An Approach to study compressible flow with strong boundary layer

Abstract: It is a classical problem in fluid dynamics about the stability and instability of different hydrodynamic patterns in various physical settings, especially in the high Reynolds number limit of laminar flow with boundary effect. In this talk, after reviewing the background and some recent main progress on boundary layer theory, we will present a new approach introduced by Yang-Zhu to study the compressible fluid both on the instability analysis for unsteady flow and high Reynolds number limit for steady problem. The talk is based on some recent joint work with Shengxin Li and Zhu Zhang.

Spherically symmetric stationary solutions for the compressible Navier-Stokes equation for outflow/inflow problems

Abstract: In the present talk, we discuss the properties and the asymptotic stability of spherically symmetric stationary solutions to the compressible Navier-Stokes equations in the exterior domain of a unit ball. We study two types of boundary conditions, i.e., inflow and outflow conditions. For both problems, we derive several properties of stationary solutions, especially convergence rates toward far fields. By using the rates, we derive a-priori estimate of the perturbation from the stationary solution in the suitable Sobolev space. It shows the asymptotic stability of the stationary solution. In this derivation, the relative energy form plays an essential role. Here we do not have to assume any smallness assumptions on the initial data for the outflow problem if it belongs to the suitable weighted Sobolev space. It is proved by the aid of the representation formula of the density.

A conservative semi-Lagrangian scheme for the ES-BGK model of the Boltzmann equation

Abstract: In this talk, we introduce finite difference high order conservative semi-Lagrangian schemes for the ellipsoidal BGK model of the Boltzmann equation. To avoid the time step restriction induced by the convection term, we adopt the semi-Lagrangian approach. For treating the nonlinear stiff relaxation operator, we employ high order $L$-stable diagonally implicit Runge-Kutta time discretization or backward difference formula. The proposed implicit scheme is designed to update solutions explicitly without resorting to any Newton solver. Numerical results show that our method is able to capture the behavior of compressible Navier-Stokes equations for moderate values of Knudsen number, and provide good approximation to Boltzmann equation for relatively large values of Knudsen number.

The Non-cutoff Boltzmann Equation in Bounded Domains and Velocity Averaging Lemma

Abstract: In this talk, we will investigate the existence of the non-cutoff Boltzmann equation near a global Maxwellian in a general $C^3$ bounded domain $\Omega$. This includes convex and non-convex cases with inflow or Maxwell reflection boundary conditions. We obtain global-in-time $L^2$--$L^\infty$ existence, which has an exponential decay rate for both hard and soft potentials. The crucial method is to extend the boundary problem to the whole space, the velocity averaging lemma, extra damping from the advection operator, and the De Giorgi iteration.

Asymptotic Stability for the 3D Vlasov-Maxwell system in $\mathbb{T}^2\times \mathbb{R}_+$

Abstract: In this talk, I will introduce recent stability results for the 3-dimensional Vlasov-Maxwell system in $\mathbb{T}^2 \times \mathbb{R}_+$. The self-consistent electromagnetic fields follow the Maxwell equations, which are equivalent to the 3D inhomogeneous wave equations in $\mathbb{T}^2 \times \mathbb{R}_+$ given the continuity equations for the inhomogeneity terms. The main difficulties for the proof of (asymptotic) stability arise from the lack of dimensions for the dispersion of waves in $\mathbb{T}^2 \times \mathbb{R}_+$ compared to those in $\mathbb{R}^3$. We assume inflow boundary conditions at $x_3 = 0$ and prove the global-in-time well-posedness of classical solutions and the asymptotic stability of the vacuum and the stability of the Maxwellian state. The initial velocity distribution profile does not need to be small in $L^\infty$; we only assume that its energy density is exponentially decaying in $x_3$. This is a joint work with Chanwoo Kim at the University of Wisconsin-Madison.

Cavity dynamics and regularity for compressible Navier-Stokes equations 

Abstract: In this talk I will talk about global existence of cavity dynamics and regularity for the solutions of the Navier-Stokes equations of compressible barotropic flows that have initially jump discontinuities along the curve grazing two non-convex corners. Rectangular cavities may produce corner shear layers and jump discontinuities between lid and cavity interior. We construct a vector field lifting the jump values into the region and sort out the corner singularity functions by the Lame system with zero boundary condition. We show that the velocity vector has the decomposition into the lifting vector plus the corner singularity plus the remainder of the twice differentiability. The fluid density and velocity gradient have jump discontinuities across the curve as predicted by Rankine-Hugoniot conditions. The interface curve remains the C^{1+a}-regularity, 1/2<a<2/3, despite the singular decomposition of the velocity vector. 

Radon measure-valued solutions to compressible Euler equations with applications  

Abstract: In recent years, we have introduced a new concept known as Radon measure-valued solutions to the compressible Euler equations for general fluids, including polytropic gases. This concept aims to address certain singularities encountered in multi-scale problems, such as inviscid hypersonic flows around bodies/the Newtonian-Busemann law, free-piston problems/singular Riemann problems, and droplets undergoing phase transitions. I will present these developments, which offer a novel approach to tackling complex fluid-structure interactions in engineering applications.

Deep adaptive sampling for numerical PDEs   canceled

Abstract: We present a deep adaptive sampling method for solving PDEs where deep neural networks are utilized to approximate the solutions. More precisely, we propose the failure informed adaptive sampling for PINNs, which can adaptively refine the training set with the goal of reducing the failure probability. Applications to both forward and inverse PDEs problems will be presented.

Fifth-Order Bound-, Positivity-, and Equilibrium- Preserving Affine-Invariant AWENO Scheme for Two-Medium gamma-based Model of Stiffened Gas

Abstract: We describe a quasi-conservative finite difference AWENO scheme with the affine-invariant Z-type nonlinear weights (Ai-AWENO) for the $\gamma$-based model. The shock-capturing scheme should always but often fail to preserve the constant velocity and pressure. One leading cause is that switching the equation of state between different mediums generates numerical oscillations around the medium interface. In the Ai-AWENO scheme, the {\it conservative variables}, instead of the primitive variables, are used, and the equilibriums of velocity and pressure are preserved. A hybrid flux-based bound- and positivity-preserving (BP-P) limiter, which is a convex combination of the high-order (for resolution) and first-order (for BP-P) numerical fluxes, is also implemented to enforce the physical constraints. The theoretical analysis yields the exact CFL conditions of the first-order Lax-Friedrichs numerical flux for the stiffened gas. The numerical diffusion coefficient depends nonlinearly on the local Mach number. Various one-, two-, and three-dimensional benchmark two-medium shock-tube problems illustrate the proposed scheme's high-order accuracy and enhanced robustness.

Suppression of Hydrodynamic Instabilities with Surface Tension and Magnetic Fields 

Abstract: In this talk, we consider the Kelvin-Helmholtz and Rayleigh-Taylor instabilities, subject to a shear flow and gravitational acceleration. These flows are fundamental instabilities in fluid dynamics and are important in various natural and engineering processes such as turbulence, fusion, and supernova. We discuss the effects of surface tension and magnetic fields on the instabilities. We present the linear stability analysis to determine the growth rates of the interfaces. We also conduct numerical simulations of the instabilities and show that both surface tension and magnetic fields give stabilizing effects on the evolution of the instabilities. Complex behaviors on the interfaces such as pinching, elongation, and Alfven waves will be addressed.

Bayesian inference for chaotic systems

Abstract: Accurate prediction of fluid systems is challenging due to their nonlinear and chaotic behaviors. Practitioners, particularly those in numerical weather prediction, have been using data to improve the statistical accuracy of fluid prediction models. This talk will cover Bayesian inference methods for chaotic systems from a numerical analysis point of view, including essential computational techniques, such as inflation and sampling error mitigation, to stabilize the inference process.

Discrete regular decompositions for vector field problems and their application to domain decomposition theories.

Abstract: In this presentation, we introduce stable discrete regular decompositions for problems posed in H(curl) and H(div). There decompositions have been shown to be an effective tool for the analysis of Maxwell's equation and Navier-Stokes equation. We then demonstrate the convergence theories of various types of domain decomposition methods for related vector field problems based on the decompositions.

Operator Network Representations for Fluid Dynamics Problems

Abstract: This talk will explore advanced operator network frameworks, including the Spectral Coefficient Learning Operator Network and the Finite Element Operator Network. These methodologies enable the solution of diverse equations, such as the Navier-Stokes equations, without reliance on data, demonstrating robust generalization capabilities. Additionally, we will discuss how these approaches can effectively address boundary layer phenomena. Built on numerical analysis foundations, these operator networks facilitate rigorous convergence studies, providing theoretical insights alongside computational efficiency. This presentation aims to highlight the potential of operator networks as a versatile and reliable tool for tackling complex fluid dynamics problems.  

Viscous Layers in Fluid Mechanics: Singular Perturbations, Analysis, and Computation

Abstract: Singular perturbation problems arise in differential equations where a small parameter multiplies the highest-order derivatives, creating distinct "fast" and "slow" scales that pose significant challenges in both mathematical analysis and numerical computation. Solutions to such problems often feature localized regions of rapid variation, such as boundary, interior, or corner layers, while remaining smooth elsewhere. In fluid mechanics, the Navier-Stokes equations governing viscous flows can be viewed as a singular perturbation of the inviscid Euler equations, with viscosity acting as the small parameter. In this talk, we discuss recent advances in the study of viscous layers in the Navier-Stokes equations and related fluid systems. We examine the complete structure of boundary layers and other viscous layer phenomena, along with the vanishing viscosity limit, using the method of correctors to refine and extend earlier results. Furthermore, we discuss the implementation of effective numerical schemes for slightly viscous fluid equations, emphasizing the critical role of viscous layer correctors in achieving accuracy and efficiency. This work highlights the synergy between rigorous analysis and computational methods in addressing key challenges in fluid mechanics.

Analysis and Approximation of Incompressible Chemically Reacting Generalized Newtonian Fluid

Abstract: We consider a system of nonlinear partial differential equations modeling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier--Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method. Key technical tools include discrete counterparts of the Bogovskii operator, De Giorgi’s regularity theorem in two dimensions, and the Acerbi-Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.

Self-similar blow up profiles for fluids via physics-informed neural networks

Abstract: In this talk I will explain a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution for different equations in fluid dynamics. The new numerical framework is shown to be both robust and readily adaptable to several situations.

Absence of anomalous dissipation in the 2D Navier Stokes equations

Abstract:  In this talk, we will discuss Leray-Hopf solutions to the two-dimensional Navier-Stokes equations with vanishing viscosity. We aim to demonstrate that when the initial vorticity is only integrable, the Leray-Hopf solutions in the vanishing viscosity limit do not exhibit anomalous dissipation. Moreover, we extend this result to the case where the initial vorticity is merely a Radon measure, assuming its singular part maintains a fixed sign. Our proof draws on several key observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler equation. This is a joint work with Luigi De Rosa (Gran Sasso Science Institute).

Local well-posedness and smoothing of MMT kinetic wave equation

Abstract: In this talk, I will prove local well-posedness of kinetic wave equation arising from MMT equation, which is introduced by Majda, Mclaughlin, and Tabak and is one of the standard toy models to study wave turbulence. Surprisingly, our result reveals a regularization effect of the collision operator, which resembles the situation of non-cutoff Boltzmann. This talk is based on a joint work with Pierre Germain (Imperial College London) and Katherine Zhiyuan Zhang (Northeastern).

From Lane-Emden Stars to Binary Stars: Stability in the gravitational Euler–Poisson System

Abstract: In astrophysical fluid dynamics, stars are modeled as isolated fluid masses governed by self-gravity. A fundamental hydrodynamic framework for describing the dynamics of Newtonian stars is provided by the gravitational Euler–Poisson (EP) system. In this talk, I will explore stability issues for both one-body and multi-body solutions of the EP system, which correspond to individual stars and star systems, respectively. 

In the first half of the talk, I will review key results on the stability of one-body solutions, specifically the so-called Lane-Emden stars. In the second half, I will present my results on the stability and instability of two-body rotating solutions of the EP system, which serve as models for binary star systems.

Sharp Hölder estimates in corner domains and the incompressible Euler equations 

Abstract:  We consider double Riesz transforms on certain corner domains, namely the domains obtained by intersections of half-spaces in $\mathbb{R}^n$. We prove sharp Hoelder estimates for functions not necessarily vanishing on the boundary. This can be applied to the incompressible Euler equations to obtain wellposedness and growth of norms of solutions in time. 

Efficient numerical method solving the penalized system of the Stokes equations on a non-convex polygon

Abstract: In this talk, we show the corner singularity expansion and its convergence result regarding the penalized system obtained by eliminating the pressure variable in the Stokes problem of incompressible flow. The penalized problem is a kind of the Lamé system, so we first discuss the corner singularity theory of the Lamé system on a non-convex polygon. Considering the inhomogeneous Dirichlet boundary condition, we try to show the decomposition of its solution, composed of singular parts and a smoother remainder near a re-entrant corner, and furthermore, we provide the explicit formulae of coefficients in singular parts. In particular, these formulae can be used in the development of highly accurate numerical scheme. Moreover, we briefly propose a locking-free nonconforming quadrilateral finite element scheme handling the corner singularities of the planar linear elasticity problem related to the penalized system of the Stokes equations.

Mathematical Approaches for Effective and Robust Scientific Machine Learning

Abstract: Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific and engineering problems, which has become an emerging field, Scientific Machine Learning (SciML). Many ML techniques, however, are very complex and sophisticated, commonly requiring many trial-and-error and tricks. These result in a lack of robustness and interpretability, which are critical factors for scientific applications. This talk centers around mathematical approaches for SciML, promoting trustworthiness. The first part will present recent efforts advancing the predictive power of physics-informed machine learning through robust training methods. This includes an effective training method for multivariate neural networks, namely, Active Neuron Least Squares (ANLS) and a two-step training method for deep operator networks. The second part is about how to embed the first principles of physics into neural networks. I will present a general framework for designing NNs that obey the first and second laws of thermodynamics. The framework not only provides flexible ways of leveraging available physics information but also results in expressive NN architectures. I will also present an intriguing phenomenon of this framework when it is applied in the context of latent space dynamics identification where an intriguing correlation is observed between an entropy related quantity in the latent space and the behaviors of the full-state solution.