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In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flow of nonhomogeneous incompressible Navier-Stokes equations. The convergence for the density and velocity…

Analysis of PDEs · Mathematics 2020-12-23 Shijin Ding , Zhilin Lin , Dongjuan Niu

Reformulating the incompressible Stokes equations with the velocity sought in H(curl) has recently emerged as a promising approach for the design of helicity-preserving schemes in magnetohydrodynamics and pressure-robust finite element…

Numerical Analysis · Mathematics 2026-03-23 Wietse M. Boon , Ralf Hiptmair , Wouter Tonnon , Enrico Zampa

We study the two-dimensional incompressible Navier-Stokes equations in a channel $\Omega=(0,L)\times(0,H)$ with small viscosity $\varepsilon\ll1$, an $\varepsilon$-Navier slip condition on the horizontal walls, and a viscous inflow…

Analysis of PDEs · Mathematics 2026-02-24 Yan Guo , Zhuolun Yang

In this work we present new wall-laws boundary conditions including microscopic oscillations. We consider a newtonian flow in domains with periodic rough boundaries that we simplify considering a Laplace operator with periodic inflow and…

Analysis of PDEs · Mathematics 2007-12-17 Didier Bresch , Vuk Milisic

In this paper, we investigate the asymptotic behavior of solutions to the Navier-Stokes equations in the half-plane under high Reynolds number conditions, where the initial vorticity belongs to the Yudovich class and is supported away from…

Analysis of PDEs · Mathematics 2025-10-22 Jingchi Huang , Chao Wang , Jingchao Yue , Zhifei Zhang

We extend the One-Way Navier Stokes (OWNS) approach to support nonlinear interactions between waves of different frequencies, which will enable nonlinear analysis of instability and transition. In OWNS, the linearized Navier-Stokes…

Fluid Dynamics · Physics 2025-02-10 Michael K. Sleeman , Matthew T. Lakebrink , Tim Colonius

We consider solutions to the Navier-Stokes equations with Navier boundary conditions in a bounded domain in the plane with a C^2-boundary. Navier boundary conditions can be expressed in the form w = (2 K - A) v . T and v . n = 0 on the…

Mathematical Physics · Physics 2007-05-23 James P. Kelliher

In this paper we show the non-uniqueness of mild solutions to the two-dimensional forced Navier-Stokes equations in the half space under the noslip boundary condition, following the program established by Albritton, Bru{\'e}, and Colombo in…

Analysis of PDEs · Mathematics 2025-07-04 Motofumi Aoki , Yasunori Maekawa

The abstract theory of critical spaces developed in [22] and [20] is applied to the Navier-Stokes equations in bounded domains with Navier boundary conditions as well as no-slip conditions. Our approach unifies, simplifies and extends…

Analysis of PDEs · Mathematics 2017-10-25 Jan Pruess , Mathias Wilke

We develop a mathematically and physically sound definition of the spectrally-hyperviscous Navier-Stokes equations (SHNSE) on general bounded domains \Omega with zero (no-slip) boundary conditions prescribed on \varGamma=\partial\varOmega.…

Analysis of PDEs · Mathematics 2019-08-30 Joel Avrin

Variational approaches have been used successfully as a strategy to take advantage from real data measurements. In several applications, this approach gives a means to increase the accuracy of numerical simulations. In the particular case…

Optimization and Control · Mathematics 2015-10-15 Telma Guerra , Adélia Sequeira , Jorge Tiago

This theoretical study deals with the Navier-Stokes equations posed in a 3D thin domain with thickness $0<\varepsilon\ll 1$, assuming power law slip boundary conditions, with an anisotropic tensor, on the bottom. This condition, introduced…

Analysis of PDEs · Mathematics 2025-12-18 María Anguiano , Francisco J. Suárez-Grau

The Stokes problem with non-homogeneous Dirichlet boundary condition is solved numerically using conforming discretizations and an approximation of the boundary datum in the corresponding trace space. Optimal discretization error estimates…

Numerical Analysis · Mathematics 2026-04-14 Thomas Apel , Katharina Lorenz , Johannes Pfefferer

Despite the physical importance, there are limited mathematical theories for the compressible Navier-Stokes equations with strong boundary layers. This is mainly due to the absence of a stream function structure, unlike the extensively…

Analysis of PDEs · Mathematics 2025-02-12 Shengxin Li , Tong Yang , Zhu Zhang

It is well-known that by placing judiciously chosen image point forces and doublets to the Stokeslet above a flat wall, the no-slip boundary condition can be conveniently imposed on the wall [Blake, J. R. Math. Proc. Camb. Philos. Soc.…

Numerical Analysis · Mathematics 2018-08-29 Wen Yan , Michael Shelley

For a general class of hyperbolic-parabolic systems including the compressible Navier-Stokes and compressible MHD equations, we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions…

Analysis of PDEs · Mathematics 2015-05-13 Olivier Gues , Guy Metivier , Mark Williams , Kevin Zumbrun

Euler-Lagrange (EL) point-particle simulations rely on hydrodynamic force closure models to accurately predict particle dynamics in flows. The closure models currently employed for dilute particle-laden flows require the undisturbed fluid…

Fluid Dynamics · Physics 2025-03-06 Akshay Chandran , Fabien Evrard , Berend van Wachem

Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the shock wave case, we study stability of compressive, or "shock-like", boundary layers of the isentropic compressible Navier-Stokes equations with gamma-law pressure by…

Analysis of PDEs · Mathematics 2017-06-12 Nicola Costanzino , Jeffrey Humpherys , Toan Nguyen , Kevin Zumbrun

We consider the motion of incompressible viscous non-homogeneous fluid described by the Navier-Stokes equations in a bounded cylinder under boundary slip conditions. Assume that the third co-ordinate axis is the axis of the cylinder.…

Analysis of PDEs · Mathematics 2012-02-07 Wojciech M. Zajaczkowski

This paper recalls a partial differential equations system, which is the linearization of a recognized fluid-elasticity interaction three-dimensional model. A collection of regularity results for the traces of the fluid variable on the…

Analysis of PDEs · Mathematics 2020-09-11 Francesca Bucci