Ohio State University Partial Differential Equations Seminar
Year 2017-2018
Time/Location: Tuesdays 1:00 - 2:00pm / MW 154 (unless otherwise noted) |
Schedule of talks:
|
| TIME |
SPEAKER |
TITLE |
HOST |
August 29
| No seminar
() |
|
|
September 5
| Eiji Yanagida
(Tokyo Institute of Technology) |
Dynamics of hot spots in the logarithmic diffusion equation |
Lam/Lou |
September 12
| No seminar
() |
|
|
September 19
| Leo Girardin
(Paris VI) |
Non-cooperative Fisher-KPP systems:
traveling waves and long-time behavior |
Lam |
September 26
| No seminar
() |
|
|
October 3
| No seminar
() |
|
|
October 10
| No seminar
() |
|
|
October 17
| No seminar
() |
|
|
October 19
11:15-12:15
CH240
| Carey Caginalp
(U. Pittsburgh) |
Conservation Laws, Shocks, and Random Initial Conditions |
Keyfitz |
October 24
| No seminar
() |
|
|
October 31
| No seminar
() |
|
|
November 9
11:15-12:15
CH240
| Yu Yuan
(U. Washington) |
Asymptotic behavior of solutions to Hessian equations over exterior domains |
Guan |
November 14
1-2pm / MW154
| Chris Henderson
(Chicago) |
A local-in-time Harnack inequality and applications to reaction-diffusion equations |
Lam/Lou |
November 21
10-11am / MW154
| Yanhui Zhang
(Beijing Tech and Business) |
Integral Representations of a Class of Harmonic Functions and the Asymptotic Behavior of Fractional Laplacians in the Half Space |
Guan |
November 28
| Gael Raoul
(Ecole Polytechnique) |
A PDE model to investigate the impact of climate change on a
population |
Lam/Lou |
December 5
| No seminar
() |
|
|
December 12
| No seminar
() |
|
|
December 19
| No seminar
() |
|
|
December 26
| No seminar
() |
|
|
January 2
|
No Seminar
() |
|
|
January 9
|
No Seminar
() |
|
|
January 16
|
No Seminar
() |
|
|
January 23
|
No Seminar
() |
|
|
January 30
|
No Seminar
() |
|
|
February 6
|
No Seminar
() |
|
|
February 13
|
No Seminar
() |
|
|
February 20
|
No Seminar
() |
|
|
February 27
|
No Seminar
() |
|
|
March 6
|
No Seminar
() |
|
|
March 13
|
No Seminar
() |
|
|
March 20
4:30-5:30pm
MA105 |
Robin Young
(U. Mass. Amherst) |
Weak* Solutions of Hyperbolic Conservation Laws |
Keyfitz |
March 27
|
No Seminar
() |
|
|
April 3
|
No Seminar
() |
|
|
April 10
|
Qiliang Wu
(Ohio Univ.) |
|
Lam |
April 17
|
Shuangjian Zhang
(Toronto) |
On concavity of the principal’s profit maximization facing agents who respond nonliearly to prices |
Guan |
April 24
|
No Seminar
() |
|
|
Abstracts
Eiji Yanagida (Sep 5, 2017)
Title: Dynamics of hot spots in the logarithmic diffusion equation
Abstract: This talk is concerned with the behavior of positive solutions to the logarithmic diffusion
equation. It is known that due to fast diffusion of the equation, the extinction of solutions
may occur in finite time. I will discuss the behavior of hot spots (maximum points) of
solutions near the extinction time in the one-dimensional case. By applying the intersection
number principle, it is shown that the hot spot typically converges to a point or goes to
spatial infinity. However, in some cases, the hot spot remains bounded but does not
converge to any particular point.
Leo Girardin, Paris VI (Sep 19, 2017)
Title: Non-cooperative Fisher-KPP systems:
traveling waves and long-time behavior
Abstract: I will present recent results about a family of reactionâÂÂdiffusion
systems whose prototype is the LotkaâÂÂVolterra competitive system with
diffusion and mutations. These systems are nonlinear and
non-cooperative, which makes their study difficult; the idea is then
to notice and to use the underlying KPP structure.
Carey Caginalp (Oct 19, 2017)
Title: Conservation Laws, Shocks, and Random Initial Conditions
Abstract: Conservation laws have a wide range of applications to problems in fluid mechanics, turbulence, kinetic theory and many others. As perhaps the simplest quasilinear PDE, Burgers' equation has an important role as a prototype or test case for analysis for qualitative properties. I will discuss two related probabilistic approaches for which one can model the evolution of shocks through objects known as n-point functions, generalized analogs of a probability density for the solution at various points. In the second approach, I will illustrate how this method leads to solutions that persist through shock interactions, without need for any sort of resetting procedure, which is required for many methods.
Yu Yuan (Nov 9, 2017)
Title: Asymptotic behavior of solutions to Hessian equations over exterior domains
Abstract: We present a unified approach to quadratic asymptote of solutions to
a class of fully nonlinear elliptic equations over exterior domains, including
Monge-Ampere equations (previously known), special Lagrangian equations, quadratic Hessian equations, and inverse harmonic Hessian equations. The argument is based on the consequence of our "exterior" Evans-Krylov--an exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian--and also certain rotation arguments developed earlier toward Hessian bound. This is joint work with Dongsheng Li and Zhisu Li.
Yu Yuan (Nov 9, 2017)
Title: Asymptotic behavior of solutions to Hessian equations over exterior domains
Abstract: We present a unified approach to quadratic asymptote of solutions to
a class of fully nonlinear elliptic equations over exterior domains, including
Monge-Ampere equations (previously known), special Lagrangian equations, quadratic Hessian equations, and inverse harmonic Hessian equations. The argument is based on the consequence of our "exterior" Evans-Krylov--an exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian--and also certain rotation arguments developed earlier toward Hessian bound. This is joint work with Dongsheng Li and Zhisu Li.
Chris Henderson (Nov 14, 2017)
Title: A local-in-time Harnack inequality and applications to reaction-diffusion equations
Abstract: The classical Harnack inequality requires one to look back in time to relate the suprema and infima of a solution to a parabolic equation. In this talk, I will introduce a Harnack-type inequality that allows us to remove this looking-back-in-time restriction at the expense of a slightly weaker bound. I will then discuss applications of this bound to (time permitting) three non-local reaction-diffusion equations arising in biology and combustion. In particular, in each case, this inequality allows us to show that solutions to these equations, which do not enjoy a maximum principle, may be compared with solutions to a related local equation, which does enjoy a maximum principle. Precise estimates of the propagation speed follow from this.
Yanhui Zhang, Beijing Technology and Business University
(Nov 21, 2017)
Title: Integral Representations of a Class of Harmonic Functions and the Asymptotic Behavior of Fractional Laplacians in the Half Space
Abstract: In this article, motivated by the classic Hadamard factorization theorem about an entire function of finite order in the complex plane, we firstly prove that a harmonic function whose positive part satisfies some growth conditions, can be represented by its integral on the boundary of the half space. By using Nevanlinnaâs representation of harmonic functions and the modified Poisson kernel of the half space, we further prove a representation formula through integration against a certain measure on the boundary hyperplane for harmonic functions not necessarily continuous on the boundary hyperplane whose positive parts satisfy weaker growing conditions than the first question. The result is further generalized by involving a parameter m dealing with the singularity at the infinity. Moreover, we will give the asymptotic behaviors of fractional Laplacians in the half space of Rn and discuss the weighted boundary limits of modified fractional Laplacians. The asymptotic behaviors hold outside an explicitly defined exceptional set, whose size can be controlled and set to be as small as required.
Gael Raoul, Ecole Polytecnhique
(Nov 28, 2017)
Title: A PDE model to investigate the impact of climate change
on a
population
Abstract: We consider a population structured by a phenotypic trait and
a spatial variable. In this talk we will only consider
asexual
populations. The model is then a parabolic equation with a non-local
competition term. We will relate the dynamics of the solutions to the
properties
of the linearized equation. For simple environments, this
connection will provide explicit formula for the survival and
propagation speed of the population.
The dynamics of populations in more complicated environments is
important for biological applications: can a natural reserve help a
population to survive?
will a mountain stop an invasive species? etc. We
will explain how the methods described above could provide an
interesting insight for such problems.
Robin Yong (U Mass, Amherst)
(Mar 20, 2018)
Title: Weak* Solutions of Hyperbolic Conservation Laws
Abstract:We introduce a new notion of solution, which we call weak* solutions,
for systems of conservation laws. These solutions can be used to
handle singular situations that standard weak solutions cannot, such
as vacuums in Lagrangian gas dynamics or cavities in elasticity. Our
framework allows us to treat the systems as ODEs in Banach space.
Rather than the usual notion of weak solution, we refer to the
integral form of the equations, and interpret the corresponding
integrals in appropriate Banach spaces, via the Gelfand integral. We
develop the necessary calculus and describe stability properties that
are built in to our definition of solutions. Our approach treats test
functions implicitly, which in turn simplifies calculations. A
critical step is the reinterpretation of nonlinear fluxes and
energies. We present several examples and discuss extension to three
dimensions, where we expect to address vexing issues of ill-posedness. This is joint work with Alexey Miroshnikov (UCLA).
Qiliang Wu (Ohio U)
(April 10, 2018)
Title:
Abstract:
Shuangjian Zhang (Toronto)
(April 17, 2018)
Title: On concavity of the principal’s profit maximization facing agents who respond nonliearly to prices
Abstract: A monopolist wishes to maximize her profits by finding an optimal price policy. After she announces a menu of products and prices, each agent will choose to buy that product which maximizes his own utility, if positive. The principal's profits are the sum of the net earnings produced by each product sold. These are determined by the costs of production and the distribution of products sold, which in turn are based on the distribution of anonymous agents and the choices they make in response to the principal's price menu. In this talk, we describe a necessary and sufficient condition for the convexity or concavity of the principal's problem, assuming each agent's disutility is a strictly increasing but not necessarily affine (i.e. quasilinear) function of the price paid. Concavity when present, makes the problem more amenable to computational and theoretical analysis; it is key to obtaining uniqueness and stability results for the principal's strategy in particular. Even in the quasilinear case, our analysis goes beyond previous work by addressing convexity as well as concavity, by establishing conditions which are not only sufficient but necessary, and by requiring fewer hypotheses on the agents' preferences. This talk represents joint work with my supervisor Robert McCann.
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