More Like this

1. Stationary solutions to the Boltzmann equation in the hydrodynamic limit. Ann. PDE 4 (2018), no. 1, Art. 1, 119 pp.

   Esposito, Raffaele; Guo, Yan; Kim, Chanwoo; Marra, Rossana (July 2018, Annals of PDE)

   Abstract Despite its conceptual and practical importance, a rigorous derivation of the steady incompressible Navier–Stokes–Fourier system from the Boltzmann theory has been an outstanding open problem for general domains in 3D. We settle this open question in the affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition.We employ a recent quantitative L2–L∞ approach with new L6 estimates for the hydrodynamic part P f of the distribution function. Our results also imply the validity of Fourier law in the hydrodynamical limit, and our method leads to an asymptotic stability of steady Boltzmann solutions as well as the derivation of the unsteady Navier–Stokes–Fourier system. 

   more » « less
2. Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

   https://doi.org/https://doi.org/10.1016/j.jde.2018.08.033  

   Larios, A; Pei, Y; Rebholz, L (February 2019, Journal of differential equations)

   The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization. 

   more » « less
3. Validity of steady Prandtl layer expansions

   https://doi.org/10.1002/cpa.22109  

   Guo, Yan; Iyer, Sameer (November 2023, Communications on Pure and Applied Mathematics)

   Abstract Let the viscosity for the 2D steady Navier‐Stokes equations in the region and with no slip boundary conditions at . For , we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. Our uniform estimates in ε are achieved through a fixed‐point scheme:for solving the Navier‐Stokes equations, where are the tangential and normal velocities at , DNS stands for of the vorticity equation for the normal velocityv, and the compatibility ODE for at . 

   more » « less
4. Ghost effect from Boltzmann theory

   https://doi.org/10.1002/cpa.70017  

   Esposito, Raffaele; Guo, Yan; Marra, Rossana; Wu, Lei (October 2025, Communications on Pure and Applied Mathematics)

   Abstract Taking place naturally in a gas subject to a given wall temperature distribution, the “ghost effect” exhibits a rare kinetic effect beyond the prediction of classical fluid theory and Fourier law in such a classical problem in physics. As the Knudsen number goes to zero, the finite variation of temperature in the bulk is determined by an infinitesimal, ghost‐like velocity field, created by a givenfinitevariation of the tangential wall temperature as predicted by Maxwell's slip boundary condition. Mathematically, such a finite variation leads to the presence of a severe singularity and a Knudsen layer approximation in the fundamental energy estimate. Neither difficulty is within the reach of any existing PDE theory on the steady Boltzmann equation in a general 3D bounded domain. Consequently, in spite of the discovery of such a ghost effect from temperature variation in as early as 1960s, its mathematical validity has been a challenging and intriguing open question, causing confusion and suspicion. We settle this open question in affirmative if the temperature variation is small but finite, by developing a new framework with four major innovations as follows: (1) a key ‐Hodge decomposition and its corresponding local ‐conservation law eliminate the severe bulk singularity, leading to a reduced energy estimate; (2) a surprising gain in via momentum conservation and a dual Stokes solution; (3) the ‐conservation, energy conservation, and a coupled dual Stokes–Poisson solution reduces to an boundary singularity; (4) a crucial construction of ‐cutoff boundary layer eliminates such boundary singularity via new Hardy's and BV estimates. 

   more » « less
5. Partial regularity of suitable weak solutions of the Navier- Stokes-Planck-Nernst-Poisoon equation

   Gong, Huajun; Zhang, Xiaotao; Wang, Changyou (January 2021, SIAM journal on mathematical analysis)

   In this paper, inspired by the seminal work by Caffarelli, Kohn, and Nirenberg [Comm. Pure Appl. Math., 35 (1982), pp. 771--831] on the incompressible Navier--Stokes equation, we establish the existence of a suitable weak solution to the Navier--Stokes--Planck--Nernst--Poisson equation in dimension three, which is smooth away from a closed set whose 1-dimensional parabolic Hausdorff measure is zero. 

   more » « less