University of Missouri
Department of Mathematics

Differential Equations Seminar
Spring Semester 2008

Every Thursday from January 24 to May 8, 2008 (excludes University Holidays)
1:00-1:50 p.m., 312 Mathematical Sciences Building (except as noted)

Schedule of Talks

Janaury 31
Stephen Montgomery-Smith (UMC, Math)
Numerical Solutions of PDE on the Sphere Using Spherical Harmonics

February 7
Alin Pogan (UMC, Math)
The Dichotomy Theorem for ill-posed equations and applications to infinite dimensional Evans function
Abstract: We prove that the operator $G$, the closure of the first-order differential operator $-d/dt + D(t)$ on $L_2(R,X)$, is Fredholm if and only if the ill-posed equation $u'(t) = D(t)u(t)$, $t\in\RR$, has exponential dichotomies on $\RR_+$ and $\RR-$ and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of $G$. Here $X$ is a Hilbert space, $D(t) = A + B(t)$, $A$ is the generator of a bi-semigroup, $B(\cdot)$ is a bounded piecewise strongly continuous operator valued function. Also, we prove some perturbations results and consider various examples of ill-posed problems. The applications to the study of the
infinite dimensional Evans function are discussed.

February 12—1:00-1:50 p.m., 208 Strickland (GCB)
Hongqiu Chen (University of Memphis)
Sharp Results of Well-posedness
Abstract: Consider the initial-value problem

where u=u(x,t) is a real-valued function, L is a Fourier multiplier operator with real symbol , that is , and g is a smooth, real-valued function of a real variable. Equations of this form arise as models of wave propagation in a variety of physical contexts. Here, fundamental issues of local and global well-posedness are established for L_p , H^s and bore-like or kink-like initial data. In the special case where wherein r > 1 and , the initial value problem (1) is locally well-posed in H^s if r and s satisfy one of the following three conditions:

In addition, if r > 1 and , then the well-posedness is global.

February 28
Stamatis Dostoglou (UMC, Math)
Towards A Mathematical Definition of Turbulent Eddies

March 6
Zhiwu Lin (UMC, Math)
Unstable Solitary Water Waves and Wave Breaking

March 13
Charles Li (UMC, Math)
The Poincaré recurrence theorem

March 20
Konstantin Makarov (UMC, Math)
The Generalized Weyl Commutation Relations and Unitarily Affine Invariant Operators
Abstract: We introduce a concept of unitarily affine invariant operators associated with an arbitrary group G of affine transformations of the real axis and its unitary representation U in a Hilbert space. If G is a continuous, one-parameter group, we show that the unitary representation U and the one V generated by a self-adjoint unitarily affine invariant operator satisfy generalized commutation relations and establish an analog of the Stone-von Neumann uniqueness result in this case. We also give a complete classification of the pairs of unitarily affine invariant operators (Å, A) where $\stackrel{<em>\AA </em>}{}$is a symmetric operator with deficiency indices (1,1) and A is its self-adjoint extension.

April 3
Yuri Latushkin (UMC, Math)
The Dichotomy Theorem and the Evans Function I
Abstract: I will give am elementary introduction to these two topics. The Dichotomy Theorem gives necessary and sufficient conditions for an abstract operator d/dx+A(x) to be Fredholm. The Evans function is a tool to detect the point spectrum of this operator.

April 10
Dongho Chae (University of Chicago and Sungkyunkwan University)
On the blow-up problem and new a priori estimates for the 3D Euler and the Navier-Stokes equations
Abstract: In this talk we discuss blow-up rates and the blow-up profiles of possible asymptotically selfsimilar singularities of the 3D Euler equations, where the sense of convergence and selfsimilarity are considered in various sense. We extend much further, in particular, the previous nonexistence results of self-similar/ asymptotically self-similar singularities obtained in [1, 2]. Some implications the notions for the 3D Navier-Stokes equations are also deduced. Generalization of the selfsimilar transforms is also considered, and by appropriate choice of the transform we obtain new a priori estimates for the 3D Euler and the Navier-Stokes equations.
References
[1] D. Chae, Nonexistence of self-similar singularities for the 3D incompressible Euler equations, Comm. Math. Phys., 273, no. 1, (2007), pp. 203-215.
[2] D. Chae, Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations, Math. Ann., 338, no. 2, (2007), pp. 435-449.

April 17
Ciprian Gal (UMC, Math)
Long time behavior of a parabolic-hyperbolic system in the theory of phase transitions
Abstract: We intend to discuss the global long time behavior (i.e. attractors) of a parabolic-hyperbolic system arising in the field of phase transitions for the Cahn-Hilliard equation. This system models the relative concentration u of a binary system governed by an hyperbolic equation (characterized by the presence of inertia and viscosity) and the (relative) temperature v which is governed by a reaction-diffusion equation suitably coupled with the equation for u. We shall also mention results about convergence to equilibria.

April 24
Mike Heitzman (UMC, Math)
Some general theory of hyperbolic conservation laws and a problem in gas dynamics

May 1
Cancelled

May 8
Stephen Schecter (North Carolina State University- Raleigh)
Stability of fronts in gasless combustion
Abstract: For gasless combustion in one space dimension, we show that the physical combustion front has a type of nonlinear stability, in a space of perturbations that are bounded behind the front and decrease exponentially ahead of the front, provided the linearized operator has no eigenvalues in the right half-plane besides a simple zero eigenvalue. Simplicity of the zero eigenvalue is shown using the Evans function. Factors that complicate the analysis are: (1) the linearized operator is not sectorial, and (2) in a space in which the linearized operator has good spectral properties, the nonlinear term is not small. I will explain why the result makes good physical sense.
This is joint work with Aparecido de Souza (Universidade Federal de Campina Grande), Anna Ghazaryan (University of North Carolina), and Yuri Latushkin (University of Missouri).

This page was last updated Monday, April 28, 2008