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In the presence of vacuum, the physical entropy for polytropic gases behaves singularly and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time…
A fundamental open problem in the theory of the multidimensional compressible Navier-Stokes equations is whether smooth solutions can develop singularities in finite time. For constant viscosity coefficients, recent remarkable results show…
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…
We develop the asymptotic behavior for the solutions to the stationary Navier-Stokes equation in the exterior domain of the 2D hyperbolic space. More precisely, given the finite Dirichlet norm of the velocity, we show the velocity decays to…
In this paper, we are concerned with the minimal regularity of both the density and the velocity for the weak solutions keeping energy equality in the isentropic compressible Navier-Stokes equations. The energy equality criteria without…
This is the first of two papers concerning the asymptotic behavior of the incompressible Navier-Stokes equations in a half-space at high Reynolds numbers, with initial data given by a point vortex. In the present work, we establish the…
We discuss optimal estimates of solutions to the compressible Navier-Stokes equations in Besov norms. In particular, we consider the estimate of the curl-free part of the solution to the linearised equations, in the homogeneous case. We…
Bounding curves in the energy,enstrophy-plane are derived for the 3D Navier-Stokes equations under an assumption on coherence of the vorticity direction. The analysis in the critical case where the direction is H\"older continuous with…
In the vanishing viscosity limit from the Navier-Stokes to Euler equations on domains with boundaries, a main difficulty comes from the mismatch of boundary conditions and, consequently, the possible formation of a boundary layer. Within a…
The asymptotic behavior of the vorticity for the steady incompressible Navier-Stokes equations in a two-dimensional exterior domain is described in the case where the velocity at infinity $\boldsymbol{u}_{\infty}$ is nonzero. It is well…
The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for…
Assuming that initial velocity and initial vorticity are bounded in the plane, we show that on a sufficiently short time interval the unique solutions of the Navier-Stokes equations converge uniformly to the unique solution of the Euler…
A linear stability analysis of the Navier-Stokes (NS) granular hydrodynamic equations is performed to determine the critical length scale for the onset of vortices and clusters instabilities in granular dense binary mixtures. In contrast to…
This paper is devoted to studying the Cauchy problem for the three-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosities given by $\mu=\rho^\alpha,\lambda=\rho^\alpha(\alpha>0)$. We establish the…
We consider the global existence and large-time asymptotic behavior of strong solutions to the Cauchy problem of the three-dimensional nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity and vacuum. We…
In the recent paper, the global-in-time inviscid limit of the three-dimensional (3D) isentropic compressible Navier-Stokes equations is considered. First, when viscosity coefficients are given as a constant multiple of density's power…
The swimming of a deformable planar slab in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations. A continuum of plane wave displacements, symmetric on both sides of the slab and characterized by a…
Starting from isentropic compressible Navier-Stokes equations with growth term in the continuity equation, we rigorously justify that performing an incompressible limit one arrives to the two-phase free boundary fluid system.
We consider the 2D incompressible Navier-Stokes equations on $\mathbb{T}\times \mathbf{R}$, with initial vorticity that is $\delta$ close in $H^{log}_xL^2_{y}$ to $-1$(the vorticity of the Couette flow $(y,0)$). We prove that if $\delta\ll…
We study the regularity criteria for weak solutions to the $3D$ incompressible Navier--Stokes equations in terms of the geometry of vortex structures, taking into account the boundary effects. A boundary regularity theorem is proved on…