In this talk, I will mostly discuss the singularity formation of Burgers equation. It is well-known that, when the initial data has negative gradient at some point, the solutions blow up in a finite time. We shall study the properties of the blow-up profile of Burgers equation by introducing the self-similar variables and the modulations, which can be used to study the blow-up for general nonlinear hyperbolic systems. If time permits, I will also discuss the singularity formation for the 1D compressible Euler equations and the related open questions.

In this talk, I will mostly discuss the singularity formation of Burgers equation. It is well-known that, when the initial data has negative gradient at some point, the solutions blow up in a finite time. We shall study the properties of the blow-up profile of Burgers equation by introducing the self-similar variables and the modulations, which can be used to study the blow-up for general nonlinear hyperbolic systems. If time permits, I will also discuss the singularity formation for the 1D compressible Euler equations and the related open questions.

In this talk, I will mostly discuss the singularity formation of Burgers equation. It is well-known that, when the initial data has negative gradient at some point, the solutions blow up in a finite time. We shall study the properties of the blow-up profile of Burgers equation by introducing the self-similar variables and the modulations, which can be used to study the blow-up for general nonlinear hyperbolic systems. If time permits, I will also discuss the singularity formation for the 1D compressible Euler equations and the related open questions.

Global wellposedness and asymptotic stability of the Boltzmann equation with specular reflection boundary condition in 3D non-convex domain is an outstanding open problem in kinetic theory. Motivated by Guo’s L^2-L^\infty theory, the problem was completely solved for general C^3 domain, but it is still widely open for general non-convex domains. The problem was solved in cylindrical domain with analytic non-convex cross section. Generalizing previous work, we study the problem in general solid torus, a solid torus with general analytic convex cross-section. This is the first results for the domain which contains essentially 3D non-convex structure. This is a joint work with Chanwoo Kim and Gyeonghun Ko.