Related papers: A posteriori estimates for Euler and Navier-Stokes…
This paper is devoted to the study of the Cauchy problem for the stratified Navier-Stokes system in space dimension three. In the first part of the paper, we prove the existence of a unique global solution $(v_\nu,\rho_\nu)$ for this system…
We consider the Cauchy problem for the incompressible Navier--Stokes equations on the whole space $\mathbb{R}^3$, with initial value $\vec u_0\in {\rm BMO}^{-1}$ (as in Koch and Tataru's theorem) and with force $\vec f=\Div \mathbb{F}$…
We establish the existence of infinitely many stationary solutions, as well as ergodic stationary solutions, to the three dimensional Navier--Stokes and Euler equations in both deterministic and stochastic settings, driven by additive…
In this paper, we prove that if $u\in C([0,\infty), \dot{H}^{1/2}_{a,1}(\mathbb{R}^3))$ is a global solution of 3D incompressible Navier-Stokes equations, then $\|u\|_{\dot{H}^{1/2}_{a,1}}$ decays to zero as time approaches infinity.…
We develop a Fourier-Chebyshev pseudospectral direct numerical simulation (DNS) to examine a potentially singular solution of the radially bounded, three-dimensional (3D), axisymmetric Euler equations [G. Luo and T.Y. Hou, Proc. Natl. Acad.…
A new discrete-velocity model is presented to solve the three-dimensional Euler equations. The velocities in the model are of an adaptive nature---both the origin of the discrete-velocity space and the magnitudes of the discrete-velocities…
We consider the Navier-Stokes Cauchy problem with an initial datum in a weighted Lebesgue space. The weight is a radial function increasing at infinity. Our study partially follows the ideas of the paper by G.P. Galdi and P. Maremonti "On…
The Cauchy problem for the Navier-Stokes equations with the Coriolis force is considered. It is proved that a similar a priori estimate, which is derived for the Navier-Stokes equations by Lei and Lin [11], holds under the effect of the…
We prove new \emph{a priori} estimates for the 3D Euler, the 3D Navier-Stokes and the 2D quasi-geostrophic equations by the method of similarity transforms.
We consider the 2D Navier-Stokes equation on $\mathbb T \times \mathbb R$, with initial datum that is $\varepsilon$-close in $H^N$ to a shear flow $(U(y),0)$, where $\| U(y) - y\|_{H^{N+4}} \ll 1$ and $N>1$. We prove that if $\varepsilon…
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to…
For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is…
We prove the existence and some moment estimates for an invariant measure $\mu$ for the two-dimensional ($2$D) deterministic Euler equations on the unbounded domain $\mathbb R^2$ and with highly regular initial data. The result is achieved…
In this paper we consider the initial value problem of the incompressible generalized Navier-Stokes equations in torus $\mathbb{T}^d$ with $d \geq 2$. The generalized Navier-Stokes equations is obtained by replacing the standard Laplacian…
We introduce a new modified Navier-Stokes model in $3$ dimensions by modifying the convection term in the ordinary Navier-Stokes equations. This is done by replacing the convective term $(\textbf{u}\cdot \nabla) \textbf{u}$ by…
In astrophysics and meteorology there exist numerous situations where flows exhibit small velocities compared to the sound speed. To overcome the stringent timestep restrictions posed by the predominantly used explicit methods for…
We study the Boltzmann equation on the $d$-dimensional torus in a perturbative setting around a global equilibrium under the Navier-Stokes linearisation. We use a recent functional analysis breakthrough to prove that the linear part of the…
In the note, various scenarios of potential Type II blowups of suitable weak solutions to the Navier-Stokes equations are studied. It is shown, that under some assumptions, such type of blowups cannot happen. In this case, corresponding…
A rigorous derivation of the incompressible Euler equations with the no-penetration boundary condition from the Boltzmann equation with the diffuse reflection boundary condition has been a challenging open problem. We settle this open…
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…