Related papers: Navier-Stokes equations, determining forms, determ…
The evolution of a determining form for the 2D Navier-Stokes equations (NSE), which is an ODE on a space of trajectories is completely described. It is proved that at every stage of its evolution, the solution is a convex combination of the…
The Navier--Stokes equations for incompressible flows past a two--dimensional sphere are considered in this article. The existence of an inertial form of the equations is established. Furthermore for the first time for fluid equations, we…
In this paper we show that the long time dynamics (the global attractor) of the 2D Navier-Stokes equation is embedded in the long time dynamics of an ordinary differential equation, named {\it determining form}, in a space of trajectories…
A remarkable feature of fluid dynamics is its relationship with classical dynamics and statistical mechanics. This has motivated in the past mathematical investigations concerning, in a special way, the "derivation" based on kinetic theory,…
We introduce a determining wavenumber for the forced 3D Navier-Stokes equations (NSE) defined for each individual solution. Even though this wavenumber blows up if the solution blows up, its time average is uniformly bounded for all…
We investigate the long-term dynamics of the three-dimensional Navier-Stokes-Voight model of viscoelastic incompressible fluid. Specifically, we derive upper bounds for the number of determining modes for the 3D Navier-Stokes-Voight…
We have developed dynamic manifold solutions for the Navier-Stokes equations using an extension of differential geometry called the calculus for moving surfaces. Specifically, we have shown that the geometric solutions to the Navier-Stokes…
We study the pointwise decay properties of solutions to the incompressible Navier-Stokes equations, both in the space and time variables. It is well known that generic global solutions on $\mathbb{R}^n$ do not decay faster at infinity than…
We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This solution family is equivalent to a fractal curve in complex…
In this paper the problem of strong solvability of the incompressible Navier-Stokes equations (INSE) is revisited, with the goal of determining the minimal assumptions for the validity of a local existence and uniqueness theorem for the…
Statistical solutions, which are time-parameterized probability measures on spaces of square-integrable functions, have been established as a suitable framework for global solutions of incompressible Navier-Stokes equations (NSE). We…
Let $F:X\to X$ be a $C^2_\loc$ map in a Banach space $X$, and $A$ be its Fr\`echet derivative at the element $w:=w_\ve$, which solves the problem $(\ast) \dotw=-A^{-1}_\ve(F(w)+\ve w)$, $w(0)=w_0$, where $A_\ve:=A+\ve I$. Assume that…
We consider the system of partial differential equations governing two-dimensional flows of a robust class of viscoelastic rate-type fluids with stress diffusion, involving a general objective derivative. The studied system generalizes the…
In this work, we consider time-fractional Navier-Stokes equations (NSE) with the external forces involving finite delay. Equations are considered on a bounded domain in 3-D space having sufficiently smooth boundary. We transform the system…
The Navier--Stokes (NS) equations describe fluid dynamics through a high-dimensional, nonlinear system of partial differential equations (PDEs). Despite their fundamental importance, their behavior in turbulent regimes remains incompletely…
We construct solutions to the randomly-forced Navier--Stokes--Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense…
We are concerned with the "transfer of regularity" phenomenon for the incompressible Navier--Stokes Equations (NSE) in dimension $n \geq 3$; that is, the strong solutions of NSE on $\mathbb{R}^n$ can be nicely approximated by those on…
In this article, we devote to the existence of an $N$-dimensional inertial manifold for the incompressible Navier-Stokes equations in $\mathbb{T}^{d}$ ($d=2,3$). Our results can be summarized as two aspects: Firstly, we construct an…
We investigate the three dimensional compressible Navier-Stokes and the continuity equations in Cartesian coordinates for Newtonian fluids. The polytropic equation of sate is used as closing condition. The key idea is the three-dimensional…
The dynamical large deviations principle for the three-dimensional incompressible Landau-Lifschitz-Navier-Stokes equations is shown, in the joint scaling regime of vanishing noise intensity and correlation length. This proves the…