The singular limit problem is an important issue in various forms of ODEs and PDEs, and it is particularly known as a fundamental problem in equations derived from fluid dynamics. In this presentation, I will introduce some general phenomena of the singular limit problem through several examples. Subsequently, I will examine how the solution of the Euler-Maxwell equations converges to the MHD equations under the assumption that the speed of light approaches infinity, and how the Boussinesq equations converge to the QG equations in certain regimes.

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.

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"

In this note, we investigate threshold conditions for global well-posedness and finite-time blow-up of solutions to the focusing cubic nonlinear Klein–Gordon equation (NLKG) on $\bbR^{1+3}$ and the focusing cubic nonlinear Schrödinger equation (NLS) on $\bbR$. Our approach is based on the Payne–Sattinger theory, which identifies invariant sets through energy functionals and conserved quantities. For NLKG, we review the Payne–Sattinger theory to establish a sharp dichotomy between global existence and blow-up. For NLS, we apply this theory with a scaling argument to construct scale-invariant thresholds, replacing the standard mass-energy conditions with a $\dot{H}^{\frac12}$-critical functional. This unified framework provides a natural derivation of global behavior thresholds for both equations.

In this talk, we will introduce vector field method for the wave equation. The key step is to establish the Klainerman-Sobolev inequality developed in [1]. Using this inequality, we will provide dispersive estimates of the linear wave equation, and prove small-data global existence for some nonlinear wave equations. The main reference will be Chapter II in [2]. 참고문헌: [1]. Sergiu Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), no. 3, 321–332. MR 784477 [2]. Christopher D. Sogge, Lectures on Nonlinear Wave Equations, Second Edition

Abstract: We consider the initial-boundary value problem (IBVP) for the 1D isentropic Navier-Stokes equation (NS) in the half space. Unlike the whole space problem, a boundary layer may appear due to the influence of viscosity. In this talk, we first briefly study the asymptotic behavior for the initial value problem of NS in the whole space. Afterwards, we will present the characterization of the expected asymptotics for the IBVP of NS in the half space. Here, we focus only on the inflow problem, where the fluid velocity is positive on the boundary. Reference: Matsumura, Akitaka. Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Methods Appl. Anal. 8 (2001), no. 4, 645–666.