Related papers: The Navier-Stokes equations in the critical Lebesg…
Consider the steady solution to the incompressible Euler equation $\bar u=Ae_1$ in the periodic tunnel $\Omega=\mathbb T^{d-1}\times(0,1)$ in dimension $d=2,3$. Consider now the family of solutions $u^\nu$ to the associated Navier-Stokes…
We show that finite-energy weak solutions to the incompressible Navier--Stokes equations on a three-dimensional bounded smooth domain are regular up to the boundary, provided that the $L^4_tL^4_x$-norm of the solution is smaller than a…
In this paper we study Liouville properties of smooth steady axially symmetric solutions of the Navier-Stokes equations. First, we provide another version of the Liouville theorem of \cite{kpr15} in the case of zero swirl, where we replaced…
We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent $\alpha$ can be larger than the Lions exponent $5/4$. It is well-known that, due to Lions [55], for any $L^2$ divergence-free initial data,…
For the N>=2 dimensional incompressible Naver-Stokes Equation, We have got its solution as a power series of time t, in which the coefficients are all known functions determined only by the initial velocity v0. We also prove that the…
In this paper we study the periodic Navier--Stokes equation. From the periodic Navier--Stokes equation and the linear equation $\partial_t u = \nu\Delta u + \mathbb{P} [v\nabla u]$ we derive the corresponding equations for the time…
In a plane polygon $P$ with straight sides, we prove analytic regularity of the Leray-Hopf solution of the stationary, viscous, and incompressible Navier-Stokes equations. We assume small data, analytic volume force and no-slip boundary…
This article offers a modern perspective which exposes the many contributions of Leray in his celebrated work on the Navier--Stokes equations from 1934. Although the importance of his work is widely acknowledged, the precise contents of his…
In this paper, we investigate the three dimensional stationary compressible Navier-Stokes equations, and obtain Liouville type theorems if a smooth solution $(\rho, \mathbf{u})$ satisfies some suitable conditions. In particular, our results…
We positively answer Question 2.2 and Question 2.3 in [Bru\`e, De Lellis, 2023] in dimension $4$ by building new examples of solutions to the forced $4d$ incompressible Navier-Stokes equations, which exhibit anomalous dissipation, related…
First, we show that if the pressure $p$ (associated to a weak Leray-Hopf solution $v$ of the Navier-Stokes equations) satisfies $\|p\|_{L^{\infty}_{t}(0,T^*; L^{\frac{3}{2},\infty}(\mathbb{R}^3))}\leq M^2$, then $v$ possesses higher…
We prove quantitative regularity and blowup theorems for the incompressible Navier-Stokes equations in $\mathbb R^d$, $d\geq4$ when the solution lies in the critical space $L_t^\infty L_x^d$. Explicit subcritical bounds on the solution are…
We consider the Cauchy problem for the incompressible Navier-Stokes equations in dimension three and construct initial data in the critical space $BMO^{-1}$ from which there exist two distinct global solutions, both smooth for all $t>0$.…
We study the Navier-Stokes equations with an extra eddy viscosity term in the whole space in three dimensions. We introduce a suitable regularized system for which we prove the existence of a regular solution defined for all time. We prove…
We consider the Navier-Stokes equations in $\mathbb R^d$ ($d=2,3$) with a stochastic forcing term which is white noise in time and coloured in space; the spatial covariance of the noise is not too regular, so It\^o calculus cannot be…
Several types of new regularity criteria for Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. Some of them are based on the third component $u_3$ of velocity under Prodi-Serrin index condition, another type is…
We study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier-Stokes equations in the super-critical spaces $L_{t}^{\gamma}L_{x}^{p}$ when $\alpha\in[1,\frac{3}{2})$, and obtain the conclusion that the…
We consider the 3-D Navier-Stokes initial value problem, $$ v_t - \nu \Delta v = -\mathcal{P} [ v \cdot \nabla v ] + f , v(x, 0) = v_0 (x), x \in \mathbb{T}^3 (*) $$ where $\mathcal{P}$ is the Hodge projection. We assume that the Fourier…
We consider any cover $\mathscr{C}$ of $\mathbb{R}^3$ by balls of radius bigger or equal $1$ satisfying two conditions: (i) any ball intersects at most $\sigma>0$ other balls, and (ii) intersecting balls have comparable sizes. We consider a…
In this paper we will prove that the vorticity belongs to L1(0; T ; L2(R3)) for the Cauchy problem of 3D incompressible Navier-Stokes equation, then the existence of a global smooth solution is obtained. Our approach is to construct a set…