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The paper explores the symbiotic relation between the Navier-Stokes equations and the associated stochastic cascades. Specifically, we examine how some well-known existence and uniqueness results for the Navier-Stokes equations can inform…

Analysis of PDEs · Mathematics 2021-12-08 Radu Dascaliuc , Tuan N. Pham , Enrique Thomann

In this note, we study the random data problem for incompressible Navier-Stokes equations in Euclidean space with arbitrary regularity, using the randomization introduced in \cite{Shen-Soffer-Wu-2021-1}.

Analysis of PDEs · Mathematics 2022-07-25 Ruobing Bai , Jia Shen

The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier--Stokes problem in a time-dependent domain. In this study, the domain's evolution is assumed to be known and independent of the…

Numerical Analysis · Mathematics 2024-08-26 Michael Neilan , Maxim Olshanskii

A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations…

Analysis of PDEs · Mathematics 2020-07-28 Zhilei Liang , Dehua Wang

In this work we consider the two dimensional instationary Navier-Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time…

Numerical Analysis · Mathematics 2026-05-20 Boris Vexler , Jakob Wagner

This paper presents an analytic solution of the incompressible Navier-Stokes equations as recurrence relations for the solution's derivatives, addressing the Clay Mathematics Institute's Millennium Prize problem on Navier-Stokes existence…

Fluid Dynamics · Physics 2025-02-28 Nathan Strange

Local behaviors near boundary are analyzed for solutions of the Stokes and Navier-Stoke equations in the half space with localized non-smooth boundary data. We construct solutions of Stokes equations whose velocity field is not bounded near…

Analysis of PDEs · Mathematics 2024-06-07 TongKeun Chang , Kyungkeun Kang

We consider the steady Navier-Stokes system with mixed boundary conditions, in subdomains of a holdall domain. We study, via the penalization method, its approximation properties. Error estimates, obtained using the extension operator,…

Optimization and Control · Mathematics 2025-08-29 Cornel Marius Murea , Dan Tiba

In this paper, we consider the inhomogeneous Dirichlet boundary value problem for the stationary Navier--Stokes equations in $n$-dimensional half spaces $\mathbb{R}^n_+= \{ x=(x',x_n)\ ;\ x' \in \mathbb{R}^{n-1}, x_n > 0 \}$ with $n \geq 3$…

Analysis of PDEs · Mathematics 2024-10-21 Mikihiro Fujii

We give a geometric approach to proving know regularity and existence theorems for the 2D Navier-Stokes Equations. We feel this point of view is instructive in better understanding the dynamics. The technique is inspired by constructions in…

Analysis of PDEs · Mathematics 2016-09-07 J. C. Mattingly , Ya. G. Sinai

In the following paper we will consider Navier-Stokes problem and it's interpretation by hyperbolic waves, focusing on wave propagation. We will begin with solution for linear waves, then present problem for non-linear waves. Later we will…

Numerical Analysis · Computer Science 2016-01-22 Erik Arakelyan , Aram Serobyan , Narek Jilavyan

Weak solutions of incompressible Navier-Stokes Equations re-obtained variationally

Analysis of PDEs · Mathematics 2020-11-20 Arkady Poliakovsky

We treat the 1D shock tube problem, establishing existence of steady solutions of full (nonisentropic) polytropic gas dynamics with arbitrary noncharacteristic data. We present also numerical experiments indicating uniqueness and…

Analysis of PDEs · Mathematics 2023-04-13 Blake Barker , Benjamin Melinand , Kevin Zumbrun

We are concerned with the existence and uniqueness issue for the inhomogeneous incompressible Navier-Stokes equations supplemented with H^1 initial velocity and only bounded nonnegative density. In contrast with all the previous works on…

Analysis of PDEs · Mathematics 2017-06-27 Raphaël Danchin , Piotr Boguslaw Mucha

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations --…

Numerical Analysis · Mathematics 2024-12-20 Robert Altmann , Jan Heiland

A hyperbolic relaxation of the classical Navier-Stokes problem in 2D bounded domain with Dirichlet boundary conditions is considered. It is proved that this relaxed problem possesses a global strong solution if the relaxation parameter is…

Analysis of PDEs · Mathematics 2018-08-01 Alexei Ilyin , Yuri Rykov , Sergey Zelik

We study well-posedness of a velocity-vorticity formulation of the Navier--Stokes equations, supplemented with no-slip velocity boundary conditions, a no-penetration vorticity boundary condition, along with a natural vorticity boundary…

Analysis of PDEs · Mathematics 2017-08-09 Maxim A. Olshanskii , Leo G. Rebholz , Abner J. Salgado

In this paper, we investigate the link between kinetic equations (including Boltzmann with or without cutoff assumption and Landau equations) and the incompressible Navier-Stokes equation. We work with strong solutions and we treat all the…

Analysis of PDEs · Mathematics 2026-05-20 Kleber Carrapatoso , Isabelle Gallagher , Isabelle Tristani

We consider evolution (non-stationary) space-periodic solutions to the $n$-dimensional non-linear Navier-Stokes equations of anisotropic fluids with the viscosity coefficient tensor variable in space and time and satisfying the relaxed…

Analysis of PDEs · Mathematics 2024-02-09 Sergey E. Mikhailov

We consider the compressible Navier-Stokes system on time-dependent domains with prescribed motion of the boundary. For both the no-slip boundary conditions as well as slip boundary conditions we prove local-in-time existence of strong…

Analysis of PDEs · Mathematics 2018-12-07 Ondřej Kreml , Šárka Nečasová , Tomasz Piasecki