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Discrete & Continuous Dynamical Systems - A

2010 , Volume 27 , Issue 4

A special issue
Dedicated to Roger Temam on the Occasion of his 70th Birthday
Part I

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Preface
Claude-Michel Brauner, Danielle Hilhorst, Alain Miranville, Shouhong Wang and Xiaoming Wang
2010, 27(4): i-iii doi: 10.3934/dcds.2010.27.4i +[Abstract](181) +[PDF](342.1KB)
Abstract:
Born in Tunis on May 19, 1940, Roger Temam moved to Paris in 1957 to study at the University of Paris, which was at that time the only university in Paris, known as La Sorbonne. He wrote his doctoral thesis under the supervision of Professor Jacques-Louis Lions and became a professor at the University of Paris-Sud XI at Orsay in 1968. There, he founded, together with Professors Jacques Deny and Charles Goulaouic, the Laboratory of Numerical and Functional Analysis which he directed from 1972 to 1988. He was also a Maître de Conférences at the famous Ecole Polytechnique from 1968 to 1986.
   In 1983, Roger Temam co-founded the SMAI, the French Applied and Industrial Mathematical Society, analogous to SIAM, and served as its first president. He initiated the ICIAM conference series and was head of the Steering Committee of its first meeting held in Paris in 1987. He was also the Editor-in-Chief of the mathematical journal M2AN from 1986 to 1997, and he is or has been on the editorial board of such journals as Asymptotic Analysis, Discrete and Continuous Dynamical Systems, Journal of Differential Equations, Physica D, Communications in PDEs and SIAM Journal of Numerical Analysis.

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The vortex core structure inside spherical ferromagnetic particles
François Alouges, Sylvain Faure and Jutta Steiner
2010, 27(4): 1259-1282 doi: 10.3934/dcds.2010.27.1259 +[Abstract](185) +[PDF](1483.8KB)
Abstract:
We analyze the vortex core structure inside spherical ferromagnetic particles through both a bifurcation analysis and numerical simulations. Based on properties of the solution and simplifying assumptions, specific numerical algorithms are developed. Numerical results are provided showing the applicability of the methods.
Some mathematical problems in a neoclassical theory of electric charges
Anatoli Babin and Alexander Figotin
2010, 27(4): 1283-1326 doi: 10.3934/dcds.2010.27.1283 +[Abstract](218) +[PDF](547.2KB)
Abstract:
We study here a number of mathematical problems related to our recently introduced neoclassical theory for electromagnetic phenomena in which charges are represented by complex valued wave functions as in the Schrödinger wave mechanics. In the non-relativistic case the dynamics of elementary charges is governed by a system of nonlinear Schrödinger equations coupled with the electromagnetic fields, and we prove that if the wave functions of charges are well separated and localized their centers converge to trajectories of the classical point charges governed by Newton's equations with the Lorentz forces. We also found exact solutions in the form of localized accelerating solitons. Our studies of a class of time multiharmonic solutions of the same field equations show that they satisfy Planck-Einstein relation and that the energy levels of the nonlinear eigenvalue problem for the hydrogen atom converge to the well-known energy levels of the linear Schrödinger operator when the free charge size is much larger than the Bohr radius.
On universal relations in 2-D turbulence
Nusret Balci, Ciprian Foias, M. S Jolly and Ricardo Rosa
2010, 27(4): 1327-1351 doi: 10.3934/dcds.2010.27.1327 +[Abstract](261) +[PDF](321.0KB)
Abstract:
A rigorous study of universal laws of 2-D turbulence is presented for time independent forcing at all length scales. Conditions for energy and enstrophy cascades are derived, both for a general force, and for one with a large gap in its spectrum. It is shown in the gap case that either a direct cascade of enstrophy or an inverse cascade of energy must hold, provided the gap modes of the velocity has a nonzero ensemble average. Partial rigorous support for 2-D analogs of Kolmogorov's 3-D dissipation law, as well as the power law for the distribution of energy are given.
On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition
Hamid Bellout, Jiří Neustupa and Patrick Penel
2010, 27(4): 1353-1373 doi: 10.3934/dcds.2010.27.1353 +[Abstract](229) +[PDF](303.9KB)
Abstract:
Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T0 > 0, V* > 0 and a unique family of strong solutions uv of the Euler or Navier-Stokes initial-boundary value problem on the time interval (0, T0), depending continuously on the viscosity coefficient $\nu$ for $0\leq\nu< $ V*. The solutions of the Navier-Stokes problem satisfy a Navier-type boundary condition. We give the information on the rate of convergence of the solutions of the Navier-Stokes problem to the solution of the Euler problem for $\nu\to 0+$.
Systems of Bellman equations to stochastic differential games with non-compact coupling
Alain Bensoussan, Jens Frehse and Jens Vogelgesang
2010, 27(4): 1375-1389 doi: 10.3934/dcds.2010.27.1375 +[Abstract](259) +[PDF](220.8KB)
Abstract:
We consider a class of non-linear partial differential systems like

-div$(a(x)\nabla u_{\nu}) +\lambda u_{\nu}=H_{\nu}(x, Du) \, $

with applications for the solution of stochastic differential games with $N$ players, where $N$ is an arbitrary but positive number. The Hamiltonian $H$ of the non-linear system satisfies a quadratic growth condition in $D u$ and contains interactions between the players in the form of non-compact coupling terms $\nabla u_{i} \cdot\nabla u_j$. A $L^{\infty}\cap H^1$-estimate and regularity results are shown, mainly in two-dimensional space. The coupling arises from cyclic non-market interaction of the control variables.

Stability of solitary-wave solutions to the Hirota-Satsuma equation
Jerry L. Bona and Didier Pilod
2010, 27(4): 1391-1413 doi: 10.3934/dcds.2010.27.1391 +[Abstract](261) +[PDF](304.6KB)
Abstract:
The evolution equation

$ u_t- $ uxxt$ +u_x-$uut$ +u_x\int_x^{+\infty}u_tdx'=0, $ (1)

was developed by Hirota and Satsuma as an approximate model for unidirectional propagation of long-crested water waves. It possesses solitary-wave solutions just as do the related Korteweg-de Vries and Benjamin-Bona-Mahony equations. Using the recently developed theory for the initial-value problem for (1) and an analysis of an associated Liapunov functional, nonlinear stability of these solitary waves is established.

A fully nonlinear equation for the flame front in a quasi-steady combustion model
Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi and Gregory I. Sivashinsky
2010, 27(4): 1415-1446 doi: 10.3934/dcds.2010.27.1415 +[Abstract](241) +[PDF](361.1KB)
Abstract:
We revisit the Near Equidiffusional Flames (NEF) model introduced by Matkowsky and Sivashinsky in 1979 and consider a simplified, quasi-steady version of it. This simplification allows, near the planar front, an explicit derivation of the front equation. The latter is a pseudodifferential fully nonlinear parabolic equation of the fourth-order. First, we study the (orbital) stability of the null solution. Second, introducing a parameter ε, we rescale both the dependent and independent variables and prove rigourously the convergence to the solution of the Kuramoto-Sivashinsky equation as ε $ \to 0$.
Numerical solution of a variational problem arising in stress analysis: The vector case
Alexandre Caboussat and Roland Glowinski
2010, 27(4): 1447-1472 doi: 10.3934/dcds.2010.27.1447 +[Abstract](234) +[PDF](1318.4KB)
Abstract:
In this article, we discuss the numerical solution of a constrained minimization problem arising from the stress analysis of elasto-plastic bodies. This minimization problem has the flavor of a generalized non-smooth eigenvalue problem, with the smallest eigenvalue corresponding to the load capacity ratio of the elastic body under consideration. An augmented Lagrangian method, together with finite element approximations, is proposed for the computation of the optimum of the non-smooth objective function, and the corresponding minimizer. The augmented Lagrangian approach allows the decoupling of some of the nonlinearities and of the differential operators. Similarly an appropriate Lagrangian functional, and associated Uzawa algorithm with projection, are introduced to treat non-smooth equality constraints. Numerical results validate the proposed methodology for various two-dimensional geometries.
Decay of solutions to a water wave model with a nonlocal viscous dispersive term
Min Chen, S. Dumont, Louis Dupaigne and Olivier Goubet
2010, 27(4): 1473-1492 doi: 10.3934/dcds.2010.27.1473 +[Abstract](187) +[PDF](363.0KB)
Abstract:
In this article, we investigate a water wave model with a nonlocal viscous term

$ u_t+u_x+\beta $uxxx$+\frac{\sqrt{\nu}}{\sqrt{\pi}}\int_0^t \frac{u_t(s)}{\sqrt{t-s}}ds+$uux$=\nu $uxx$. $

The wellposedness of the equation and the decay rate of solutions are investigated theoretically and numerically.

Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time
Vladimir V. Chepyzhov and Mark I. Vishik
2010, 27(4): 1493-1509 doi: 10.3934/dcds.2010.27.1493 +[Abstract](292) +[PDF](268.3KB)
Abstract:
We consider a non-autonomous reaction-diffusion system of two equations having in one equation a diffusion coefficient depending on time ($\delta =\delta (t)\geq 0,t\geq 0$) such that $\delta (t)\rightarrow 0$ as $t\rightarrow +\infty $. The corresponding Cauchy problem has global weak solutions, however these solutions are not necessarily unique. We also study the corresponding "limit'' autonomous system for $\delta =0.$ This reaction-diffusion system is partly dissipative. We construct the trajectory attractor A for the limit system. We prove that global weak solutions of the original non-autonomous system converge as $t\rightarrow +\infty $ to the set A in a weak sense. Consequently, A is also as the trajectory attractor of the original non-autonomous reaction-diffusions system.
A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions
Laurence Cherfils, Madalina Petcu and Morgan Pierre
2010, 27(4): 1511-1533 doi: 10.3934/dcds.2010.27.1511 +[Abstract](213) +[PDF](471.4KB)
Abstract:
We consider a finite element space semi-discretization of the Cahn-Hilliard equation with dynamic boundary conditions. We prove optimal error estimates in energy norms and weaker norms, assuming enough regularity on the solution. When the solution is less regular, we prove a convergence result in some weak topology. We also prove the stability of a fully discrete problem based on the backward Euler scheme for the time discretization. Some numerical results show the applicability of the method.
Semilinear wave equations of viscoelasticity in the minimal state framework
Monica Conti, Elsa M. Marchini and Vittorino Pata
2010, 27(4): 1535-1552 doi: 10.3934/dcds.2010.27.1535 +[Abstract](202) +[PDF](237.4KB)
Abstract:
A semilinear integrodifferential equation of hyperbolic type is studied, where the dissipation is entirely contributed by the convolution term accounting for the past history of the variable. Within a novel abstract framework, based on the notion of minimal state, the existence of a regular global attractor is proved.
Asymptotic behavior of stochastic PDEs with random coefficients
Giuseppe Da Prato and Arnaud Debussche
2010, 27(4): 1553-1570 doi: 10.3934/dcds.2010.27.1553 +[Abstract](197) +[PDF](219.6KB)
Abstract:
We study the long time behavior of the solution of a stochastic PDEs with random coefficients assuming that randomness arises in a different independent scale. We apply the obtained results to $2D$- Navier-Stokes equations.
Group foliation of equations in geophysical fluid dynamics
Jeffrey J. Early, Juha Pohjanpelto and Roger M. Samelson
2010, 27(4): 1571-1586 doi: 10.3934/dcds.2010.27.1571 +[Abstract](207) +[PDF](228.5KB)
Abstract:
The method of group foliation can be used to construct solutions to a system of partial differential equations that, as opposed to Lie's method of symmetry reduction, are not invariant under any symmetry of the equations. The classical approach is based on foliating the space of solutions into orbits of the given symmetry group action, resulting in rewriting the equations as a pair of systems, the so-called automorphic and resolvent systems, involving the differential invariants of the symmetry group, while a more modern approach utilizes a reduction process for an exterior differential system associated with the equations. In each method solutions to the reduced equations are then used to reconstruct solutions to the original equations. We present an application of the two techniques to the one-dimensional Korteweg-de Vries equation and the two-dimensional Flierl-Petviashvili (FP) equation. An exact analytical solution is found for the radial FP equation, although it does not appear to be of direct geophysical interest.
Long time behavior and attractors for energetically insulated fluid systems
Eduard Feireisl
2010, 27(4): 1587-1609 doi: 10.3934/dcds.2010.27.1587 +[Abstract](180) +[PDF](293.8KB)
Abstract:
We study the long time behavior, and, in particular, the existence of attractors for the Navier-Stokes-Fourier system under energetically insulated boundary conditions. We show that the attractor consists of static solutions determined uniquely by the total mass and energy of the fluid.
Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations
Ciprian Foias, Ricardo Rosa and Roger Temam
2010, 27(4): 1611-1631 doi: 10.3934/dcds.2010.27.1611 +[Abstract](241) +[PDF](291.0KB)
Abstract:
The three-dimensional incompressible Navier-Stokes equations are considered along with its weak global attractor, which is the smallest weakly compact set which attracts all bounded sets in the weak topology of the phase space of the system (the space of square-integrable vector fields with divergence zero and appropriate periodic or no-slip boundary conditions). A number of topological properties are obtained for certain regular parts of the weak global attractor. Essentially two regular parts are considered, namely one made of points such that all weak solutions passing through it at a given initial time are strong solutions on a neighborhood of that initial time, and one made of points such that at least one weak solution passing through it at a given initial time is a strong solution on a neighborhood of that initial time. Similar topological results are obtained for the family of all trajectories in the weak global attractor.
A model for shape memory alloys with the possibility of voids
Michel Frémond and Elisabetta Rocca
2010, 27(4): 1633-1659 doi: 10.3934/dcds.2010.27.1633 +[Abstract](187) +[PDF](361.4KB)
Abstract:
The paper is devoted to the study of a mathematical model for the thermomechanical evolution of metallic shape memory alloys. The main novelty of our approach consists in the fact that we include the possibility for these materials to exhibit voids during the phase change process. Indeed, in the engineering paper [60] has been recently proved that voids may appear when the mixture is produced by the aggregations of powder. Hence, the composition of the mixture varies (under either thermal or mechanical actions) in this way: the martensites and the austenite transform into one another whereas the voids volume fraction evolves. The first goal of this contribution is hence to state a PDE system capturing all these modelling aspects in order then to establish the well-posedness of the associated initial-boundary value problem.

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