Welcome to the research page of Margaret Beck
For her home page, click here.
Research Interests
My primary interest is determining the nonlinear stability and large-time behavior of solutions to
dissipative PDEs, such as reaction-diffusion equations and viscous conservation laws. This includes studying
nonlinear waves such as traveling waves and spatially and/or temporally periodic patterns. I typically view these PDEs as
infinite-dimensional dynamical systems, and I analyze them using a variety of mathematical techniques,
for example invariant manifolds, similarity variables, geometric singular
perturbation theory, exponential dichotomies, and pointwise estimates.
In much of my work, the main mathematical difficulty arises from the fact
that the linear operator lacks a so-called spectral gap. For example, the spectrum could be as in the above figure on the left,
where a zero
eigenvalue is embedded in the continuous spectrum. As a result, one cannot use standard spectral decomposition methods to separate
the solutions that decay from those that simply remain bounded. Furthermore,
estimates necessary for the related nonlinear analysis can become extremely delicate, as can the properties of associated bifurcations.
One particular topic I've been interested in recently is known as "metastability."
Roughly speaking, this refers to long transients in the dynamics. For example,
solutions could spend large periods of time near unstable states before
settling down to their stable, asymptotic limit. This is interesting
mathematically, because there are far fewer techniques available for analyzing
transient behaviors than there are for analyzing asymptotic behaviors. Also,
in real world systems, the transient time-scale may be so long that one will
never actually see the limiting behavior. Thus, metastability is important in
applications, as well. The type of behavior arises, for example, in the
two-dimensional Navier-Stokes equation.
The above picture on the right is of me in the Catalan town of Vic. It doesn't really have anything to do with my research. However
the statue, known as "Merma," is said to come alive during the festival of de Patron and chase the children while carrying a lash.
Occaisionally, while working on my research, I feel as if Merma is chasing me around in circles.
"Rigorous justification of Taylor dispersion via center manifolds
and hypocoercivity," with O. Chaudhary and
C. E. Wayne. Submitted. arXiv
"Grassmannian flows and applications to nonlinear partial differential
equations," with A. Doikou, S.J.A. Malham, and
I. Stylianidis. Submitted.
arXiv
"Selection of quasi-stationary states in the Navier-Stokes equation on
the torus," with E. Cooper and K. Spiliopoulos. Submitted. arXiv
"Partial differential systems with nonlocal nonlinearities: Generation
and solutions," with A. Doikou, S.J.A. Malham, and
I. Stylianidis. Philosophical Transactions A, 376, no 2117 (2018). arXiv
"Isolas versus snaking of localized rolls," with T. Aougab, P. Carter,
S. Desai, B. Sandstede, M. Stadt, and A. Wheeler. Journal of
Dynamics and Differential Equations (2017), https://doi.org/10.1007/s10884-017-9624-0. [.pdf]
"Stability of nonlinear waves: pointwise estimates." Discrete and
Continuous Dynamical Systems Series S, 10, 191-211 (2017). Part of a special
issue associated with the 2015 Bremen winter school and symposium
entitled "Diffusion on fractals and non-linear dynamics". [.pdf]
"Analysis of enhanced diffusion in Taylor dispersion via a model problem,"
with
O. Chaudhary and
C. E. Wayne. "Hamiltonian
PDEs and Applications," Fields Institute Communications, 75, 31-71
(2015) . arXiv
"Computing the Maslov Index for large systems," with S.J.A. Malham. Proceedings of the
AMS 143, no. 5, 2159-2173 (2015). arXiv
"Superadiabaticity in Reaction Waves as a Mechanism for Energy
Concentration," with S. G. Mahajan, J. T. Abrahamson, S. Birkhimer,
E. Friedman, Q. H. Wang, and M. S. Strano.
Energy Environ. Sci., 7, 3391-3402 (2014)
[Journal]
"Nonlinear stability of source defects in the complex Ginzburg-Landau equation," with T. Nguyen, B.
Sandstede and K.
Zumbrun. Nonlinearity, 27, 739-786 (2014). [.pdf]
"Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations," with C. E. Wayne. Proc. Roy. Soc. Edinburgh Sect. A., 143, 905-927 (2013). arXiv
"Toward nonlinear stability of sources via a modified Burgers equation,''
with T. Nguyen, B.
Sandstede and K.
Zumbrun. Physica D, 241, 382-392 (2012). arXiv
"Using global invariant manifolds to understand metastability in Burgers equation with small viscosity,"
with C. E.
Wayne. SIAM Rev. 53, no. 1, 129-153 (2011). [.pdf]
"Stability of traveling wave solutions for coupled surface and grain
boundary motion," with
Z. Pan and B. Wetton. Phys. D 239, 1730-1740 (2010). [.pdf]
"Nonlinear stability of time-periodic viscous shocks,'' with
B.
Sandstede and K.
Zumbrun. Arch. Ration. Mech. Anal. 196, 1011-1076 (2010). arXiv
"Nonlinear stability of semi-discrete shocks for two sided
schemes," with H. J. Hupkes,
B. Sandstede, and K. Zumbrun.
SIAM J. Math. Anal. 42, no. 2, 857-903 (2010). [.pdf]
"Using global invariant manifolds to understand metastability in Burgers equation with small viscosity,"
with C. E.
Wayne . SIAM J. Appl. Dyn. Syst. 8, no. 3, 1043-1065 (2009). arXiv
"Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities," with A. Ghazaryan, and B. Sandstede. J. Differential Equations 246, 4371-4390 (2009).
[.pdf]
"Invariant manifolds and the stability of
traveling waves in scalar viscous conservation laws," with C. E. Wayne
. J. Differential Equations 244, 87-116 (2008).
[.pdf]
"A geometric construction of
traveling waves in a bioremediation model," with A. Doelman
and
T. J. Kaper . J. Nonlinear Sci.
16, 329-349 (2006).
[.pdf]
"A geometric theory of chaotic phase synchronization," with K. Josic . Chaos 13, 247-258 (2003).
[.pdf]
Lecture Notes, Thesis, Slides for Talks, Poster Presentations, and Miscellany
"Vortices and the Navier-Stokes equation: understanding solutions of
equations that we can't actually solve": Hour-long public lecture, given as
part of the Third Bremen
Winter School and Symposium, in March 2015. [.pdf]
There were embedded videos in this talk that do not play in the above .pdf
version. Links to two of them can be found here:
"Nonlinear stability of coherent structures via pointwise estimates":
Presenatation for an hour-long talk. [.pdf]
"Rapid convergence to quasi-stationary states in the 2D Navier-Stokes
equation": Presentation for an hour-long talk. [.pdf]
"Nonlinear stability of semi-discrete shocks for two sided schemes":
Presentation for a half-hour long talk. [.pdf]
"Nonlinear convective stability of travelling fronts near Turing
and Hopf instabilities": Presentation for an hour long talk. [.pdf]
"Using global invariant manifolds to understand metastability in Burgers equation with small viscosity":
Persentation for an hour long talk. [.pdf]
"Nonlinear stability of time-periodic viscous shocks":
Presentation for an hour long talk.
[.pdf]
"Snakes, ladders, and isolas of localised patterns":
This talk was presented at the SIAM student conference at the University of Oxford, April 25, 2008.
[.pdf]
Poster presentations
"Nonlinear stability of time-periodic viscous shocks": This poster was presented at the conference
"Geometric Analysis, Elasticity and PDE" at Heriot-Watt University, June 23-27, 2008.
[.pdf]
"A geometric analysis of traveling waves in a bioremediation model": This poster was presented at the
conferences "SIAM conference on Application of Dynamical Systems," in
Snowbird, UT, May 22-26, 2005, and "Connections for Women: Dynamical Systems" at MSRI, January 18-19, 2007.
[.pdf]
Miscellany
Wrote review of the book "Multiple Time Scale Dynamics" by Christian Kuehn for SIAM Review (2017).
Watch a video
of the talk I gave in the Topology Seminar at the University of Edinburgh on
November 14, 2013, on the potential use of the Maslov Index when studying
stability in PDEs.
Entry on "Burgers equation" in the Encyclopedia of
Applied and Computational Mathematics, Springer (print version 2015 -
published online in 2013).
Oberwolfach report for the workshop I spoke
at entitled "Dynamics of
Patterns," organized by Wolf-Jürgen Beyn, Bernold Fiedler, and
Björn Sandstede, in December 2012. Published in the European Mathematical
Society, Volume 9, Issue 4 (2012), pg 3585-3588.