Related papers: On weak solutions to the Navier-Stokes inequality …
This paper concerns the Cauchy problem in R^d for the stochastic Navier-Stokes equation \partial_tu=\Delta u-(u,\nabla)u-\nabla p+f(u)+ [(\sigma,\nabla)u-\nabla \tilde p+g(u)]\circ \dot W, u(0)=u_0,\qquad divu=0, driven by white noise \dot…
This paper focuses on the regularity of the Navier-Stokes equations in critical space. Let $ u(x,t) $ and $ p(x,t) $ denote suitable weak solution of the Navier-Stokes equations in $Q_T=\mathbb{R}^3\times(-T, 0)$. We prove that if $u(x,t)$…
We study the Cauchy problem for the (generalized) incompressible Navier-Stokes equations \begin{align} u_t+(-\Delta)^{\alpha}u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0. \nonumber \end{align} We show the analyticity of…
In this paper, inspired by the seminal work by Caffarelli-Kohn-Nirenberg \cite{CKN} on the incompressible Navier-Stokes equation, we establish the existence of a suitable weak solution to the Navier-Stokes-Planck-Nernst-Poisson equation in…
In this paper, we intend to reveal how the fractional dissipation $(-\Delta)^{\alpha}$ affects the regularity of weak solutions to the 3d generalized Navier-Stokes equations. Precisely, it will be shown that the $(5-4\alpha)/2\alpha$…
In this paper we consider the Navier-Stokes equations supplemented with either the Dirichlet or vorticity-based Navier boundary conditions. We prove that weak solutions obtained as limits of solutions to the Navier-Stokes-Voigt model…
A forced solution $v$ of the Navier-Stokes equation in any open domain with no slip boundary condition is constructed. The scaling factor of the forcing term is the critical order $-2$. The velocity, which is smooth until its final blow up…
The purpose of this paper is to prove the existence of global in time local energy weak solutions to the Navier-Stokes equations in the half-space $\mathbb R^3_+$. Such solutions are sometimes called Lemari\'e-Rieusset solutions in the…
In 1934, Leray proved the existence of global-in-time weak solutions to the Navier-Stokes equations for any divergence-free initial data in $L^2$. In the 1980s, Giga and Kato independently showed that there exist global-in-time mild…
The weak solution to the Navier-Stokes equations in a bounded domain $D \subset \mathbb{R}^3$ with a smooth boundary is proved to be unique provided that it satisfies an additional requirement. This solution exists for all $t \geq 0$. In a…
Different authors had received a lot of results regarding the Euler and Navier-Stokes equations. Existence and smoothness of solution for the Navier-Stokes equations in two dimensions have been known for a long time. Leray showed that the…
Classical vorticity solution branches of the three dimensional incompressible Euler equation are constructed where a velocity component can blow up at some point after finite time for regular data in H2. Furthermore, vorticity can blow up…
Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of solutions for the Navier-Stokes equations in two dimensions have been known for a long time. Leray showed…
We prove existence of global-in-time weak solutions of the incompressible Navier-Stokes equations in the half-space $\mathbb{R}^3_+$ with initial data in a weighted space that allow non-uniformly locally square integrable functions that…
In 1934 Leray proved that the Navier-Stokes equations have global weak solutions for initial data in $L^2(\mathbb{R}^N)$. In 1990 Calder\'on extended this result to the initial value spaces $L^p(\mathbb{R}^N)$ ($2\leq p<\infty$). In the…
We prove non-uniqueness for a class of weak solutions to the Navier-Stokes equations which have bounded kinetic energy, integrable vorticity, and are smooth outside a fractal set of singular times with Hausdorff dimension strictly less than…
In this paper, by invoking the appropriate decomposition of pressure to exploit the energy hidden in pressure, we present some new $\varepsilon$-regularity criteria for suitable weak solutions of the 3D Navier-Stokes equations at one scale:…
In this paper, we consider the energy conservation and regularity of the weak solution $u$ to the Navier-Stokes equations in the endpoint case. We first construct a divergence-free field $u(t,x)$ which satisfies $\lim_{t\to…
Based on Dou Huashu's energy gradient theory, this paper focuses on the weak singularity of the incompressible Navier-Stokes (NS) equations in steady, fully developed flows. When the gradient of total mechanical energy is perpendicular to…
Onsager's conjecture for the 3D Navier-Stokes equations concerns the validity of energy equality of weak solutions with regards to their smoothness. In this note we establish energy equality for weak solutions in a large class of function…