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Singularity formation in Landau equation with very hard potential
2025-12-22
Chinese Academy of Sciences, PDE and Applications Seminar
Abstract
Abstract: In this talk, we consider the inhomogeneous Landau equation with softness parameter $\gamma$ between $\sqrt3$ and $2$. We show the existence of solutions that arise from smooth, strictly positive initial data but that develop a finite time singularity. The singularity manifests as a Type II implosion for the mass, momentum, and energy of the system. The hydrodynamic fields blow up with a self-similar profile that asymptotically matches smooth implosion singularities of the compressible Euler equations. To our knowledge, this provides the first example of a collisional kinetic model which is globally well-posed in the homogeneous setting, but admits finite time singularities for inhomogeneous data. This talk is based on an ongoing joint work with several collaborators.
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Conditional Liouville theorem for the stationary Navier-Stokes equations
2025-12-16
Chinese Academy of Sciences, PDE and Applications Seminar
Abstract
Abstract: We present a novel approach to the Liouville problem for the stationary Navier-Stokes equations. As an application of our method, we prove conditional Liouville theorems with assumptions on the antiderivative of the velocity that represent substantial improvements on what was heretofore known.
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Conditional Liouville theorem for the stationary Navier-Stokes equations
2025-10-05
Tulane University, AMS Fall Southeastern Sectional Meeting
Abstract
Abstract: We present a novel approach to the Liouville problem for the stationary Navier-Stokes equations. Under assumptions on the growth rate of the antiderivative of the velocity, we prove there exists no nontrivial solution to the three-dimensional stationary Navier-Stokes equation.
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The flexibility and rigidity of incompressible fluid equations
2025-09-04
Johns Hopkins University, Applied Mathematics and Statistics Seminar
Abstract
Abstract: In this talk, we will showcase a few recent developments in the mathematical analysis of incompressible fluid equations, highlighting both their flexibility and rigidity. For evolutionary Euler and Navier-Stokes equation, we discuss when an initial condition corresponds to a unique solution or weak solution. For stationary Euler and Navier-Stokes equations, we explore the conditions under which nontrivial solutions exist or do not exist.
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Conditional Liouville theorem for the stationary Navier-Stokes equations
2025-08-24
University of Denver, AMS Fall Western Sectional Meeting
Abstract
Abstract: We present a novel approach to the Liouville problem for the stationary Navier-Stokes equations. Under assumptions on the growth rate of the antiderivative of the velocity, we prove there exists no nontrivial solution to the three-dimensional stationary Navier-Stokes equation.
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Beyond absolute continuity: a new class of dynamic risk measures
2025-07-24
University of Southern California, ICCOPT 2025
Abstract
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.
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Duality results for Wasserstein Robust Optimization
2025-05-17
University of Wisconsin, Madison, 2025 AWM Research Symposium
Abstract
Abstract: Distributionally robust optimization is a popular framework in data-driven decision making. We present a general duality result for Wasserstein distributionally robust optimization that holds for any Kantorovich transport cost, measurable loss function, and nominal probability distribution. Assuming an interchangeability principle inherent in existing duality results, our proof only uses one-dimensional convex analysis. In this talk, I will present the proof, and motivate the results using example with geometric interpretations.
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Quantitative Convergence Guarantees for the Mean-field Dispersion Process
2025-05-13
Denver, SIAM Conference on Applications of Dynamical Systems (DS25)
Abstract
Abstract: We study the discrete Fokker-Planck equation associated with the mean-field dynamics of a particle system called the dispersion process. For different regimes of the average number of particles per site, we establish various quantitative long-time convergence guarantees toward the global equilibrium, which is also confirmed by numerical simulations.
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Energy dissipation near the outflow boundary in the vanishing viscosity limit
2025-04-15
University of California, Los Angeles, PDE and Analysis Seminar
Abstract
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 tangential component of the difference between Euler and Navier-Stokes on the outflow boundary. Moreover, we show that the enstrophy within a layer of order $\nu / \bar U$ is comparable with the total enstrophy. The rate of enstrophy production near the boundary is inversely proportional to $\nu$. This is based on joint work with Vincent Martinez, Anna Mazzucato, and Alexis Vasseur.
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Energy dissipation near the outflow boundary in the vanishing viscosity limit
2025-04-09
The University of Texas at Austin, Analysis Seminar
Abstract
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 tangential component of the difference between Euler and Navier-Stokes on the outflow boundary. Moreover, we show that the enstrophy within a layer of order $\nu / \bar U$ is comparable with the total enstrophy. The rate of enstrophy production near the boundary is inversely proportional to $\nu$. This is based on joint work with Vincent Martinez, Anna Mazzucato, and Alexis Vasseur.
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Energy dissipation near the outflow boundary in the vanishing viscosity limit
2025-03-27
Princeton University, Analysis of Fluids and Related Topics
Abstract
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 tangential component of the difference between Euler and Navier-Stokes on the outflow boundary. Moreover, we show that the enstrophy within a layer of order $\nu / \bar U$ is comparable with the total enstrophy. The rate of enstrophy production near the boundary is inversely proportional to $\nu$.
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New Estimates for Navier–Stokes and the Inviscid Limit Problem
2024-11-26
Institute for Advanced Study, Analysis and Mathematical Physics
Abstract
Abstract: In this talk, I will present several a priori interior and boundary trace estimates for the 3D incompressible Navier–Stokes equation, which recover and extend the current picture of higher derivative estimates in the mixed norm. Then we discuss the applications in the inviscid limit problem, with both characteristic and noncharacteristic boundary conditions. In particular, we provide estimates on layer separation and energy dissipation in the zero viscosity limit. This talk will be based on several work in collaboration with Alexis Vasseur, Vincent Martinez, and Anna Mazzucato.
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Energy dissipation near the outflow boundary in the vanishing viscosity limit
2024-11-22
CUNY Graduate Center, Harmonic Analysis & PDE Seminar
Abstract
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.
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Beyond absolute continuity: a new class of dynamic risk measures
2024-10-22
Seattle, Informs Annual Meeting 2024
Abstract
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.
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Energy dissipation near the outflow boundary in the vanishing viscosity limit
2024-10-04
Brown University, PDE Seminar
Abstract
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.
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Energy dissipation of Navier-Stokes equation with non-characteristic boundary condition
2024-10-02
Institute for Advanced Study, Short Talks by Postdoctoral Members
Abstract
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.
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The role of risk measure in multistage distributionally robust stochastic optimization
2024-08-22
University of Illinois Urbana-Champaign, Probabaility Seminar
Abstract
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.
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Layer separation and energy dissipation for 3D NSE at high Reynolds number
2024-06-11
Karlstad University, EquaDiff 2024
Abstract
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.
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A discrete Fokker-Planck equation for the dispersion process
2024-05-14
University of Minnesota, Recent Advances in Nonlinear Partial Differential Equations
Abstract
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.
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Layer separation and energy dissipation for 3D NSE at high Reynolds number
2024-05-07
New York University Abu Dhabi (zoom talk), SITE Research Center Talk Series
Abstract
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.
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Vorticity interior trace estimates and higher derivative estimates via blow-up method
2024-03-23
Florida State University, AMS Spring Southeastern Sectional Meeting
Abstract
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.
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Layer separation and energy dissipation for 3D NSE at high Reynolds number
2024-02-19
Johns Hopkins University (zoom talk), Simons Turbulence Seminar
Abstract
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.
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Higher regularity and trace estimates for Navier-Stokes equation
2023-11-13
University of Chicago, Calderón-Zygmund Analysis Seminar
Abstract
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.
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Higher regularity and trace estimates for Navier-Stokes equation
2023-11-02
Purdue University, PDE Seminar
Abstract
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.
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Recent developments in the Navier-Stokes equation
2023-10-12
Johns Hopkins University, Applied Mathematics and Statistics Seminar
Abstract
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.
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New a propri interior trace estimates on the 3D incompressible Navier-Stokes equation
2023-10-06
University of Nebraska-Lincoln, 8th Annual Meeting of the SIAM Central States Section
Abstract
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.
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Vorticity estimates for the 3D incompressible Navier-Stokes equation
2023-08-21
Waseda University (zoom talk), 10th International Congress on Industrial and Applied Mathematics
Abstract
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.
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Trace estimates of 3D NSE via blow-up
2023-07-07
Chinese Academy of Sciences, PDE and Applications Seminar
Abstract
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.
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Layer separation, anomalous dissipation, and drag force in the inviscid limit for 3D NSE
2023-05-06
University of Notre Dame, Midwest PDE Seminar
Abstract
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.
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Layer separation for the 3D Navier-Stokes equation in a bounded domain
2023-01-09
University of Chicago, Calderón-Zygmund Analysis Seminar
Abstract
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.
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Layer separation for the 3D Navier-Stokes equation in a bounded domain
2022-12-16
Chinese Academy of Sciences (zoom talk), PDE and Applications Seminar
Abstract
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.
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Layer separation for the 3D Navier-Stokes equation in a bounded domain
2022-11-03
Princeton University, Analysis of Fluids and Related Topics
Abstract
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.
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Distributionally Robust End-to-end Learning with Side Information
2022-10-16
Indianapolis, Informs Annual Meeting 2022
Abstract
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.
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Partial regularity results for the three-dimensional incompressible Navier-Stokes equation
2022-03-25
The University of Texas at Austin (hybrid talk), Ph.D. Defense Talk
Abstract
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]
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Inviscid Limit and Boundary Layer Separation
2021-10-15
The University of Texas at Austin (zoom talk), Junior Analysis Seminar
Abstract
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.
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New Estimates on the Second Derivatives of the 3D Navier-Stokes Equation
2021-05-02
San Francisco State University (zoom talk), AMS Spring Western Sectional Meeting
Abstract
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.
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Shapley Extension and Application
2021-04-09
The University of Texas at Austin (zoom talk), Junior Analysis Seminar
Abstract
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.
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New Estimates on the Second Derivatives of the 3D Navier-Stokes Equation
2021-03-20
Brown University (zoom talk), AMS Spring Eastern Sectional Meeting
Abstract
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.
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Distributionally Robust Optimization
2020-10-09
The University of Texas at Austin (zoom talk), Junior Analysis Seminar
Abstract
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.
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Oscillatory Integral and Decay of Dispersive Equation
2020-04-10
The University of Texas at Austin (zoom talk), Junior Analysis Seminar
Abstract
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.
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A New Covering Lemma and its Application in 3D Incompressible Navier-Stokes Equations
2020-03-27
The University of Texas at Austin (zoom talk), Ph.D. Candidacy Talk
Abstract
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]
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Introduction to Hardy Space
2019-11-11
The University of Texas at Austin, Harmonic Analysis Reading Seminar
Abstract
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.
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A Covering Lemma for Fluid and Application to Navier-Stokes Equation
2019-11-01
The University of Texas at Austin, Junior Analysis Seminar
Abstract
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 $p < 4/3$.
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Manifold-valued Semimartingales and their Quadratic Variation
2019-10-11
The University of Texas at Austin, Stochastic Calculus in Manifold Reading Seminar
Abstract
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.
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Regularity for a Local-Nonlocal Transmission Problem
2019-04-12
The University of Texas at Austin, Junior Analysis Seminar
Abstract
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 $C^\alpha$ regularity, and show how he used a reflection method to deal with the main difficulty, the discrepancy in scaling.
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An Introduction to Stochastic Integrals
2018-04-20
The University of Texas at Austin, Sophex Seminar
Abstract
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.
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Stability and instability of shear flow in a rotating system
2017-09-22
The University of Texas at Austin, Sophex Seminar
Abstract
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.
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Inviscid Damping of Couette Flow in a Stratified Fluid
2017-04-26
Georgia Institute of Technology, Fluid Mechanics Seminar
Abstract
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]