Ohio State University Partial Differential Equations Seminar
Year 2015-2016
Time/Location: Tuesdays 4:10 - 5:10pm / MA 105 (unless otherwise noted) |
Schedule of talks:
|
| TIME |
SPEAKER |
TITLE |
HOST |
| September 8
| Adrian Lam
(Ohio State University) |
An integro-differential equation from evolution of random dispersal. |
|
September 15
| No seminar
() |
|
|
September 22
| No seminar
() |
|
|
September 29
| John Holmes
(Ohio State) |
Minimum regularity solutions for a Burgers type equation. |
|
October 6
|
Tong Li
(Iowa) |
Global entropy solutions to a quasilinear hyperbolic system modeling blood flow |
Lam |
October 13
|
Yang Yang
(Purdue) |
Coupled Physics Inverse Problems in Photo-acoustic and Thermo-acoustic Tomography |
|
October 20
|
Katarina Jegdi
(U. Houston-Downtown) |
Semi-hyperbolic patches for the unsteady transonic small disturbance equation |
Keyfitz |
October 27
|
No Seminar
() |
|
|
November 3
|
Xiaoqiang Zhao
(Mem. Univ. of New Foundland) |
Propagation Phenomena for A Reaction-Advection-Diffusion Competition
Model in A Periodic Habitat |
Lam |
November 10
|
Gerard Misiolek
(Notre Dame) |
Continuity properties of the solution map of Euler equations in Holder spaces |
Tiglay |
November 17
11:30-12:30, CH240 |
Eun Heui Kim
(California State, Long Beach) |
Transonic problems in multidimensional conservation laws |
Keyfitz/Hao |
November 24
Joint with Analysis
1:50-2:50, CH240 |
Paul Eloe
(Dayton) |
Comparisons of Green's functions for families of boundary value problems for higher order differential equations |
|
December 1
Joint with Analysis
4:10-5:10, CH240 |
Alex Iosevich
(Rochester) |
Tiling and exponential bases in ${\Bbb Z}_p \times {\Bbb Z}_p$. |
Taylor |
December 8
|
No Seminar
() |
|
|
December 15
|
No Seminar
() |
|
|
December 22
|
No Seminar
() |
|
|
December 29
|
No Seminar
() |
|
|
January 5
|
No Seminar
() |
|
|
January 12
|
No seminar
() |
|
|
January 19
|
No Seminar
() |
|
|
January 26
|
No Seminar
() |
|
|
February 2
|
Xiao-Biao Lin
(North Carolina State) |
Spatial Dynamics and Concatenated Wave Patterns Consisting of Non-intersecting Traveling Waves |
Keyfitz/T.-H. Hsu |
February 9
|
Adrian Lam
(Ohio State) |
Nash Equilibria and Concentration in Reaction-Diffusion Equations: A Hamilton-Jacobi Approach |
|
February 16
|
John Holmes
(Ohio State) |
A note on the Fornberg-Whitham equation |
|
February 23
Joint with Analysis |
Peter Takac
(Universität Rostock) |
On Compact Support Solutions to Parabolic Problems with
the p-Laplacian for p>2 and Their "Counterparts" for p<2. |
Jan Lang |
March 1
|
No Seminar
() |
|
|
March 8
|
No Seminar
() |
|
|
March 15
|
No Seminar
() |
|
|
March 22
With Appl Math
1:50-2:50pm, CH240 |
Glenn Webb
(Vanderbilt) |
Mathematical Analysis of a Clonal Evolution Model of Tumor Cell Proliferation |
Adrian Lam |
March 29
|
No Seminar
() |
|
|
April 5
With Appl Math
1:50-2:50pm, CH240 |
Horst Thieme
(Arizona State) |
Persistence and minimal habitat size of sexually reproducing populations
|
Adrian Lam |
April 12
|
No seminar
() |
|
|
April 19
|
Javier Gomez-Serrano
(Princeton) |
|
Tanveer |
April 26
2-3pm, CH240 |
Tianran Zhang
(Southwest Univ., China) |
Minimal wave speed for a class of non-cooperative reaction-diffusion systems |
Lam |
Abstracts
Adrian Lam, Ohio State University (Sep 8, 2015)
Title: An integro-differential equation from evolution of random dispersal
We consider a mutation-selection model of a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We focus on the asymptotic profile of positive steady state solutions. Our result shows that in the limit of small mutation rate, the solution remains regular in the spatial variables and yet concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. [A. Hastings, Theor. Pop. Biol., 24, 244-251, 1983] and [Dockery et al., J. Math. Biol., 37, 61-83, 1998] showed that for two competing species in spatially heterogeneous but temporally constant environment, the slower diffuser always prevails, if all other things are held equal.
Our result suggests that their findings may hold for arbitrarily many traits. This is joint work with Y. Lou (Ohio State and Renmin Univ.).
John Holmes, Ohio State University (Sep 29, 2015)
Title: Minimum regularity solutions for a Burgers type equation.
We study the Cauchy problem in the non-periodic case for a nonlocal perturbation of Burgers equation. In particular, well posedness (existence, uniqueness*, and continuous dependence) for classical solutions (solutions taking values in $C([0,T]; C^1)$) for a nonlocal perturbation of Burgers equation having $k+1$-degree nonlinearities is shown. This result shows that the Camassa-Holm, Degasperi-Precesi and the Novikov equations are well posed in the aforementioned space, which improves upon earlier results in Sobolev and Besov spaces.
Tong Li, University of Iowa (Oct 6, 2015)
Title: Global entropy solutions to a quasilinear hyperbolic system modeling blood flow
This talk is concerned with an initial-boundary value problem on bounded domains for a one dimensional quasilinear hyperbolic model of blood flow with viscous damping. It is shown that Lâ?? entropy weak solutions exist globally in time when the initial data are large, rough and contains vacuum states. Furthermore, based on entropy principle and the theory of divergence measure field, it is shown that any Lâ?? entropy weak solution converges to a constant equilibrium state exponentially fast as time goes to infinity. The physiological relevance of the theoretical results obtained in this paper is demonstrated. This is a joint work with Kun Zhao.
Yang Yang, Purdue University (Oct 13, 2015)
Title: Coupled Physics Inverse Problems in Photo-acoustic and Thermo-acoustic Tomography
Photo-acoustic tomography (PAT) and Thermo-acoustic tomography (TAT) are newly developed hybrid imaging modalities. These modalities are able to generate images with both high resolution and high contrast by coupling radiation of different frequencies and acoustic waves. In this talk we will formulate the inverse problems and discuss our recent progress in PAT and TAT. These results are based on the previous joint works with J. Chen and P. Stefanov.
Katarina Jegdic, University of Houston - Downtown (Oct 20, 2015)
Title: Semi-hyperbolic patches for the unsteady transonic small disturbance equation
We consider a two-dimensional Riemann problem for the unsteady transonic small disturbance equation resulting in diverging rarefaction waves. We write the problem in self-similar coordinates and we obtain a mixed type (hyperbolic-elliptic) system. Resolving the one-dimensional discontinuities in the far field, where the system is hyperbolic, and using characteristics, we formulate the problem in a semi-hyperbolic patch that is between the hyperbolic and the elliptic regions. A semi-hyperbolic patch is known as a region where one family out of two nonlinear families of characteristics starts on a sonic curve and ends on a transonic shock. We obtain existence of a smooth local solution in this semi-hyperbolic patch and we prove various properties of global smooth solutions based on a characteristic decomposition using directional derivatives. This is joint work with I. Jegdic.
Xiaoqiang Zhao, Memorial University of Newfoundland (Nov 3, 2015)
Title: Propagation Phenomena for A Reaction-Advection-Diffusion Competition
Model in A Periodic Habitat
In this talk, I will report our recent research on a reaction-advection-diffusion
competition model in a periodic habitat. We first investigate the global
attractivity of a semi-trivial steady state (i.e., the competitive exclusion)
for the periodic initial value problem. Then we establish the existence of the
rightward spreading speed and its coincidence with the minimal wave speed for
spatially periodic rightward traveling waves. Further, we obtain a set of sufficient
conditions for the rightward spreading speed to be linearly determinate. Finally,
we apply the obtained results to a prototypical reaction-diffusion model. Our
method involves monotone semiflows, principal eigenvalues, lower and upper
solutions. We also extend this work to the time and space periodic case.
Gerard Misiolek, University of Notre Dame (Nov 10, 2015)
Title: Continuity properties of the solution map of Euler equations in Holder spaces
The study of the incompressible Euler equations in Holder spaces goes back to the work of Gyunter, Lichtenstein and Wolibner. Although many refined existence and uniqueness results have been obtained since, continuity properties of the associated data-to-solution map received less attention. We will describe a simple counterexample to continuity in C^{1,\alpha} for any 0<\alpha<1 and argue that a natural setting for wellposendess in the sense of Hadamard is the separable subspace c^{1,\alpha}.
(Joint work with Tsuyoshi Yoneda).
Eun Heui Kim, Cal State Long Beach (Nov 17, 2015)
Title: Transonic problems in multidimensional conservation laws
We discuss the recent progress in multidimensional conservation laws. In particular we consider self-similar two dimensional Riemann problems where the problems are transonic, meaning, hyperbolic in far field and mixed type near the origin. We present existence and numerical results on a specific system, the nonlinear wave system, for certain configurations.
Eun Heui Kim, Cal State Long Beach (Nov 17, 2015)
Title: Transonic problems in multidimensional conservation laws
We discuss the recent progress in multidimensional conservation laws. In particular we consider self-similar two dimensional Riemann problems where the problems are transonic, meaning, hyperbolic in far field and mixed type near the origin. We present existence and numerical results on a specific system, the nonlinear wave system, for certain configurations.
Xiao-Biao Lin, North Carolina State (Feb 2, 2016)
Title: Spatial Dynamics and Concatenated Wave Patterns Consisting of Non-intersecting Traveling Waves
We consider a reaction-diffusion equation in one space dimension whose
initial condition is approximately a sequence of widely separated traveling waves
with increasing velocity, each of which is individually asymptotically stable. We
show that the sequence of traveling waves is itself asymptotically stable: as t goes to infinity,
the solution approaches the concatenated wave pattern, with different shifts of each
wave allowed.
Essentially the same result was proved by Beyn, Selle, Th\"ummler, and
Wright. They regarded the concatenated wave pattern as a sum
of traveling waves. On the contrary, we regard the pattern as a sequence of
traveling waves restricted to subintervals of R and separated by
small jump discontinuities.
Adrian Lam, Ohio State (Feb 9, 2016)
Title: Nash Equilibria and Concentration in Reaction-Diffusion Equations: A Hamilton-Jacobi Approach
We consider an integro-PDE model for a population structured by
the spatial variables and a trait variable affecting the dispersal coefficients. Competition
for resource is local in spatial variables, but nonlocal in the trait variable.
We focus on the asymptotic profile of positive steady state solutions. Our result
shows that in the limit of small mutation rate, the solution remains regular
in the spatial variables and yet concentrates in the trait variable and forms Dirac concetrations (i) at one boundary point; (ii) the interior; or (iii) at both boundary points.
The main techniques are the perturbed test function approach, a Liouville result on a cylinder, and elliptic DeGiorgi-Nash-Moser estimates for the obligue derivative problem.
Finally, connections to notions and concepts in evolutionary game theory will be discussed. This is joint work with Wenrui Hao (MBI) and Yuan Lou(Ohio State).
John Holmes, Ohio State (Feb 16, 2016)
Title: A note on the Fornberg-Whitham equation
The Fornberg-Whitham equation may be the simplest nonlinear PDE which admits peaked traveling wave solutions. Derived in 1978 as a model for breaking waves in shallow water, this equation was meant to be a competitor of the Korteweg-de Vries (KdV) equation. We show that this equation is well-posed in periodic Sobolev spaces $H^s$ when $s>3/2$. We also show that the data-to-solution map is not better than continuous.
Glenn Webb, Vanderbilt (Mar 22, 2016)
Title: Mathematical Analysis of a Clonal Evolution Model of Tumor Cell Proliferation
An analysis is given for a partial differential equation model of a cancer cell population. The model structures the population with respect to cell age and cell telomere length. A continuous telomere length structure is assumed, which corresponds to the clonal model of tumor cell growth. This assumption leads to a model with a non-standard non-local boundary condition. The global existence and qualitative behavior of solutions are investigated. An analysis is made of the effect of telomere restoration on cancer cell dynamics. The results indicate that without telomere restoration, the cell population typically extinguishes. With telomere restoration, exponential growth is observed in the linear model. The effect of crowding induced mortality on the qualitative behavior of solutions is investigated in a nonlinear version of the model. Numerical simulations are given for various examples of the models.
Horst Thieme, Arizona State (Apr 5, 2016)
Title: Persistence and minimal habitat size of sexually reproducing populations
Whether a structured population dies out or persists is
typically determined by the spectral radius of the first order
approximation of the turnover map at the zero vector (the extinction state).
If mating is ignored, this first order approximation is a positive
bounded linear operator; it mating is included, it is a homogeneous
order-preserving map on the cone.
We introduce the cone spectral radius of bounded homogeneous maps
and give various characterizations. We discuss conditions
for the existence of positive eigenvectors and eigenfunctionals
that are associated with the spectral radius. Their existence
helps to establish the persistence of the populations that are
considered.
Applications are presented for diffusing populations with a
very short mating and reproduction season,
and the minimal domain size for
population persistence is considered.
Tianran Zhang, Southwest Univ. China (Apr 26, 2016)
Title: Minimal wave speed for a class of non-cooperative reaction-diffusion systems
In this talk, I consider a class of non-cooperative reaction–diffusion systems, which include predator–prey models and SI disease-transmission models. The concepts of weak traveling wave solutions and weak minimal wave speed are proposed to study the population invasion or spread of infectious diseases. The necessary and sufficient conditions for the existence of such solutions are obtained by the Schauder’s fixed-point theorem and persistence theory. The LaSalle’s invariance principle is applied to show that weak traveling wave solutions connect two equilibria. The nonexistence of traveling wave solutions is proved by introducing a negative one-sided Laplace transform. The above results are applied to a predator-prey and a disease-transmission model with specific interaction functions. We find that the profile of traveling wave solutions may depend on different eigenvalues according to the corresponding condition, which is a new phenomenon.
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