We briefly introduce the restriction theory in harmonic analysis and its connections with PDEs through Strichartz estimate. We then discuss the Kakeya and multilinear Kakeya estimates, which naturally arise from restriction theory. The main part of the talk will focus on Larry Guth’s proof of the multilinear Kakeya estimate via the induction on scales method.

The celebrated Fredholm alternative theorem works for the setting of identity compact operators. This idea has been widely used to solve linear partial differential equations. In this talk, we demonstrate a generalized Fredholm theory in the setting of identity power compact operators, which was suggested in Cercignani and Palczewski to solve the existence of the stationary Boltzmann equation in a slab domain. We carry out the detailed analysis based on this generalized Fredholm theory to prove the existence theory of the stationary Boltzmann equation in bounded three-dimensional convex domains. To prove that the integral form of the linearized Boltzmann equation satisfies the identity power compact setting requires the regularizing effect of the solution operators. Once the existence and regularity theories for the linear case are established, with suitable bilinear estimates, the nonlinear existence theory is accomplished. This talk is based on a collaborative work with Daisuke Kawagoe and Chun-Hsiung Hsia.

In this talk, I will present the local existence theory for quasilinear symmetric hyperbolic systems, based on Sections 1.3 and 2.1 of [1]. I will begin by reviewing the framework of symmetric systems and then explain how it is applied to establish local-in-time existence of classical solutions. The main focus will be on the iteration scheme, energy estimates, and convergence arguments. We aim to understand how regularity and a priori bounds are used to construct solutions from smooth initial data.

In this talk, we will discuss Leray-Hopf solutions to the incompressible Navier-Stokes equations with vanishing viscosity. We explore important features of turbulence, focusing around the anomalous energy dissipation phenomenon. As a related result, I will present a recent result proving that for two-dimensional fluids, assuming that the initial vorticity is merely a Radon measure with nonnegative singular part, there is no anomalous energy dissipation. Our proof draws on several key observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler equation. We will also discuss possible extensions to the viscous SQG equation in the context of Hamiltonian conservation and existence of weak solutions for a rough initial data.

For many variant kinetic equations, we choose an appropriate approx- imatation equations. Also, this approximation equations are solvable more easier than the original equations and it retains the expected a priori bounds. Then, we use the variant compactness theorem to pass to the limit in the sense of distributions in the approximation equations. In the kinetic theory, this compactness theorem is called the averaing lemma, that is, the averaging in velocity improves regularity in the space and time variables. For this PDE seminar, we study the basic averaging lemma. In other words, we investigate the basic properties of the free transport operator ∂t + v · ∇x.

The Lyapunov-Schmidt reduction is a powerful tool to solve PDEs. This method reduces the equations, which are essentially infinite-dimensional, to finite-dimensional ones. In this talk, we illustrate the reduction by showing the existence of a positive solution to the singularly perturbed problem in for positive smooth and appropriate . To show the existence, we first construct an -dimensional surface of approximate solutions. Then, we reduce the problem onto that surface by the Lyapunov-Schmidt reduction. The key to the reduction is proving the invertibility of a certain operator, which in turn, is proved by a certain uniqueness result. After the reduction, we end the proof by solving the equation on the -dimensional surface.

We study support properties of solutions to stochastic heat equations $\partial_t u = \Delta u + \sigma(u) \xi$ where $\xi$ is Gaussian noise. For $\sigma(u) = u^\lambda$ with colored noise, we show the compact support property holds if and only if $\lambda \in (0, 1)$. Here, the compact support property refers to the property that if the initial function has compact support, then so does the solution for all time. For space-time white noise with general $\sigma$, we characterize when solutions maintain compact support versus become strictly positive. We also discuss how the initial function influences these support properties. This is based on joint work with Beom-Seok Han and Jaeyun Yi.

In this talk, we prove that the inviscid surface quasi-geostrophic (SQG) equation is strongly ill-posed in critical Sobolev spaces: there exists an initial data $H^2(\mathbb{R}^2)$ without any solutions in $L^{\infty}_tH^2$. Then, we introduce similar ill-posedness results for $\alpha$-SQG and two-dimensional incompressible Euler equations. This talk is based on joint works with In-Jee Jeong(SNU), Young-Pil Choi(Yonsei Univ.), Jinwook Jung(Hanyang Univ.), and Min Jun Jo(Duke Univ.).