Related papers: On the possible time singularities for the 3D Navi…

We prove existence of a weak solution to the three-dimensional Beris--Edwards system in the whole space under the stable bulk assumption $c>0$. The solution satisfies the natural bounds $Q\in L^\infty_tH^1_x\cap L^2_tH^2_x$ and $u\in…

We consider the 2-D incompressible Euler equations in a bounded domain and show that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity…

We prove, with a more geometric approach, that the solutions to the Navier-Stokes equations are regular up to a set of Hausdorff dimension 1. The main tool for the proof is a new compactness lemma and the monotonicity property of harmonic…

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…

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…

In this paper, we improve some known uniqueness results of weak solutions for the 3D Navier-Stokes equations. The proof uses the Fourier localization technique and the losing derivative estimates.

We study the regularity of a distributional solution $(u,p)$ of the 3D incompressible evolution Navier-Stokes equations. Let $B_r$ denote concentric balls in $\mathbb{R}^3$ with radius $r$. We will show that if $p\in L^{m} (0,1; L^1(B_2))$,…

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…

Higher moments of the vorticity field $\Omega_{m}(t)$ in the form of $L^{2m}$-norms ($1 \leq m < \infty$) are used to explore the regularity problem for solutions of the three-dimensional incompressible Navier-Stokes equations on the domain…

Navier-Stokes equations in the whole space R^3 subject to an anisotropic viscosity and a random perturbation of multiplicative type is described. By adding a term of Brinkman-Forchheimer type to the model, existence and uniqueness of global…

In this paper, we are concerned with the partial regularity of the suitable weak solutions to the fractional MHD equations in $\mathbb{R}^{n}$ for $n=2,\,3$. In comparison with the work of the 3D fractional Navier-Stokes equations obtained…

We study upper bounds on the box-counting dimension of the set of potential singular points in suitable weak solutions to the 3D incompressible hyperdissipative Navier-Stokes system \begin{equation*} \partial_t u +…

In this paper, we study the regularity problem of the 3D incompressible Navier\~nStokes equations. We prove that the strong solution exists globally for new regularity criteria. For negligible forces, we give an improvement of the known…

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…

We investigate the three-dimensional incompressible Navier-Stokes equations. The equations are discretized with Fourier spectral method and a fourth-order Runge-Kutta scheme in time. The spectral accuracy, resolution conditions, and an…

In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop…

The uniqueness of Leray-Hopf solutions to the incompressible Navier-Stokes equations remains a significant open question in fluid mechanics. This paper proposes a potential mechanism for non-uniqueness, illustrated in a natural dyadic shell…

We prove that every weak solution $u$ to the 3D Navier-Stokes equation that belongs to the class $L^3L^{9/2}$ and $\n u$ belongs to $L^3L^{9/5}$ localy away from a 1/2-H\"{o}lder continuous curve in time satisfies the generalized energy…

We prove a general result that implies that very weak solutions to the Cauchy problem for the Navier-Stokes equations must be, in fact, Leray-Hopf solutions if only their initial data are (solenoidal) with finite kinetic energy.

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…