Research

Abstract: The modern theory of risk measures copes with uncertainty by considering multiple probability measures. While it is often assumed that a reference probability measure exists, under which all relevant probability measures are absolutely continuous, there are examples where this assumption does not hold, such as certain distributional robust functionals. In this talk, we introduce a novel class of dynamic risk measures that do not rely on this assumption. We will discuss its convexity, coherence, and time consistency properties.

Abstract: We consider the incompressible Navier-Stokes and Euler equations in a bounded domain with non-characteristic boundary condition, and study the energy dissipation near the outflow boundary in the zero-viscosity limit. We show that in a general setting, the energy dissipation rate is proportional to $\bar U \bar V ^2$, where $\bar U$ is the strength of the suction and $\bar V$ is the strength of the shear. Moreover, we show that the rate of enstrophy production near the boundary is proportional to the Reynolds number.

Abstract: In this talk, I will discuss the inviscid limit problem. I will start with a general introduction of the Euler and Navier-Stokes system, and describe the problem of inviscid limit. I will cite the result of Kato (1984) which gives a necessary and sufficient condition for inviscid limit to hold. This will lead to a discussion on the zero-viscosity limit of energy dissipation. I will present a theorem of Vasseur—Yang (2023,2024) which gives an upper bound in the case of the impermeable and non-slip boundary condition, and an ongoing work which shows upper and lower bound in the case of the inflow-outflow condition.

Abstract: In this talk, we discuss the concept of risk measure, where the realms of optimization and finantial mathematics meets. New challenges arise in a dynamic environment and in data-driven settings. We will introduce a new class of dynamic risk measures with a good interpretability and time consistency. While it is often assumed that a reference probability measure exists, under which all relevant probability measures are absolutely continuous, there are examples where this assumption does not hold, such as certain distributional robust functionals. Our construction do not rely on this assumption.

Abstract: In this talk, we consider the 3D incompressible Navier-Stokes equation in a bounded domain, with a canonical example of Poiseuille flow in mind. We provide an unconditional upper bound for the boundary layer separation and energy dissipation of Leray–Hopf weak solutions, uniformly in high Reynolds numbers. We estimate layer separation by measuring the energy norm of the discrepancy between a (turbulent) low-viscosity Leray–Hopf solution and a fixed (laminar) regular Euler solution with similar initial conditions and body forces. This is accomplished by a new nonlinear boundary vorticity estimate, achieved along with several new trace estimates and higher derivative estimates using blow-up methods.

Abstract: We study the dispersion process on the complete graph introduced in the recent work under the mean-field framework. In contrast to the probabilistic approach, our focus is on the investigation of the large time behavior of solutions of the associated kinetic mean-field system of nonlinear ordinary differential equations (ODEs). We establish various analytical and quantitative convergence results for the long time behaviour of the mean-field system and related numerical illustrations are also provided.

Abstract: In this talk, we consider the 3D incompressible Navier-Stokes equation in a bounded domain, with a canonical example of Poiseuille flow in mind. We provide an unconditional upper bound for the boundary layer separation and energy dissipation of Leray–Hopf weak solutions, uniformly in high Reynolds numbers. We estimate layer separation by measuring the energy norm of the discrepancy between a (turbulent) low-viscosity Leray–Hopf solution and a fixed (laminar) regular Euler solution with similar initial conditions and body force. This is accomplished by a new nonlinear boundary vorticity estimate.

Abstract: We derive several nonlinear a priori trace estimates for the 3D incompressible Navier-Stokes equation, which extend the current picture of higher derivative estimates in the mixed norm. The main ingredient is the blow-up method and a novel averaging operator, which could apply to PDEs with scaling invariance and -regularity, possibly with a drift.

Abstract: In this talk, we consider the 3D incompressible Navier-Stokes equation in a bounded domain, with a canonical example of Poiseuille flow in mind. We provide an unconditional upper bound for the boundary layer separation and energy dissipation of Leray–Hopf weak solutions, uniformly in high Reynolds numbers. We estimate layer separation by measuring the energy norm of the discrepancy between a (turbulent) low-viscosity Leray–Hopf solution and a fixed (laminar) regular Euler solution with similar initial conditions and body force. This is accomplished by a new nonlinear boundary vorticity estimate.

Abstract: We derive several nonlinear a priori trace estimates for the 3D incompressible Navier-Stokes equation. They recover and extend the current picture of higher derivative estimates in the mixed norm. The main ingredient is the blow-up method and a novel averaging operator, which could apply to PDEs with scaling invariance and quantitative one-scale ε-regularity.

Abstract: We derive several nonlinear a priori trace estimates for the 3D incompressible Navier-Stokes equation. They recover and extend the current picture of higher derivative estimates in the mixed norm. The main ingredient is the blow-up method and a novel averaging operator, which could apply to PDEs with scaling invariance and quantitative one-scale ε-regularity.

Abstract: In this presentation, I will discuss recent progress regarding the regularity of the three-dimensional Navier-Stokes equation, a system of partial differential equations that models the behavior of fluids. While the full regularity of the 3D incompressible Navier-Stokes equation remains an outstanding open question, recently there have been significant breakthroughs in the fluid dynamics community. I will present new mathematical tools that provide deeper insights into the partial regularity of the Navier-Stokes equation and general supercritical systems. We derive nonlinear a priori estimates and trace estimates for the 3D incompressible Navier-Stokes equation, which extend the current picture of higher derivative estimates in the mixed norm. Additionally, I will demonstrate an intriguing application to the inviscid limit problem, which questions to what extent ideal fluids can model slightly viscous fluid.

Abstract: We derive several nonlinear a priori trace estimates for the 3D incompressible Navier-Stokes equation, which extend the current picture of higher derivative estimates in the mixed norm. The main ingredient is the blow-up method and a novel averaging operator, which could apply to PDEs with scaling invariance and ε-regularity, possibly with a drift.

Abstract: We show some a priori regularity estimates for the vorticity and its trace in the three-dimensional incompressible Navier-Stokes equation. These a priori estimates are obtained via the blow-up method and a novel averaging operator. The averaging operator can be used to provide regularity and trace estimates for PDEs with ε-regularity.

Abstract: We derive several nonlinear a priori trace estimates for the 3D incompressible Navier-Stokes equation, which extend the current picture of higher derivative estimates in the mixed norm. The main ingredient is the blow-up method and a novel averaging operator, which could apply to PDEs with scaling invariance and ε-regularity, possibly with a drift.

Abstract: We provide an unconditional upper bound for the boundary layer separation, anomalous dissi- pation, and the work of drag force in the zero viscosity limit, of Leray–Hopf weak solutions to the 3D incompressible Navier-Stokes equation in a smooth bounded domain. Layer separation refers to the discrepancy between a (turbulent) low-viscosity Leray–Hopf solution and a fixed (laminar) regular Euler solution with similar initial conditions and body force. In addition, we show the boundedness of the drag coefficient with a high Reynolds number.

Abstract: We provide an unconditional $L^2$ upper bound for the boundary layer separation of 3D Leray-Hopf solutions in a smooth bounded domain. By layer separation, we mean the discrepency between a (turbulent) low-viscosity Leray-Hopf solution $u^\nu$ and a fixed (laminar) regular Euler solution $\bar u$ with initial conditions close in $L^2$. Layer separation appears in physical and numerical experiments near the boundary, and we bound it asymptotically by $C \lVert\bar u\rVert_{L^\infty}^3 t$. This extends the previous result when the Euler solution is a regular shear in a finite channel. The key estimate is to control the boundary vorticity in a way that does not degenerate in the vanishing viscosity limit. This is joint work with Alexis Vasseur.

Abstract: We provide an unconditional $L^2$ upper bound for the boundary layer separation of 3D Leray--Hopf solutions in a smooth bounded domain. By layer separation, we mean the discrepancy between a (turbulent) low-viscosity Leray--Hopf solution $u^\nu$ and a fixed (laminar) regular Euler solution $\bar u$ with initial conditions close in $L^2$. Layer separation appears in physical and numerical experiments near the boundary, and we bound it asymptotically by $C \|\bar u\|_{L^\infty}^3 t$. This extends the previous result when the Euler solution is a regular shear in a finite channel. The key estimate is to control the boundary vorticity in a way that does not degenerate in the vanishing viscosity limit.

Abstract: We provide an unconditional $L^2$ upper bound for the boundary layer separation of 3D Leray-Hopf solutions in a smooth bounded domain. By layer separation, we mean the discrepency between a (turbulent) low-viscosity Leray-Hopf solution $u^\nu$ and a fixed (laminar) regular Euler solution $\bar u$ with initial conditions close in $L^2$. Layer separation appears in physical and numerical experiments near the boundary, and we bound it asymptotically by $C \lVert\bar u\rVert_{L^\infty}^3 t$. This extends the previous result when the Euler solution is a regular shear in a finite channel. The key estimate is to control the boundary vorticity in a way that does not degenerate in the vanishing viscosity limit. This is joint work with Alexis Vasseur.

Abstract: We consider data-driven decision-making under uncertainty with side information, in which exogenous covariates data are available to reveal partial information about random problem parameters and thus facilitate better decisions. The decision maker's goal is to find an optimal end-to-end policy that directly outputs a decision for any new context. We propose a distributionally robust formulation based on the causal transport distance, which preserves the conditional information structure of random problem parameters given the value of covariates. We derive a dual reformulation and study its regularization effect. We prove that the infinite-dimensional policy optimization admits a finite-dimensional convex programming equivalent reformulation. This result renders a new class of optimal policy for adjustable robust optimization.

Abstract: We show a series of works of some regularity results on the incompressible Navier--Stokes equation in dimension three. Using the blow-up method, we estimate the higher regularity in the Lorentz norm for smooth solutions to the Navier--Stokes equation. In particular, we show a second derivative estimate for suitable weak solutions, which improves the currently known regularity. We construct a maximal function associated with geometric objects that we call skewed cylinders, appearing in inviscid flows like the Eulerian cylinders around the Lagrangian trajectories. We also apply the blow-up method to estimate the boundary vorticity, which enables us to achieve an unconditional control of the layer separation of Leray--Hopf solutions from a steady shear flow in a finite periodic channel. [Slides]

Abstract: We discuss the limiting behavior of a Leray-Hopf solution to the Navier-Stokes equation in dimension 2 and 3 as viscosity vanishes and initial velocity profile converges to a constant shear flow. Without further assumption, it is a major open question whether a solution to the Navier-Stokes equation converges to a solution to the Euler equation. In this talk, we will present a new estimate on the discrepency between a Navier-Stokes solution and a static Euler solution albeit the inviscid limit may fail. This is the first unconditional result from the positive side, and the bound is near sharp as suggested by convex integration.

Abstract: In this talk we present $L ^{4/3,q}$ local integrability of the second spatial derivatives of suitable solutions to the Navier-Stokes equation for any $q > \frac43$. This joint work with A. Vasseur improves the current result $L ^{4/3, \infty}$ (Lions, 1996), and it is based on a blow-up technique using the universal scaling along approximated Lagrangian trajectories. Locally, we can obtain any regularity of the vorticity without any a priori knowledge of the pressure. The local-to-global step uses a recently constructed maximal function for transport equations.

Abstract: In this talk, I will introduce a recently constructed extension operator, called the "Shapley extension". This operator gives a Lipschitz extension for a function defined on a discrete subset of metric space to the whole space. The extension is based on the Shapley theorem, which comes from the Nash equilibrium in the game theory. We will give an application of the Shapley extension in 1-Wasserstein robust optimization problem. This is a joint work with Gao and Zhang.

Abstract: In this talk we present $L ^{4/3,q}$ local integrability of the second spatial derivatives of suitable solutions to the Navier-Stokes equation for any $q > \frac43$. This joint work with A. Vasseur improves the current result $L ^{4/3, \infty}$ (Lions, 1996), and it is based on a blow-up technique using the universal scaling along approximated Lagrangian trajectories. Locally, we can obtain any regularity of the vorticity without any a priori knowledge of the pressure. The local-to-global step uses a recently constructed maximal function for transport equations.

Abstract: We want to estimate the expectation of a random variable when the underlying probability measure is unknown. The true probability can differ from some nominal distribution by a distance R, where the distance between distributions can be characterized by the optimal mass transport, for instance Wasserstein distance. In this talk, we will explore the duality principle for this estimate, and we can discuss the worst case distribution for the discrete case if time permits.

Abstract: In this talk, we will introduce the oscillatory integral of the first kind $\int \exp(i t \phi (x)) \psi (x) dx$. We present the Van der Corput lemma, and the method of stationary phase. As an application in PDE, we will analyze the decay rate for a family of 1-dimensional linear dispersive equations. If time permits, we can look at a stability result for the inviscid Boussinesq system.

Abstract: We want to give second derivative estimate for the suitable weak solutions to three dimensional Navier-Stokes equations with $L ^2$ initial data. Right now it is known that $\nabla ^2 u$ is in the space of $L ^{\frac43, \infty}$ homogeneous in space-time. The limiting space $L ^{\frac43}$, however, is unknown. In 2014, Choi and Vasseur gave an alternating proof for $L ^\frac43$ weak integrability, where they used blow-up techniques along trajectories and De Giorgi methods. To do local analysis, they investigate solutions near a Eulerian cylindrical neighborhood of a given point. In this talk, I present a covering lemma for these Eulerian cylinders, which could bridge the gap between $L ^\frac43$ weak and $L ^\frac43$. We are now working on a refinement of the local analysis, which will be able to give the strict $L ^\frac43$ integrability locally. [Slides]

Abstract: I will present several equivalent definition of Hardy space. Then I will try to prove their equivalence as much as I can. Reference: Grafakos, Section 6.4.1, 6.4.2; Stein, Section 3.1.

Abstract: For a fluid equation, one way to obtain higher derivatives estimate is by scaling. In particular, we smoothen the flow, zoom in and center at the regularized trajectories, and obtain interior regularity in local frame, say in a unit cube. These cubes may look very different from each other in the global frame. I will present an interesting Vitali-type covering lemma for these cubes, and as a consequence I will show $W^{2,p}$ boundedness of suitable solutions to Navier-Stokes equation for $p < 4/3$.

Abstract: I will go through the Chapter III of Stochastic Calculus in Manifold by Michel Emery. I will introduce the definition of semimartingales in manifold, and define quadratic variation for semimartingale.

Abstract: I would like to share the thesis of Dennis Kriventsov when he graduated from here four years ago. In his paper, he proposed a model for an SQG equation with different dissipation operators on two domains, corresponding to the ocean and the land, that partition the whole space. He constructed admissible weak solutions, and used De Giorgi method to show their Hölder continuity. He also bootstrapped to the optimal regularity in some cases. In this talk, I will focus on his De Giorgi part that proves the initial $C^α$ regularity, and show how he used a reflection method to deal with the main difficulty, the discrepancy in scaling.

Abstract: As the only one taking the probability prelim courses in our cohort, I want to share with everyone interesting things that are happening in the world of randomness. This talk will start with the intuition of stochastic integral, then introduce very informally the idea of martingale, and finally move to the core of SDE (stochastic differential equation): the famous Itô's lemma.

Abstract: When studying the large-scale movement of the ocean water and atmosphere, earth rotation may affect the stability of the fluid significantly. Here we discuss the linear stability of an homogeneous inviscid incompressible shear flow on a rotating frame near equator. Coriolis effect will be taken into consideration. The goal of this seminar is to illustrate how to show linear stability and instability of such a flow. As an example, we present a thorough analysis of the sinusoidal shear flow, and point out a mistake in Kuo’s (1973) book. This is a joint work with Dr. Hao Zhu and our advisor Prof. Zhiwu Lin at Georgia Tech.

Abstract: We study the inviscid damping of Couette flow with an exponentially stratified density. The optimal decay rates of the velocity field and the density are obtained for general perturbations with minimal regularity. For Boussinesq approximation model, the decay rates we get are consistent with the previous results in the literature. We also study the decay rates for the full Euler equations of stratified fluids, which were not studied before. For both models, the decay rates depend on the Richardson number in a very similar way. Besides, we also study the dispersive decay due to the exponential stratification when there is no shear. [Paper] [Slides]