Diffusion Induced Chaos in a Closed Loop Thermosyphon

This is the published version, also available here: http://dx.doi.org/10.1137/S0036139996304184.The dynamics of a closed loop thermosyphon are considered. The model assumes a prescribed heat flux along the loop wall and the contribution of axial diffusion. The well-posedness of the model which consists of a coupled ODE and PDE is shown for both the case with diffusion and without diffusion. Boundedness of solutions, the existence of an attractor, and an inertial manifold is proven, and an exact reduction to a low-dimensional model is obtained for the diffusion case. The reduced systems may have far fewer degrees of freedom than the reduction to the inertial manifold. For the three mode models, equivalence with the classical Lorenz equations is shown. Numerical results are presented for five mode models

Reduction of dimension of approximate intertial manifolds by symmetry

In this paper, we study approximate inertial manifolds for nonlinear evolution partial differential equations which possess symmetry. The relationship between symmetry and dimensions of approximate inertial manifolds is established. We demonstrate that symmetry can reduce the dimensions of an approximate inertial manifold. Applications for concrete evolution equations are given

A note on the Cahn-Hilliard equation in H1(RN)H^1(\mathbb R^N) involving critical exponent

summary:We consider the Cahn-Hilliard equation in $H^1(\mathbb R^N)$ with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as $|u|\to \infty$ and logistic type nonlinearities. In both situations we prove the $H^2(\mathbb R^N)$-bound on the solutions and show that the individual solutions are suitably attracted by the set of equilibria. This complements the results in the literature; see J. W. Cholewa, A. Rodriguez-Bernal (2012)

Diffusion Induced Chaos in a Closed Loop Thermosyphon

The dynamics of a closed loop thermosyphon are considered. The model assumes a prescribed heat flux along the loop wall and the contribution of axial di#usion. The well-posedness of the model which consists of a coupled ODE and PDE is shown for both the case with di#usion and without di#usion. Boundedness of solutions, the existence of an attractor, and an inertial manifold is proven, and an exact reduction to a low-dimensional model is obtained for the di#usion case. The reduced systems may have far fewer degrees of freedom than the reduction to the inertial manifold. For the three mode models, equivalence with the classical Lorenz equations is shown. Numerical results are presented for five mode models

Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations

In this well-written paper, the authors consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors. \par In the first part of the paper, some notions concerning dissipative systems in ordered space are recalled. Then follow results on the existence of extremal solutions and global attractors and finally on the inclusion of the global attractor in an order interval formed by the minimal and the maximal equilibria. \par In the second part of the paper, they then show some applications of the abstract scheme to various evolutionary problems, from ODEs and retarded functional differential equations to parabolic and hyperbolic PDEs. In particular, they exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in $\Bbb R^N$ with nonlinearities depending on the gradient of the solution. \par The authors consider as well systems of reaction-diffusion equations in $\Bbb R^N$ and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in $\Bbb R^N$. They further discuss some nonlocal and quasilinear problems, as well as the fourth order Cahn-Hilliard equation

Attractors for partly dissipative reaction diffusion systems in R-n

In this paper, we study the asymptotic behavior of solutions for the partly dissipative reaction diffusion equations in R-n. We prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in L-2(R-n) X L-2(R-n)