Day 2 (June 11)

 Multidimensional conservation laws is an active research area with open questions about existence, uniqueness, and stability of properly defined weak solutions, even for fundamental models such as the compressible Euler system. Understanding particular classes of weak solutions, such as Riemann problems, is crucial in this context. This talk focuses on self-similar solutions to two-dimensional Riemann problems involving transonic shocks for compressible Euler systems. Examples include regular shock reflections, Prandtl reflection, and four-shocks Riemann problem. We first review the results on existence, regularity, geometric properties and uniqueness of global self-similar solutions of regular relfection structure in the framework of potential flow equation. A significant open problem is to extend these results to compressible Euler system, i.e. to understand the effects of vorticity. We show that for the isentropic Euler system, solutions of regular reflection structure have low regularity. We further discuss existence, uniquenss and stability of renormalized solutions to the transport equation for vorticity in this low regularity setting.

 We study the long-time asymptotic behavior of the Sticky Particles dynamics on the real line. The time average of the Sticky Particles Lagrangian map has a limit which arises as a general property of projections onto closed convex cones in Hilbert spaces. More notably, we prove that the map itself has an asymptotic limit in the case where the Sticky Particles dynamics is confined to a compact set. We also demonstrate that there is no universal rate of decay, i.e. sticky particle solutions can converge to equilibrium at an arbitrarily slowly. 

 We are concerned with the existence and compactness of entropy solutions of the compressible Euler system for two dimensional steady potential flow around an obstacle for a polytropic gas with supersonic farfield velocity. This problem was first formulated by Morawetz in 1985 and has remained open since then. In this paper, we develop a complete compactness framework that allows for cavitation and show how it can be applied to obtain an existence theorem for the Morawetz problem by developing a new entropy analysis, in combination with a vanishing viscosity method and compensated compactness ideas. The main difficulty is that the problem becomes singular as the flow approaches cavitation, resulting in a loss of strict hyperbolicity and a singularity of the entropy equation for the case of adiabatic exponent 𝛾=3. Our analysis provides a complete description of the entropy and entropy-flux pairs via the Loewner-Morawetz relations, which leads to the establishment of the compensated compactness framework. As direct applications of our entropy analysis and the compensated compactness framework, we further obtain the compactness of entropy solutions and the weak continuity of the compressible Euler system in the supersonic regime. 

 This talk is concerned with our recent developments on the time asymptotic stability of composite wave with Oleinik shock and rarefaction wave to both scalar cubic nonconvex viscous conservation laws and the physical system for nonconvex viscoelasticity.

 The Sadovskii vortex patch is a traveling wave for the two-dimensional incompressible Euler equations consisting of an odd symmetric pair of vortex patches touching the symmetry axis. Its existence was first suggested by numerical compuations of Sadovskii in [J. Appl. Math. Mech., 1971], and has gained significant interest due to its relevance in the inviscid limit of planar flows via Prandtl-Batchelor theory and as the asymptotic state for vortex ring dynamics. In this talk, I will sketch a proof of the existence of such a vortex and stability in the class using an energy maximization approach under the exact impulse condition and an upper bound on the circulation. (For reference, a completely different proof of the same existence result with more information via a fixed point method appeared around the same time by Huang and Tong. The uniquenss of such a vortex remians open.) This talk is based on joint work with In-Jee Jeong(SNU), Youngjin Sim(UNIST), and Kwan Woo(SNU).