In a Sea of Solutions, a Search for Uniqueness
Jiuyi Zhu, assistant professor at the LSU Department of Mathematics, recently received a $166K NSF EPSCoR Research Infrastructure Improvement grant for his work on uniqueness and quantitative uniqueness of solutions to partial differential equations. The goal of the grant is to enhance his research capacity, strengthen LSU’s research program, and benefit undergraduate and graduate education:

Jiuyi Zhu makes frequent visits to the University of Chicago.
“My research is about uniqueness and quantitative uniqueness of solutions to partial
differential equations. Math is everywhere. All phenomena in our world can be described
in math terms. Partial differential equations can be used to describe and investigate
numerous real-life problems, such as sound, light, heat, electrodynamics, and quantum
mechanics. There can be so many solutions to these equations. Hence, it’s impossible
for us to solve all of them. But if we can prove some qualitative and quantitative
properties such as uniqueness and quantitative uniqueness, then we simplify the problem.
If we can prove that the solution is unique, it simplifies the model significantly.
“So, what is quantitative uniqueness? It’s a way of quantifying the rate of vanishing
of solutions. We want to know how a solution decays to zero. If a solution to some
equation vanishes in infinite order, then the solution is trivial. But if it isn’t
trivial, we want to quantify how fast the solution vanishes. Quantitative uniqueness
is a developing area of research related to many problems in mathematics.”
“I do pure mathematics. There are many beautiful and important equations that uncover
the mystery of nature and our world. Whereas applied research has immediate implications,
what I’m doing is fundamental research. This is not like engineering or computer science
where you can apply your research results to industry and get benefits directly and
more or less immediately. Math is ahead of that. By working on partial differential
equations, we try to understand how things work, understand nature more.
“For this project, I chose to collaborate with a world-renowned leader in the study
of partial differential equations at the University of Chicago, Professor Carlos Kenig.
He’s also the leading expert on uniqueness and quantitative uniqueness.
“Math is a vast subject nowadays. In our math department here at LSU, we have around
52 faculty. Each is working in his or her own area. Even mathematicians don’t necessarily
understand the research other mathematicians are doing. Through the support of this
grant, however, I look forward to integrating my research into my classroom teaching
and increasing our research capacity overall.”
Elsa Hahne
LSU Office of Research & Economic Development
225-578-4774
ehahne@lsu.edu