Recent
Publications
· Y.
Frenkel and V. Roytburd, Traveling solitary waves
for Maxwell-Duffing media: Computation via simulated annealing, submitted to Physics
Lettr A (2008) (preprint)
· M. Frankel and V. Roytburd, Dissipative Dynamics for a Class of Nonlinear Pseudo-Differential Equations, to appear, Journal of Evolution Equations (2008) (pdf text)
For the evolution in spaces of periodic functions with zero mean we demonstrate existence of a universal absorbing set and compact attractor. Furthermore, we show that the attractor is of a finite Hausdorff dimension. The dissipation mechanism for the class of equations studied in the paper is akin to the nonlinear saturation in the Kuramoto-Sivashinsky equation. A similar generalization of the Kuramoto-Sivashinsky equation was studied by Nicolaenko et al. under the assumption of a purely quadratic nonlinearity and reflection invariance of both: the equation and solutions.
· M. Frankel and V. Roytburd, Dynamics of thermally-insulated nonequilibrium Stefan problem, Journal of Evolution Equations 7 (2007), 317-345 (pdf text)
We study a two-phase
Stefan problem with kinetics. Our results obtained earlier rely on an
artificial damping represented by the heat losses. The evidence from DNS,
however, suggests that this additional damping is unnecessary. Here we prove
existence of a finite-dimensional attractor for the problem without heat
losses. We use a more elegant technique of energetic type estimates in
appropriately defined weighted Sobolev spaces as opposed to the parabolic
potentials for the earlier work. We demonstrate existence of compact attractors
in the Sobolev spaces and prove that the attractor consists of sufficiently
regular functions. This allows us to show that the Hausdorff dimension of the
attractor is finite.
· M. Frankel and V. Roytburd, Stability for a class of nonlinear pseudo-differential equations, Appl. Math. Letters 21 (2008), 425-430. (pdf text)
We study a class of
nonlinear evolutionary equations generated by a pseudo-differential operator
with the elliptic principal symbol and with nonlinearities of the form G(ux) where cη2≤G(η)≤Cη2
for large |η|. We demonstrate existence
of a universal absorbing set, and a compact attractor, and show that the
attractor is of a finite Hausdorff dimension. The stabilization mechanism is
similar to the nonlinear saturation well known for the Kuramoto–Sivashinsky
equation.
· M. Frankel and V. Roytburd, Dynamics
of SHS in periodic media, Nonlinear Analysis, Theory, Methods &
Applications 63 (2005), pp. e1507-e1515. (Complete text in pdf format)
·
C.-M. Brauner, M. Frankel, J.
Hulschoff, and V. Roytburd, Stability and attractors for quasi-steady
evolution of cellular flames, Interfaces and Free Boundaries 8
(2006), 301-316. (Complete text in pdf
format)
We continue to study a simple integro-differential equation: the Quasi-Steady equation (QS) of flame front dynamics. This equation is dynamically similar to the Kuramoto-Sivashinsky (KS) equation. We demonstrate that QS possesses a universal absorbing set, and a compact attractor. Furthermore we show that the attractor is of a finite Hausdorff dimension, and give an estimate on it. We discuss relationship with the Kuramoto-Sivashinsky and Burgers-Sivashinsky equations.
· V. Roytburd, Working Analysis by Jeffery
Cooper, a book review, SIAM
· M. Frankel and V. Roytburd, Finite-Dimensional Attractor for a Nonequilibrium Stefan Problem with Heat Losses, Discrete Cont. Dynamical Syst. 13 (2005) 35-62 (Text in pdf)
We study a two-phase modified Stefan problem modeling
solid combustion and nonequilibrium phase transition. The problem is known to
exhibit a variety of non-trivial dynamical scenarios. We develop a priori
estimates and establish well-posedness of the problem in weighted spaces of
continuous functions. The estimates secure sufficient decay of solutions that
allows for an analysis in Hilbert spaces. We demonstrate existence of compact
attractors in the weighted spaces and prove that the attractor consists of
sufficiently regular functions. This allows us to show that the Hausdorff
dimension of the attractor is finite.
· M. Frankel and V. Roytburd, Dynamical structure of one-phase model of solid combustion, Discrete and Continuous Dynamical Systems, Suppl. (2005), pp.287-296. (Complete text in pdf format)
· M. Frankel and V. Roytburd, Frequency Locking for Combustion Synthesis in Periodic Medium, Physics Lettr A 329 (2004) 68-75 complete text
Solutions of a 1-D
free-interface problem modeling solid combustion front propagating in
combustible mixture with periodically varying concentration of reactant exhibit
classical phenomenon of mode locking. Numerical simulation shows a variety of
locked periodic, quasi-periodic and chaotic solutions.
· Y. Deng, R. Kersting, V. Roytburd, J. Xu, R. Ascazubi, K. Liu, S. L. Rumyantsev, X.-C. Zhang, and M. S. Shur, Spectrum Determination of Terahertz Sources Using Fabry-Perot Interferometer and Bolometer Detector, International Journal of Infrared and Millimeter Waves, Vol. 25, No. 2, (2004) 215-226.
· M. Frankel and V. Roytburd, On Attractors
for a Sharp-Interface Model of Exothermic Phase Transitions, Advances Mathematical Sci. Applications 14
(2004) 25-40. (Complete
text in pdf format)
We study a free interface problem related to combustion of
condensed matter and some non-equilibrium exothermic phase transitions. In
spite of a variety of non-trivial dynamical scenarios exhibited by the model
the solutions are uniformly bounded and the interface velocity is a smooth
function. The main result of the paper establishes existence of a compact
connected attractor for the classical solutions of the problem. Numerical
evidence leads to the conjecture that the fractal dimension of the attractor is
finite.
· M. Frankel and V. Roytburd,
Finite-dimensional Attractor for a 1-Phase Stefan Problem with Kinetics, J.
Dynamics Diff. Equations (2003) ( Complete text
in pdf format)
For a one-phase free-boundary problem with kinetics, which is known
to generate a rich dynamics, we study evolution of the infinitesimal volume along
the trajectories in the attractor. We demonstrate that for sufficiently large m
that is defined solely by the properties of the kinetic function the
m-dimensional volume decays exponentially. This property combined with the
uniform differentiability of the semigroup leads to the conclusion that the
Hausdorff dimension of the attractor is finite.
· M. Frankel and V. Roytburd,
Low fractal dimension of attractors for a one-phase nonequilibrium Stefan
problem, Proc. Fourth Internat. Conf. Dynamical Syst. Diff. Equations (2002)
281-287 (Complete
text in pdf format)
· M. Frankel and V. Roytburd,
Compact Attractors for a Stefan Problem with Kinetics, Electron. J. Diff.
Eqns. (2002) No 15 pp. 1-27. ( Complete text in pdf format)
For a one-phase free-boundary problem with kinetics, a proof of
existence of unique bounded classical solutions for continuous initial
conditions is presented. The main result of the paper establishes existence of
a compact attractor for the classical solutions of the problem.
· M. Frankel and V. Roytburd,
Finite-Dimensional Attractors for a Free-Boundary Problem with Kinetic
Condition, Appl. Math. Lettr. 15 (2002)
83-87. ( Complete text in pdf format)
For a one-phase free-boundary problem with kinetics, a proof of
existence of a compact attractor of finite Hausdorff dimension is outlined.
· M. Frankel, G. Kovacic, V.
Roytburd, and I. Timofeyev, Finite-dimensional dynamical system modeling
thermal instabilities, Physica D 137 (2000), 295-315. ( Complete text in pdf format)
We describe a three-dimensional dynamical system, which is obtained
as a pseudo-spectral approximation to a free boundary problem modeling solid
combustion and rapid solidification, and is capable of generating its major
dynamical patterns. These patterns include a Hopf bifurcation followed by a
sequence of secondary period doubling and a transition to chaos, reverse
sequences, and sequences followed by Shilnikov type trajectories. A
computer-assisted bifurcation analysis uncovers some novel mechanisms of
stability exchange. The most striking of them is an infinite period bifurcation
which resembles the classical Shilnikov bifurcation, but instead of a
funnel-shaped spiral along which the period is continually increasing, the
continuation produced a series of isolas. Each isola is a closed branch of
solutions of roughly the same period, and with the same number of oscillations.
The isolas corresponding to consecutive numbers of low amplitude oscillations
about the equilibrium are adjacent to each other, and appear to accumulate on
saddle-focus homoclinic connection of Shilnikov type.
· M. Frankel and V. Roytburd,
On dynamics of exothermic interfaces, Contemporary Math. 255 (2000),
Proceedings of the 1998 AMS-IMS-SIAM Joint Summer Research Conference on
Nonlinear PDEs.
We investigate dynamical behavior of a version of the Stefan free
boundary problem, with kinetic and surface tension effects. This free boundary
model is related to a rapid crystallization of amorphous films and to the
self-propagating high-temperature synthesis (solid combustion). The unifying
feature of these diverse physical phenomena is the existence of a uniformly
propagating wave of phase transition whose stability is controlled by the
balance between the energy production at the interface and the energy dissipation
into the medium. We show that in the radially symmetric case, the problem in
the whole space possesses a bounded classical solution, globally in time. We
present numerical illustrations of complex thermokinetic oscillations exhibited
by the solutions.
· M. Frankel, L. K. Gross and V.
Roytburd, Thermo-kinetically controlled pattern selection, Interfaces
and Free Boundaries 2 (2000) 313-330. ( Complete text in pdf format)
Through a combination of asymptotic and numerical approaches we
investigate bifurcation and pattern formation for a free boundary model related
to a rapid crystallization of amorphous films and to the self-propagating
high-temperature synthesis (solid combustion). The unifying feature of these
diverse physical phenomena is the existence of a uniformly propagating wave of
phase transition whose stability is controlled by the balance between the
energy production at the interface and the energy dissipation into the medium.
For the propagation on a two-dimensional strip with thermally insulated edges,
we develop a multi-scale weakly-nonlinear analysis that results in a system of
ordinary differential equations for the slowly varying amplitudes. The
information derived from the amplitude system for different values of kinetic
parameters is utilized for predicting the evolving patterns. The pattern
selection is confirmed by direct numerical simulations on the free boundary
problem. Some numerical results on strongly nonlinear regimes are also
presented.
Last
updated 06/20/2008
