Predator-Prey model with Hassell-Varley Type functional response
Sze-Bi Hsu, Tzy-Wei Hwang and Yang Kuang
2008, 10(4): 857-871 doi: 10.3934/dcdsb.2008.10.857 +[Abstract](109) +[PDF](279.3KB)
Predator-prey models with Hassell-Varley type functional response are appropriate for interactions where predators form groups and have applications in biological control. Here we present a systematic global qualitative analysis to a general predator-prey model with Hassell-Varley type functional response. We show that the predator free equilibrium is a global attractor only when the predator death rate is greater than its growth ability. The positive equilibrium exists if the above relation reverses. In cases of practical interest, we show that the local stability of the positive steady state implies its global stability with respect to positive solutions. For terrestrial predators that form a fixed number of tight groups, we show that the existence of an unstable positive equilibrium in the predator-prey model implies the existence of an unique nontrivial positive limit cycle.
Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems
Dongho Kim and Eun-Jae Park
2008, 10(4): 873-886 doi: 10.3934/dcdsb.2008.10.873 +[Abstract](85) +[PDF](198.1KB)
We construct a posteriori error estimators for approximate solutions of linear parabolic equations. We consider discretizations of the problem by modified discontinuous Galerkin schemes in time and continuous Galerkin methods in space. Especially, finite element spaces are permitted to change at different time levels. Exploiting Crank-Nicolson reconstruction idea introduced by Akrivis, Makridakis & Nochetto [2], we derive space-time a posteriori error estimators of second order in time for the Crank-Nicolson-Galerkin finite element method.
Analysis of a dynamic Elastic-Viscoplastic contact problem with friction
Stanislaw Migórski, Anna Ochal and Mircea Sofonea
2008, 10(4): 887-902 doi: 10.3934/dcdsb.2008.10.887 +[Abstract](62) +[PDF](230.7KB)
We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with an elastic-viscoplastic constitutive law and the frictional contact is modeled with subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving an integral equation coupled with an evolutionary hemivariational inequality. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract second order evolutionary inclusions with monotone operators and a fixed point theorem.
On a Hamiltonian PDE arising in magma dynamics
Gideon Simpson, Michael I. Weinstein and Philip Rosenau
2008, 10(4): 903-924 doi: 10.3934/dcdsb.2008.10.903 +[Abstract](92) +[PDF](364.4KB)
In this article we discuss a new Hamiltonian PDE arising from a class of equations appearing in the study of magma, partially molten rock in the Earth's interior. Under physically justifiable simplifications, a scalar, nonlinear, degenerate, dispersive wave equation may be derived to describe the evolution of $\phi$, the fraction of molten rock by volume, in the Earth. These equations have two power nonlinearities which specify the constitutive realitions for bulk viscosity and permeability in terms of $\phi$. Previously, they have been shown to admit solitary wave solutions. For a particular relation between exponents, we observe the equation to be Hamiltonian; it can be viewed as a generalization of the Benjamin-Bona-Mahoney equation. We prove that the solitary waves are nonlinearly stable, by showing that they are constrained local minimizers of an appropriate time-invariant Lyapunov functional. A consequence is an extension of the regime of global in time well-posedness for this class of equations to (large) data which includes a neighborhood of a solitary wave. Finally, we observe that these equations have compactons, solitary traveling waves with compact spatial support.
Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow
Rafael Vázquez, Emmanuel Trélat and Jean-Michel Coron
2008, 10(4): 925-956 doi: 10.3934/dcdsb.2008.10.925 +[Abstract](81) +[PDF](383.6KB)
We consider the problem of generating and tracking a trajectory between two arbitrary parabolic profiles of a periodic 2D channel flow, which is linearly unstable for high Reynolds numbers. Also known as the Poiseuille flow, this problem is frequently cited as a paradigm for transition to turbulence. Our procedure consists in generating an exact trajectory of the nonlinear system that approaches exponentially the objective profile. Using a backstepping method, we then design boundary control laws guaranteeing that the error between the state and the trajectory decays exponentially in $L^2$, $H^1$, and $H^2$ norms. The result is first proved for the linearized Stokes equations, then shown to hold locally for the nonlinear Navier-Stokes system.
Very weak solutions for the magnetohydrodynamic type equations
Elder Jesús Villamizar-Roa, Henry Lamos-Díaz and Gilberto Arenas-Díaz
2008, 10(4): 957-972 doi: 10.3934/dcdsb.2008.10.957 +[Abstract](72) +[PDF](230.6KB)
We consider the magnetohydrodynamic type equations with non-smooth Dirichlet boundary conditions for the velocity and the magnetic fields. We prove the existence of a kind of distributional solutions called very weak solutions and the continuous dependence of these solutions regarding the data; as a consequence, the uniqueness of very weak solutions is also obtained.
Stability of synchronization in a shift-invariant ring of mutually coupled oscillators
R. Yamapi and R.S. MacKay
2008, 10(4): 973-996 doi: 10.3934/dcdsb.2008.10.973 +[Abstract](68) +[PDF](543.7KB)
This paper treats synchronization dynamics in a shift-invariant ring of N mutually coupled self-sustained electrical units. Via some qualitative theory for the Lyapunov exponents, we derive the regimes of coupling parameters for which synchronized oscillation is stable or unstable in the ring.
A generalized projective dynamic for solving extreme and interior eigenvalue problems
Lei-Hong Zhang and Li-Zhi Liao
2008, 10(4): 997-1019 doi: 10.3934/dcdsb.2008.10.997 +[Abstract](56) +[PDF](319.5KB)
In [18] (Golub and Liao), a continuous-time system which is based on the projective dynamic is proposed to solve some concave optimization problems (with the unit ball constraint) resulted from extreme and interior eigenvalue problems. The convergence inside the unit ball is established; however, neither further convergence result outside the unit ball nor the stability analysis is available. Moreover, preliminary numerical experience indicates that this method is sensitive to a parameter whose optimal value is still difficult to determine. After analyzing the stability of this dynamic, in this paper, we develop a generalized model and analyze the convergence of the new model both inside and outside the unit ball. The flow of the generalized model is proved to converge almost globally to some eigenvector corresponding to the smallest eigenvalue, and share many surprisingly analogous properties with the Rayleigh quotient gradient flow. Links of our generalized projective dynamical system with other related works are also discussed. The efficiency of our new model is both addressed in theory and verified in numerical testing.

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