Abstract : When a plane shock hits a wedge head on, it experiences a reflection diffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. In particular, the C^{1,1}-regularity is optimal for the solution across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock where the wedge has large-angle. Also, one can obtain the C^{2,\alpha} regularity of the solution up to the pseudo-sonic circle in the pseudo-subsonic region. Reference : Myoungjean Bae, Gui-Qiang Chen, and Mikhail Feldman. "Regularity of solutions to regular shock reflection for potential flow." (2008) Gui-Qiang Chen and Mikhail Feldman. "Global Solutions of Shock Reflection by Large-Angle Wedges for Potential Flow"

Abstract: In this talk, we discuss the global-in-time existence of strong solutions to the one-dimensional compressible Navier-Stokes system. Classical results establish only local-in-time existence under the assumption that the initial data are smooth and the initial density remains uniformly positive. These results can be extended to global-in-time existence using the relative entropy and Bresch-Desjardins entropy under the same hypotheses. This approach allows for possibly different end states and degenerate viscosity. Reference: A. Mellet and A. Vasseur, Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. SIAM J. Math. Anal., 39(4):1344–1365, 2007/08.

We prove global well-posedness and scattering for the massive Dirac-Klein-Gordon system with small and low regularity initial data in dimension two, under non-resonance condition. We introduce new resolutions spaces which act as an effective replacement of the normal form transformation.

We discuss the fine gradient regularity of nonlinear kinetic Fokker-Planck equations in divergence form. In particular, we present gradient pointwise estimates in terms of a Riesz potential of the right-hand side, which leads to the gradient regularity results under borderline assumptions on the right-hand side. The talk is based on a joint work with Ho-Sik Lee (Bielefeld) and Simon Nowak (Bielefeld).

In this talk, we will discuss some global regularity results for weak solutions to fractional Laplacian type equations. In particular, the operator under consideration involves a weight function satisfying appropriate ellipticity conditions. Under suitable assumptions on the weight function and the right hand side, we show some sharp global regularity results for the function u/d^s in the sense of Lebesgue, Sobolev and H¨older, where d(x) = dist(x, ∂Ω) is the distance to the boundary function. This talk is based on a joint work with S.-S. Byun and K. Kim.

In this talk, we consider the dispersion-managed nonlinear Schrödinger equation (DM NLS), which naturally arises in modeling of fiber-optic communication systems with periodically varying dispersion profiles. We discuss the well-posedness of the DM NLS and the threshold phenomenon related to the existence of minimizers for its ground states.

Abstract: In this talk, we consider the Navier-Stokes-Poisson (NSP) system which describes the dynamics of positive ions in a collision-dominated plasma. The NSP system admits a one-parameter family of smooth traveling waves, known as shock profiles. I will present my research on the stability of the shock profiles. Our analysis is based on the pointwise semigroup method, a spectral approach. We first establish spectral stability. Based on this, we obtain pointwise bounds on the Green's function for the associated linearized problem, which yield linear and nonlinear asymptotic orbital stability.

Abstract:The logistic diffusive model provides the population distribution of a species according to time under a fixed open domain in R^n, a dispersal rate, and a given resource distribution. In this talk, we discuss the solution of the model and its equilibrium. First, we show the existence, uniqueness, and regularity results of the solution and the equilibrium. Then, we investigate two contrasting behaviors of the equilibrium with respect to the dispersal rate by applying two methods for each case: sub-super solution method and asymptotic expansion. Finally, we introduce an optimizing problem of a total population of the equilibrium with respect to resource distribution and prove a significant property of an optimal control called bang-bang. References: [1] Cantrell, R.S., Cosner, C. Spatial ecology via reaction-diffusion equation. Wiley series in mathematical and computational biology, John Wiley & Sons Ltd (2003) [2] I. Mazari, G. Nadin, Y. Privat, Optimization of the total population size for logistic diffusive equations: Bang-bang property and fragmentation rate, Communications in Partial Differential Equation 47 (4) (Dec 2021) 797-828