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, and exponential dichotomies.
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.
Publications and Preprints
"Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations," with C. E. Wayne. Accepted for publication at Proc. Roy. Soc. Edinburgh Sect. A. [.pdf]
"Toward nonlinear stability of sources via a modified Burgers equation,''
with T. Nguyen , B.
Sandstede and K.
Zumbrun. Physica D, 241, 382-392 (2012). [.pdf]
"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). [.pdf]
"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). [.pdf]
"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]
"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]
Slides for talks
"Rapid convergence to quasi-stationary states in the 2D Navier-Stokes equation": Presentation for a 50 minute talk at the IMA. [.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]
Miscellany
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.
