In this talk, I will present the global existence and decay rates of the solutions near Maxwellian for non-linear Fokker--Planck equations and non-linear Vlasov–Poisson–Fokker–Planck equations. The global existence is proved by combining uniform-in-time energy estimates and local solution constructed by Picard type iteration sequence. The decay rates of the nonlinear model is obtained by using the precise spectral analysis of the linear Fokker--Planck operator as well as the energy method. The nonlinearity in the model brings new difficulty for the energy estimates, which is resolved by additional tailored weighted-in-$v$ energy estimates suitable for Fokker--Planck operators.

Inspired by the work of Douglas and Foias in 2006 on the structure of the square of a C0(1) operator, we form a conjecture about a certain reducibility of any power N of a C0(1) operator. We then prove this conjecture for N = 3 by determining explicitly the relevant reducing subspaces.

Most cancer clinical trials fail: 70% fail in Phase 1, and 60% fail in Phase 2. This is particularly so in the case of combination therapy with two or more drugs. One of the reasons for this failure is that not enough thought is given to the potential interactions between the different drugs. In this talk I will consider the case of two drugs, say A and B. I will present mathematical models of cancer and cancer therapy by systems of PDEs, and use them to address the following questions : (i)Are A and B positively correlated? That is, if A or B is increased, does the tumor volume decrease? Or, are there “zones or resistance,” that is, regions in the (A,B) plane where an increase in A or B actually increases the tumor volume?

We extend norm estimates of composition operators of Michael Jury in 2007 on weighted Hardy and Bergman spaces of the unit ball to more general reproducing kernel spaces on the unit ball and the polydisk, for example, on the Dirichlet space of the unit ball and the Hardy space of the polydisk. For the composition operators with affine symbols on the unit ball, we find their exact norms which extend the corresponding results of Cowen in 1988 and Hurst in 1997.

In this talk we show that an observability inequality with a compact remainder (equivalently, an observability inequatility on a finite co-dimensional subspace) implies an explicit spectral description of the set of exactly reachable states. This shows in particular that the compact remainder can be removed if the Fattorini-Hautus test is satisfied. This result gathers and extends many results of the literature, including the compactness-uniqueness method used for instance in the book of J.L. Lions (1988). We apply our result to the boundary controllability of many partial differential equations such as that a Schrödinger equation, a beam equation, a Korteweg-de Vries equation, a perturbed wave equation and an integro-differential transport equation. This talk is based on a joint work with Michel Duprez.

We propose a new finite difference MAC scheme based on the staggered mesh for the compressible Navier-Stokes equations. The scheme is designed for both Dirichlet and periodic boundary conditions. The key features are the application of upwind flux and artificial diffusion. We show that the scheme is energy stable and convergent to the dissipative measure-valued solution of the system. Using the argument of weak-strong uniqueness [E.Feireisl, P.Gwiazda, A.Swierczewska-Gwiazda, and E.Wiedemann. Dissipative measure-valued solutions to the compressible Navier-Stokes system.

Fourier restriction estimate is one of the core topics in harmonic analysis. The Fourier restriction conjecture posed by Stein in 1960s is still open today. In this talk I will give a survey on the various types of the restriction estimates and their applications to PDEs.

In this report we study two-sample functional linear regression with a scaling transformation of regression functions. We consider estimation for the intercept, the slope function and the scalar parameter based on the functional principal component analysis. We also establish the rates of convergence for the estimator of the slope function, which is shown to be optimal in a minimax sense under certain smoothness assumptions. We further investigate semiparametric efficiency for the estimation of the scalar parameter and hypothesis testing.