# Mini workshop on PDEs

## 22nd - 23rd September 2017, Department of Mathematics, Inha University

#### The purpose of this workshop is to bring together young researchers in PDEs from Suwon and Incheon so that they can report their recent progress, exchange ideas and enjoy fruitful discussions.
This workshop is sponsored by the BRL(Basic Research Lab) grant from Institute of Applied Mathematics at Department of Mathematics, Inha University. Graduate students and postdocs are particularly encouraged to participate in.
Please send an email to ypchoi at inha.ac.kr if you need further information.

## Invited speakers:

Sun-Ho Choi(Kyunghee University, Suwon), sunhochoi@khu.ac.kr

Woocheol Choi(Incheon National University, Incheon), choiwc@inu.ac.kr

Young-Pil Choi(Inha University, Incheon), ypchoi@inha.ac.kr

Jihoon Ok(Kyunghee University, Suwon), jihoonok@khu.ac.kr

Jinmyoung Seok(Kyonggi University, Suwon), jmseok@kgu.ac.kr

Seok-Bae Yun(Sungkyunkwan University, Suwon), sbyun01@skku.edu

## Talk title/abstract and Schedule:

#### September 22, 2017 (15:00-16:00, 5E102)

#### Partial Hölder regularity for elliptic systems with non-standard growth

Jihoon Ok

Department of Applied Mathematics, Kyunghee University

** Abstract:** In this talk, we discuss about partial Hölder regularity for elliptic systems with non-standard growth. We mainly consider systems with Orlicz growth and, moreover, discontinuous coefficient factor, for which we prove that their weak solutions are Hölder continuous for every Hölder exponents except a Lebesgue measure zero set. In addition, we also present similar results for systems with double phase growth.

#### September 22, 2017 (16:00-17:00, 5E102)

#### A review on the optimal control problems related to the Maxwell equation

Woocheol Choi

Deparment of Mathematics Education, Incheon National University

** Abstract:** In this talk, we review a few optimal control problmes which arise in situations involving the electromagnetic effect.

#### September 22, 2017 (17:00-18:00, 5E102)

#### Quantum mean field limit and nonlinear PDEs of Schrödinger or Hartree type

Jinmyoung Seok

Deparment of Mathematics, Kyonggi University

** Abstract:** Through a mean field limit, the dynamics of a system consisting of a huge number of relativistic bosons, such as a boson star, is described by the following mean field equation
given by
\begin{equation}\label{pseudo-relativistic NLS}
i\partial_t\psi=\sqrt{-c^2\Delta + m^2 c^4}\psi - (V*|\psi|^2)\psi,
\end{equation}
where $V$ is a two-body interaction potential. If we choose $V$ as the Newtonian potential, we obtain the nonlinear Hartree equation, also so-called boson star equation.
If we choose $V$ as the Dirac-delta distribution, the pseudo-relativistic nonlinear Schrödinger equation is derived.

In this talk, I will discuss about existence and qualitative properties of ground states to such pseudo-relativistic mean field equations.
We are particularly interested in the non-relativistic limit, which takes the speed of light $c$ to infinite so that non-relativistic physics is recovered.
We shall explicitly calculate the order and rate of convergence of the non-relativistic limit between ground states to the pseudo-relativistic and non-relativistic mean field equations.

#### September 23, 2017 (10:00-11:00, 5E102)

#### A dichotomy in the dissipation estimates for the polatomic BGK model

Seok-Bae Yun

Deparment of Mathematics, Sungkyunkwan University

** Abstract:** In this talk, we consider a dichotomy observed in the dissipation estimates for the polyatomic
BGK model. By dissipation estimate, we mean either the entropy-entropy production
estimate of the nonlinear polyatomic BGK model, or the coercive estimate of the linearized
polyatomic relaxation operator. In the former case, we observe a jump in the coefficient and
the target equilibrium state as a relaxation parameter tends to zero. In the latter case, we
show that the coefficient and the degeneracy of the coercive estimate see a sudden jump as the
same relaxation parameter reaches zero. We also discuss how these two results are related.

#### September 23, 2017 (11:00-12:00, 5E102)

#### Allee effect and discontinuity reaction

Sun-Ho Choi

Department of Applied Mathematics, Kyunghee University

** Abstract:** We study a stationary reaction diffusion equation with discontinuous reaction function.
Motivation of the discontinuous reaction function is the strong Allee effect in the ecology. The
Allee effect means that there is a critical threshold such that if initial population density is less
than this critical threshold, then this species will extinct. We construct a radial symmetric solution
for the three dimensional stationary solution to the reaction diffusion equation with discontinuous
corresponding to the critical threshold. In order to obtain the stationary solution, we employ a
change of variable to obtain equivalent initial value problem of a nonlinear ordinary differential
equation with parameter. For each parameter, there is one solution for the system. However,
most of all solutions are blow-up if variable tend to a given parameter. Our aim is to prove the
existence of a value among parameters such that corresponding bounded solution. The existence
of this solution implies that the existence of radial symmetric solution for the three dimensional
stationary solution to the reaction diffusion equation with discontinuous.

#### September 23, 2017 (12:00-13:00, 5E102)

#### On the analysis of mathematical modelling of sprays

Young-Pil Choi

Department of Mathematics and Institute of Applied Mathematics, Inha University

** Abstract:** Sprays are complex flows which are constituted of dispersed particles such as droplets, dust, etc, in an underlying gas. The interactions between particles and fluid have received a bulk of attention due to a number of their applications in the field of, for example, biotechnology, medicine, and in the study of sedimentation phenomenon, compressibility of droplets of the spray, cooling tower plumes, and diesel engines, etc. In this talk, we present coupled kinetic-fluid equations.
The proposed equations consist of Vlasov-Fokker-Planck equation with local alignment forces and the incompressible Navier-Stokes equations.
For the equations, we establish the global existence of weak solutions, hydrodynamic limit, and large-time behavior of solutions.
We also remark on blow-up of classical solutions in the whole space.

## Organizers:

Young-Pil Choi (Inha University), Jinmyoung Seok(Kyonggi University), and Seok-Bae Yun(Sungkyunkwan University)