Related papers: On the critical one component regularity for 3-D N…

In this paper, we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum in $\mathbb{R}$, where the viscosity depends on the density in a super-linear power law(i.e.,…

We are concerned with the construction of global-in-time strong solutions for the incompressible Vlasov-Navier-Stokes system in the whole three-dimensional space. One of our goals is to establish that small initial velocities with critical…

We prove the nonexistence of local self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The local self-similar solutions we consider here are different from the global self-similar solutions. The…

We study the dyadic model of the Navier-Stokes equations introduced by Katz and Pavlovi\'c. They showed a finite time blow-up in the case where the dissipation degree $\alpha$ is less than 1/4. In this paper we prove the existence of weak…

In the present paper, we study the fractional incompressible Stochastic Navier-Stokes equation on $\mathbb{R}^2$, formally defined as \[ \partial_t v = -\tfrac12 (-\Delta)^\theta v - \lambda v \cdot \nabla v + \nabla p - \nabla^{\perp}…

Considering initial data in $\dot{H}^s$, with $\frac{1}{2} \textless{} s \textless{} \frac{3}{2}$, this paper is devoted to the study of possible blowing-up Navier-Stokes solutions such that $(T*(u\_{0}) -t)^{\frac{1}{2} (s- \frac{1}{2})}…

The Navier-Stokes equation on the Euclidean space $\mathbf{R}^3$ can be expressed in the form $\partial_t u = \Delta u + B(u,u)$, where $B$ is a certain bilinear operator on divergence-free vector fields $u$ obeying the cancellation…

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…

We study the finite time blow up of smooth solutions to the Compressible Navier-Stokes system when the initial data contain vacuums. We prove that any classical solutions of viscous compressible fluids without heat conduction will blow up…

We demonstrate finite-time blow-up in a simple, realistic shell model of the 3D Navier-Stokes equations, equipped with "smooth" (i.e., rapidly decaying in frequency) initial data and forcing. Previously studied models either exhibit a…

In this paper we consider the Cauchy problem for the 3D Navier-Stokes equations for incompressible flows. The initial data are assumed to be smooth and rapidly decaying at infinity. A famous open problem is whether classical solutions can…

We consider Cauchy problem for Fourier transformation of 3-dimensional Navier-Stokes system with zero external force. Using initial data purposed by Dong Li and Ya.G.Sinai we implement self-similar regime producing fast growing behavior of…

We consider the conditional regularity of mild solution $v$ to the incompressible Navier-Stokes equations in three dimensions. Let $e \in \mathbb{S}^2$ and $0 < T^\ast < \infty$. J. Chemin and P. Zhang \cite{CP} proved the regularity of $v$…

Under assumption that $T^{\ast}$ is the maximal time of existence of smooth solution of the 3D Navier-Stokes equations in the Sobolev space $H^{s}$, we establish lower bounds for the blow-up rate of the type$\ \left( T^{\ast }-t\right)…

Perturbed, helical trefoil vortex knots and a set of anti-parallel vortices are examined numerically to identify the scaling of their helicity and vorticity norms during reconnection. For the volume-integrated enstrophy $Z=\int\omega^2 dV$,…

Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the…

We revisit the regularity theory of Escauriaza, Seregin, and \v{S}ver\'ak for solutions to the three-dimensional Navier-Stokes equations which are uniformly bounded in the critical $L^3_x(\mathbf{R}^3)$ norm. By replacing all invocations of…

In the studies of the Navier-Stokes (NS) regularity problem, it has become increasingly clear that a more realistic path to improved a priori bounds is to try to break away from the scaling of the energy-level estimates in the realm of the…

In this paper we prove a blow-up criterion for the compressible Navier-Stokes-Fourier system for general thermal and caloric equations of state with inhomogeneous boundary conditions for the velocity and the temperature. Assuming only that…

We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in $L^1(R^2)$ for positive times is entirely determined by the trace of the vorticity at $t = 0$, which is a finite…