# Mini workshop on PDEs

## Invited speakers:

Sun-Ho Choi(Kyunghee University, Suwon)
Woocheol Choi(Incheon National University, Incheon)
Young-Pil Choi(Inha University, Incheon)
Jihoon Ok(Kyunghee University, Suwon)
Jinmyoung Seok(Kyonggi University, Suwon)
Seok-Bae Yun(Sungkyunkwan University, Suwon)

## Talk title/abstract and Schedule:

#### 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.

#### 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.

#### 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 $$\label{pseudo-relativistic NLS} i\partial_t\psi=\sqrt{-c^2\Delta + m^2 c^4}\psi - (V*|\psi|^2)\psi,$$ 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.

#### 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.

#### 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.

#### 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)