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The Mathematics of Climate Seminar will meet every Tuesday (staring September 11) at 11:15a.m. in 305 Lind Hall. The first few lectures will offer an introduction to the subject. The level of mathematical sophistication will gradually increase until we are examining current research by the middle of the semester.
Climate modeling is a rapidly growing area. The mathematics involved includes ordinary and partial differential equations, dynamical systems, stochastic processes, statistics, and numerical methods. Even if you have little or no interest in climate, you might be interested to see some applications of the mathematics.
The first lecture will be "The Scientific Case for Anthropogenic Warming," by Richard McGehee. The atmosphere is informal, and we usually continue the discussions over lunch after the seminar. All are welcome to attend.
The seminars are held as part of the 2012-2013 IMA Annual Thematic Program on Infinite Dimensional and Stochastic Dynamical Systems and their Applications. Audience members are strongly encouraged to ask questions or to make comments/criticism. Please note that seminars will be held every Thursday, but will not be held during IMA workshops.
A natural first impression is that the dynamics in infinite dimensions should be more complicated than the dynamics in finite dimensions. This is true in some respects, but at the same time the infinite dimension can potentially simplify the ``macroscopic" behavior, in a way similar to the simplifications conjectured by the ergodic hypothesis for many classical systems of Statistical Mechanics. Ergodicity-type assumptions are notoriously difficult to prove or disprove (and we have nothing new to say in this direction) , but there are many other interesting questions, some of which we will address. The talk will be based on joint works with Nathan Glatt-Holtz and Vlad Vicol, and with Geordie Richards and Ofer Zeitouni.
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n linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, we first summarize key results on the exponential separation and related concepts, the principal
Floquet bundle and principal Lyapunov exponent. Then we show how they can
be effectively used in studies of some nonlinear parabolic problems on
$R^N$. In particular, we shall discuss the instability of localized
solutions, and a Liouville-type theorem for radial solutions.
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The purpose of this talk is to demonstrate how a variational approach may be used to obtain many of the known results of scalar reaction-diffusion equations. It turns out that traveling wave solutions of such equations correspond to minima of the functional. By solving the Euler-Lagrange equation, one can find, or estimate, the minimum wave speed, which is the same as the asymptotic speed of propagation of solutions of the reaction-diffusion equations. Convergence to traveling wave solutions will also be considered. It is unclear if this method can be used to handle the multi-dimensional case, but it should be able to handle reaction-diffusion systems, where stacked-fronts may occur. This is an ongoing joint work with Professor Hirokazu Ninomiya from Meiji University, Japan.
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Control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. In this talk, we shall establish analytical results on the existence and stability of steady states for a general class of pump-leak models. The key to these results is that pump-leak models possess a natural thermodynamic structure. We shall then use this thermodynamic structure as a guiding principle to obtain a spatial generalization of pump leak models - a PDE system that describes three-dimensional electrodiffusion and osmosis in cellular systems.
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In this talk, we will present an overview of recent theoretical, numerical and experimental work concerning the static, stability, bifurcation and dynamic properties of coherent structures that can emerge in one-and higher-dimensional settings within Bose-Einstein condensates at the coldest temperatures in the universe (i.e., at the nanoKelvin scale). We will discuss how this ultracold quantum mechanical setting can be approximated at a mean-field level by a deterministic PDE of the nonlinear Schrodinger type and what the fundamental nonlinear waves of the latter are, such as dark solitons and vortices. Then, we will try to go to a further layer of simplified description via nonlinear ODEs encompassing the dynamics of the waves within the traps that confine them, and the interactions between them. Finally, we will attempt to compare the analytical and numerical implementation of these reduced descriptions to recent experimental results and speculate towards a number of interesting future directions within this field
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A realistic molecular-level description of reaction-diffusion processes is often provided by spatially-discrete stochastic interacting particle systems (IPS). Spatially discreteness may be innate to the system (e.g., for reactions on 2D crystalline catalyst surfaces), or may just be a convenient modeling tool (e.g., for reactions in 1D nanoporous materials). Spatially homogeneous and heterogeneous state in these IPS can be described exactly by hierarchical master equations. These reduce to lattice differential equations, i.e., to discrete reaction-diffusion equations (RDE), after making a hierarchical truncation approximation, and the discrete RDE become continuum RDE in the hydrodynamic limit of rapid diffusion. However, the correct description of diffusion is invariably non-trivial (with concentration-dependent and tensorial diffusion coefficients), and not captured by simple approximations. Examples are provided for simple models for reaction fronts in bistable reactions such as CO-oxidation on 2D surfaces, and for catalytic conversion inside 1D linear nanopores.
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This talk is concerned with a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles. The first main result is on the classification of solutions; every solution falls into one of the three categories, namely, area expanding, area bounded and area shrinking types. We then study in detail the asymptotic behavior of solutions in each category. Among other things we show that solutions are asymptotically self-similar both in the area expanding and the area shriknking cases, while solutions converge to either a stationary solution or a traveling wave in the area bounded case. Thus the renormalized curve converges to some profile in each of the three cases, but the proof of the convergence is totally different among the three cases. This is joint work with Jong-Shenq Guo, Masahiko Shimojo and Chang-Hong Wu.
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Abstract: This lecture will focus on the legacy of Lyapunov in the area of
Dichotomies. While the use of the Lyapunov-Perron theory of exponential dichotomies in the study of ODEs and PDEs is well understood, there is a potential for more insight to be gained by combining the Lyapunov-Perron theory with the related recent work of others in the area of Lyapunov Exponents and the
Multiplicative Ergodic Theorem. As we will show, there is much to be gained
on problems where both theories are applicable.
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Abstract: Modeling problems in environmental and physical
sciences naturally leads to consider equations with
discontinuous coefficients In this talk I will survey
results obtained with colleagues at Oregon State
University on the large scale effect on quantities
of natural interest by the interface conditions required
at the location of the discontinuities of the coefficients
in the model. Examples that motivate this study
include breakthrough curves in hydrology, occupation
time in ecology and costal upwelling in oceanography.
The analysis of these problems illustrates and builds on
the profound connections between stochastic processes
and PDE's.
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Abstract: Optimal filtering problems are important in signal processing and related fields. Most of the time the optimal filter does not admit a closed-form expression, and various numerical methods are developed to solve the filtering problem. Among these methods, particle filter is a widely used tool for numerical prediction of complex systems when observation data are available. In this talk I will first give a brief introduction on particle filtering and present error analysis on the numerical formulation of particle filter. Then I will introduce an implicit filtering method, along with its convergence results.
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Abstract: We study a sixth order convective Cahn-Hilliard type equation that describes the faceting of a growing surface. It is considered with periodic boundary conditions. We deal with the problem in one and two dimensions. We establish the existence and uniquness of weak solutions. Our goal is study the long time behavior in the one- and two-dimensional cases. We show existence of a global attractor. The numerical simulations suggest that this result is optimal.
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Abstract: In this talk we analyze the dynamics of flows generated by a class of dissipa-
tive semilinear parabolic problems when some parameters of the equation vary in a
topological space. We establish abstract results and apply them to partial differen-
tial equations with nonlinear boundary conditions when (i) the domain of definition
of the solutions vary with respect to the action of diffeomorphisms, and (ii) when
some reaction and potential terms of the equation are concentrating in a narrow
strip of a portion of the boundary of the domain of the solutions. Our main goal
is to discuss the continuity of the nonlinear semigroup, as well as, the upper and
lower semicontinuity of the family of attractors.
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Abstract: In this talk I will present some results
concerning the integral $\int_a^b f(t) dY(t)$
and the differential equation: $dx(t)=f(x(t)) dY(t)$,
where $Y(t)$ is a finite dimensional Holder continuous functions.
We will also expand the solution $x(t)$ in term of
Volterra series of $Y(t)$. When $Y(t)$ is a fractional Brownian
motion, we also expand $X(t)$ as multiple Wiener-Ito integrals
of $Y$ (orthogonal expansion).
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I have been do research (1) in mathematical neuroscience
(rigorous mathematical analysis of existence and stability of
traveling wave solutions)
and (2) in nonlinear systems of fluid dynamical equations
(exact limits and decay estimates with sharp rates of $L^{2}$-norms of
global strong solutions, global weak solutions of Cauchy problems). In this lecture, I will talk about my recent results.
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Inside the analysis of equations of hydrodynamic, statistical solutions
have been investigated.
In fact, the individual solutions may give a detailed and too complicated
picture of the fluid,
while one could be interested in the behavior of some global quantity
related to the fluid,
where the microscopic picture is replaced by the macroscopic one. This is
the statistical approach to turbulence.
From the mathematical point of view, we are interested in distributions
invariant for these flows. Gaussian measures
of Gibbsian type are associated with some shell models of 3D turbulence
and 2D turbulence (GOY and SABRA models).
We prove the existence of a unique global flow for a stochastic viscous
shell model with the property that these
Gibbs measures are invariant for these flows.
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We show the existence of a unique stochastic flow of Sobolev diffeomorphisms for
stochastic differential equations (SDEs) with bounded measurable drift
coefficients. This result is counter-intuitive: The dominant 'culture' in
dynamical systems is that the flow `inherits' its spatial regularity from the
driving vector fields. Spatial regularity of the stochastic flow yields
existence and uniqueness of a Sobolev differentiable weak solution of the
stochastic transport equation with singular coefficients (cf. work by Kunita
(1990); and Flandoli-Gubinelli-Priola (2010)). The corresponding deterministic
transport equation does not in general have a solution (Ambrosio (2004)). If
time permits, we will construct a Sobolev differentiable stochastic flow of
diffeomorphisms for one-dimensional SDEs driven by bounded measurable diffusion
coefficients. No uniqueness of solutions to the SDE is presumed!
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For a general time-dependent linear competitive-cooperative tridiagonal system of differential equations, we obtain canonical Floquet invariant bundles which are exponentially separated in the framework of skew-product flows. The obtained Floquet theory is applied to study the dynamics on the hyperbolic omega-limit sets for the nonlinear competitive-cooperative tridiagonal systems in time-recurrent structures including almost periodicity and almost automorphy.
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In 1958, Pekka Myrberg was the first to discover that as a parameter c is varied in the iterated quadratic map, periodic orbits occur with the progression of periods k, 2k, 4k, 8k, ... for a large variety of k values. Subsequently, this phenomenon of period-doubling cascades has been seen in a large variety of parameter-dependent dynamical systems, be they iterated maps, ordinary differential equations, partial differential equations, delay differential equations, or even experimental observations. It has often been observed that cascades occur in a dynamical system as the system transitions from ordered to chaotic. In this talk, I will be presenting some recent results based on work of Alligood, Mallet-Paret, and Yorke to explain the link between chaos and cascades.
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We consider a nonlinear Stefan problem, which may be used
to describe the spreading of invasive species, with the free boundary
representing the invasing front. In one space dimension and in the
radially symmetric case with a logistic nonlinear term, it is known that
this model exhibits a spreading-vanishing dichotomy. In this talk we
discuss the non-radially symmetric case. By establishing suitable
regularity of the free boundary, we show that the spreading-vanishing
dichotomy still holds. Moreover, when spreading happens, the normalized
free boundary approaches the unit sphere as time goes to infinity, and the
spreading speed is the same as in the radially symmetric case. This is
joint work with Hiroshi Matano (Univ of Tokyo) and Kelei Wang (Wuhan Inst
of Physics and Math).
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In this talk I will present Hamiltonian identities for elliptic
PDEs and systems of PDEs. I will then show some interesting
applications of these identities to problems related to solutions of some
nonlinear elliptic equations in the entire space. In particular, I will
use the identities to show symmetry results for solutions of stationary
Schrodinger equations and stationary and traveling wave solutions of
the Allen-Cahn equation.
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We discuss properties of a class of dissipative dynamical systems
which display rather special long time dynamics. This class is
referred to as quasi-stable systems. It turns out that this is quite
large class of systems naturally occurs in nonlinear PDE models
arising in wave dynamics, plasma physics, thermoelasticity of plates
and gas/fluid-structure interaction models. The interest in this class
of systems stems from the fact that quasi-stability inequality almost
automatically implies number of desirable properties such as
asymptotic smoothness, finite dimensionality of attractors, regularity
of attractors, exponential attraction, etc. The notion of
quasi-stability is rather natural from the point of view of long-time
behavior. It pertains to decomposition of the flow into exponentially
stable and compact part. This represents some sort of analogy with the
"splitting" method, however, the decomposition refers to the
difference of two trajectories, rather than a single trajectory.
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Abstract:
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We will be offering a series of graduate-level lectures on the Dynamics of Evolutionary Equations during the fall 2012 semester.
These lectures are intended to meet the interest and needs of students at the graduate level at the University of Minnesota, as well as the related interests of the IMA postdoctoral fellows and other IMA visitors. In particular, the lectures are intended to serve as a companion for the 2012-2013 program thematic.
Because of this connection with the IMA program, the lectures for Math 8571 will pause during the weeks of the major IMA workshops. The first of these workshops is scheduled for the week of September 17-21, 2012.
The main reference for these lectures is the book, Dynamics of Evolutionary Equations, by George R. Sell and Yuncheng You. Copies of this book are available through both the Mathematics and the IMA Libraries.
Topics to be covered include: basic theory of semiflows, including local and global attractors; linear semigroups with applications to PDEs, with special emphasis on analytic semigroups and nonautonomous PDEs; basic theory of evolutionary equations, including exponential dichotomies and robustness theories; well-posed problems for nonlinear equations, including mild solutions and strong solutions.
In the last four to five weeks, we will focus on the dynamics of the Navier-Stokes equations, with applications to the flow fluids. An addendum to this syllabus will be prepared at that time.
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