Special Session 3: Recent Mathematical Progress in Boundary Layer Problems

Boundary Layer Analysis in Diffusive Limits of Radiative Heat Transfer System

Mohamed Ghattassi New York University Abu Dhabi United Arab Emirates Co-Author(s):    Xiaokai Huo, Nader Masmoudi

Abstract: This work investigates the diffusive limit of nonlinear radiative heat transfer systems, focusing on boundary layers under various conditions, including reflective radiative, Dirichlet, Robin, and curved boundaries. The global existence of weak solutions is demonstrated using the Galerkin method, and the convergence of these solutions to a nonlinear diffusion model in the diffusive limit is established through compactness techniques, Young measure theory, and the Banach fixed-point theorem. This work also addresses the nonlinear Milne problem, where the nonlinearity of the Stefan-Boltzmann law introduces additional mathematical challenges. Existence, exponential decay, and uniqueness of solutions are proven using uniform estimates, monotonicity properties, and spectral assumptions. Furthermore, the coupling between elliptic and kinetic transport equations is resolved via combined \( L^2 \)-\( L^\infty \) estimates. The extension to curved boundary domains includes a novel geometric correction for boundary layers, ensuring stability and convergence of solutions. These results significantly extend the existing mathematical framework for radiative heat transfer systems, providing a rigorous analysis of diffusive limits in complex geometries.

Incompressible limit of compressible systems in $\R^3$

Xianpeng Hu The Hong Kong Polytechnic University Peoples Rep of China Co-Author(s):    Guochun Wu, Xin Zhong

Abstract: We will discuss the incompressible limit of compressible Navier-Stokes equations. As the volume viscosity tends to infinity, the limit system of compressible Navier-Stokes equations with discontinuous initial data is the density-dependent incompressible Navier-Stokes equation. In this setting, the global convergence to the limit system around an equilibrium is justified.

On the hydrostatic approximation of Navier-Stokes-Maxwell system with Gevrey data

Ning Liu The Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Marius Paicu, Ping Zhang

Abstract: In this talk, we establish the local well-posedness of a scaled anisotropic Navier-Stokes-Maxwell system in a 2-D striped domain with initial data around some nonzero background magnetic field in Gevrey-2 class. Then we rigorously justify the limit from the scaled anisotropic equations to the associated hydrostatic system and provide with the precise convergence rate. Finally, with small initial data in Gevrey-3/2 class, we also extend the lifespan of thus obtained solutions to a longer time interval.

On The Hydrostatic Approximation Of Navier-Stokes-Maxwell System With 2D Electronic Fields

Faiq Raees New York University United Arab Emirates Co-Author(s):    Weiren Zhao

Abstract: In this talk, we establish the well-posedness of a scaled anisotropic Navier-Stokes-Maxwell system in a 2D striped domain with a transverse magnetic field around (0,0,1) in Gevrey-2 class. Then we justify the limit from the scaled anisotropic equations to the associated hydrostatic system and establish the precise convergence rate. Finally, we prove the global stability of the state (0,0,1) and show that small perturbations decay down exponentially. We will conclude the talk by giving some evidence that the Gevrey-2 class is optimal.

INTERACTIVE BOUNDARY LAYER THEORY

Marco Sammartino University of Palermo Italy Co-Author(s):    R.Caflisch, F.Gargano, V.Sciacca

Abstract: The main aim of this talk is to give a mathematical justification for the interactive boundary layer theory. In the first part of the talk, we shall discuss the behavior of the Navier-Stokes solutions for a high Reynolds number flow interacting with a boundary. To describe these flows, one usually introduces the boundary layer and the Prandtl equations. A rigorous mathematical justification of this setting has been given in the analytic setting. However, the Euler--Prandtl matching does not allow the interaction between the Boundary Layer and the outer flow: this imposes severe limitations on the theory, ultimately leading to the lack of a sound description of the transition phenomena occurring close to the boundary. In the second part of the talk, we shall describe a different asymptotic approach that seems able to overcome the above-described difficulties. The well-posedness of the resulting equations in the analytic setting is the main result of the talk.

On the characterization, existence and uniqueness of steady solutions to the hydrostatic Euler equations in a nozzle

Tak Kwong Wong The University of Hong Kong Hong Kong Co-Author(s):    Wang Shing Leung and Chunjing Xie

Abstract: As a variant of Prandtl boundary layer equations, the hydrostatic Euler equations describe the leading-order behavior of ideal flows passing through narrow domains. In this talk, we will discuss recent results about steady solutions to the hydrostatic Euler equations in nozzles. When expressed in terms of stream function formulation, the steady hydrostatic Euler equations become a degenerate elliptic equation, so the classical estimates for uniformly elliptic equations cannot directly apply. One of the key ingredients for the mathematical analysis is a new transformation that combines a change of variable and Euler-Lagrange transformation. With the aid of this new transformation, the solutions in the new coordinates enjoy explicit representations so that the regularity of steady solutions with respect to the horizontal variable can be gained in a clear way.

Mack modes in supersonic boundary layer

Di Wu South China University of Technology Peoples Rep of China Co-Author(s):    N. Masmoudi, Y. Wang, Z. Zhang

Abstract: Understanding the transition mechanism of boundary layer flows is of great significance in physics and engineering, especially due to the current development of supersonic and hypersonic aircraft. In this talk, we construct multiple unstable acoustic modes so-called Mack modes, which play a crucial role during the early stage of transition in the supersonic boundary layer. To this end, we develop an inner-outer gluing iteration to solve a hyperbolic-elliptic mixed type and singular system.

Stability analysis of the subsonic boundary layers at the high Reynolds number

ZHU ZHANG The Hong Kong Polytechnic University Hong Kong Co-Author(s):

Abstract: In this talk, we will present a recent result on the stability of subsonic boundary layers for compressible Navier-Stokes equations at the high Reynolds numbers.