A note on regularity of weak solutions of the Navier-Stokes equations in R^n

In this paper we consider the n dimensional Navier-Stokes equations and we prove a new regularity criterion for weak solutions. More precisely, if n = 3,4 we show that the “smallness” of at least n-1 components of the velocity in L^infty(0,T;L_w(R^n)) is sufficient to ensure regularity of the weak solutions

An elementary approach to the 3D Navier-Stokes equations with Navier boundary conditions: Existence and uniqueness of various classes of solutions in the flat boundary case

We study with elementary tools the stationary 3D Navier-Stokes equations in a flat domain, equipped with Navier (slip without friction) boundary conditions. We prove existence and uniqueness of weak, strong, and very weak solutions in appropriate Banach spaces and most of the result hold true without restrictions on the size of the data. Results are partially known, but our approach allows us to give rather elementary and self-contained proofs

Some results on the two-dimensional dissipative Euler equations

We make a review of some recent results concerning special solutions and behavior at infinity for 2D dissipative Euler equations. In particular, we give a simplified proof --in the space-periodic setting-- of the uniform space/time boundedness of the first derivatives of the velocity, under suitable assumptions on the external force and on the dissipation (damping) coefficient. This is used to sketch the proof of existence of almost-periodic solutions

ASHEE: a compressible, equilibrium-Eulerian model for volcanic ash plumes

A new fluid-dynamic model is developed to numerically simulate the non-equilibrium dynamics of polydisperse gas-particle mixtures forming volcanic plumes. Starting from the three-dimensional N-phase Eulerian transport equations for a mixture of gases and solid particles, we adopt an asymptotic expansion strategy to derive a compressible version of the first-order non-equilibrium model, valid for low concentration regimes and small particles Stokes $St<0.2$. When $St < 0.001$ the model reduces to the dusty-gas one. The new model is significantly faster than the Eulerian model while retaining the capability to describe gas-particle non-equilibrium. Direct numerical simulation accurately reproduce the dynamics of isotropic turbulence in subsonic regime. For gas-particle mixtures, it describes the main features of density fluctuations and the preferential concentration of particles by turbulence, verifying the model reliability and suitability for the simulation of high-Reynolds number and high-temperature regimes. On the other hand, Large-Eddy Numerical Simulations of forced plumes are able to reproduce their observed averaged and instantaneous properties. The self-similar radial profile and the development of large-scale structures are reproduced, including the rate of entrainment of atmospheric air. Application to the Large-Eddy Simulation of the injection of the eruptive mixture in a stratified atmosphere describes some of important features of turbulent volcanic plumes, including air entrainment, buoyancy reversal, and maximum plume height. Coarse particles partially decouple from the gas within eddies, modifying the turbulent structure, and preferentially concentrate at the eddy periphery, eventually being lost from the plume margins due to the gravity. By these mechanisms, gas-particle non-equilibrium is able to influence the large-scale behavior of volcanic plumes.Comment: 29 pages, 22 figure

Weak solution to the Navier-Stokes equations constructed by semi-discretization are suitable

We consider the three dimensional Navier-Stokese quations and we prove that weak solutions constructed by approximating the time-derivative by finite differences are suitable. The so-called method of semi-discretization is of fundamental importance in the numerical analysis and it is one of the basic building blocks for the full discretization of the equations

Convergence of second-order in time numerical discretizations for the evolution Navier-Stokes equations

We prove the convergence of certain second-order numerical methods to weak solutions of the Navier-Stokes equations satisfying in addition the local energy inequality, and therefore suitable in the sense of Scheffer and Caffarelli-Kohn-Nirenberg. More precisely, we treat the space-periodic case in three space-dimensions and we consider a full discretization in which the the classical Crank-Nicolson method ($heta$-method with $heta=1/2$) is used to discretize the time variable, while in the space variables we consider finite elements. The convective term is discretized in several implicit, semi-implicit, and explicit ways. In particular, we focus on proving (possibly conditional) convergence of the discrete solutions towards weak solutions (satisfying a precise local energy balance), without extra regularity assumptions on the limit problem. We do not prove orders of convergence, but our analysis identifies some numerical schemes providing also alternate proofs of existence of ``physically relevant'' solutions in three space dimensions

On the Boussinesq equations with anisotropic filter in a vertical pipe

We propose a new Large Eddy Simulation (LES) model for the Boussinesq equations. We consider the motion in a three-dimensional domain with solid walls, and in a particular geometric setting we look for solutions which are periodic in the vertical direction and satisfy homogeneous Dirichlet conditions on the lateral boundary. We are thus modeling a vertical pipe and one main difficulty is that of considering regularizations of the equation which are well behaved also in presence of a boundary. The LES model we consider is then obtained by introducing a vertical filter, which is the natural one for the setting that we are considering. The related interior closure problem is treated in a standard way with a simplified-Bardina deconvolution model. The most technical analytical point is related to the fact that anisotropic filters provide less regularity than the isotropic ones and, in principle, the density term appearing in the Boussinesq equations may behave very differently from the velocity. We are able to define an appropriate notion of regular weak solution, for which we prove existence, uniqueness, and we also show that the energy associated to the model is exactly preserved

Convergence of approximate deconvolution models to the mean Navier-Stokes Equations

We consider a 3D Approximate Deconvolution Model ADM which belongs to the class of Large Eddy Simulation (LES) models. We aim at proving that the solution of the ADM converges towards a dissipative solution of the mean Navier-Stokes equations. The study holds for periodic boundary conditions. The convolution filter we first consider is the Helmholtz filter. We next consider generalized convolution filters for which the convergence property still hold

On the existence of Leray-Hopf Weak Solutions to the Navier-Stokes equations

We give a rather short and self contained presentation of the global existence results for Leray-Hopf weak solutions to the three dimensional incompressible Navier-Stokes equations. We give a unified treatment in terms of the domains and the relative boundary conditions and in terms of the approximation methods. Precisely, we consider the case of the whole space, the flat torus, and the case of a general bounded domain with smooth boundary and we consider as approximation schemes the Leray approximation method, the Faedo-Galerkin method, the semi-discretization in time and the approximation by adding a Smagorinsky term. We focus mainly on developing a unified treatment especially in the compactness argument needed to show that approximations converge to the weak solutions