The study of the regularity of solutions to parabolic PDEs is quite extensive. We consider a particular 1-dimensional parabolic differential equation with non-constant Hölder diffusion coefficient that appears in studying the 1-dimensional full Ericksen-Leslie model for nematic liquid crystals. In this talk, we will discuss the existence and regularity of weak solutions for which we explicitly make use of the Hölder continuity of the coefficient.

We consider the nonlinear Schrödinger equation $$iu_t + u_{xx} + |u|^2 u = 0$$ with a nonzero boundary condition of Robin type $u_x(0, t) + \gamma u(0, t) = \varphi(t)$ and establish its Hadamard well-posedness in one as well as in two spatial dimensions (half-line and half-plane, respectively). The Neumann problem is also discussed as a special case. The results are proved by using the solution formulae for the linear Schrödinger equation obtained via the unified transform of Fokas. These formulae allow us to establish suitable linear estimates which are then combined with a contraction mapping argument to yield well-posedness for the nonlinear problems.

We generalize the Unified Transform Method (UTM) for solving linear, constant-coefficient partial differential equations (PDEs) to linear, variable-coefficient PDEs by breaking the domain into subdomains, on which we solve a constant-coefficient interface problem. We use the UTM to solve the resulting interface problem and take the limit as the number of interfaces approaches infinity. This produces an explicit representation of the solution, from which we can determine properties of the solution directly. We demonstrate this for the heat equation on the whole line and we show how the solution on a finite interval can be used to characterize the eigenvalues of the underlying eigenvalue problem.

On Euclidean spaces, a Schwartz function has all partial derivatives decaying faster than (reciprocal of absolute value of) any polynomial. In this talk, assuming geometry existence of ansatz, Tian-Yau solvability, and Hein’s decay estimate etc, we show Schwartz of an ALG Ricci flat Kahler metric on an isotrivial Calabi-Yau fibration. This relies on an $L^2$-decay of harmonic functions and non-concentration of eigen-functions.

In this talk, I will introduce the vortex front problem for quasi-geostrophic shallow water equation, which is also known as Hasegawa-Mima equation in plasma science. The contour dynamic equation of the vortex front will be derived, which is a nonlocal, nonlinear dispersive equation. The existence of global solutions will be proved when the initial data is small. This is a joint work with Fangchi Yan.

We consider the Strichartz inequality for a fourth-order Schrödinger equation on $\mathbb{R}^{2+1}$. We show that extremizers exist using a linear profile decomposition which follows from the endpoint version decomposition and the stationary phase method. Based on the existence of extremizers, we use the associated Euler-Lagrange equation to show that the extremizers have exponential decay and consequently must be analytic.