### A Combined Finite Element and Marker and Cell Method for Solving Navier-Stokes Equations.

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Let $X$ be a compact subset of a separable Hilbert space $H$ with finite fractal dimension ${d}_{F}\left(X\right)$, and ${P}_{0}$ an orthogonal projection in $H$ of rank greater than or equal to $2{d}_{F}\left(X\right)+1$. For every $\delta >0$, there exists an orthogonal projection $P$ in $H$ of the same rank as ${P}_{0}$, which is injective when restricted to $X$ and such that $\parallel P-{P}_{0}\parallel <\delta $. This result follows from Mañé’s paper. Thus the inverse ${\left(P{|}_{X}\right)}^{-1}$ of the restricted mapping ${P|}_{X}\phantom{\rule{0.222222em}{0ex}}X\to PX$ is well defined. It is natural to ask whether there exists a universal modulus of continuity for the inverse of Mañé’s...

For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present...

In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair (u,p) of suitable weak solutions to the Navier-Stokes equations in ℝ³ × ]0,∞[ the velocity field u satisfies the following property of partial regularity: The velocity u is Lipschitz continuous in a neighbourhood of a point (x₀,t₀) ∈ Ω × ]0,∞ [ if $limsu{p}_{R\to 0\u207a}1/R{\int}_{{Q}_{R}(x\u2080,t\u2080)}|curlu\times u/|u\left|\right|\xb2dxdt\le {\epsilon}_{*}$ for a sufficiently small ${\epsilon}_{*}>0$.

We investigate a finite element discretization of the Stokes equations with nonstandard boundary conditions, defined in a bounded three-dimensional domain with a curved, piecewise smooth boundary. For tetrahedral triangulations of this domain we prove, under general assumptions on the discrete problem and without any additional regularity assumptions on the weak solution, that the discrete solutions converge to the weak solution. Examples of appropriate finite element spaces are given.

A standard method for proving the inf-sup condition implying stability of finite element approximations for the stationary Stokes equations is to construct a Fortin operator. In this paper, we show how this can be done for two-dimensional triangular and rectangular Taylor-Hood methods, which use continuous piecewise polynomial approximations for both velocity and pressure.

We study a two-grid scheme fully discrete in time and space for solving the Navier-Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and linearized around the velocity uH computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of uH...

We generalize a classical result of T. Kato on the existence of global solutions to the Navier-Stokes system in C([0,∞);L3(R3)). More precisely, we show that if the initial data are sufficiently oscillating, in a suitable Besov space, then Kato's solution exists globally. As a corollary to this result, we obtain a theory of existence of self-similar solutions for the Navier-Stokes equations.

We deal with a suitable weak solution $(\mathbf{v},p)$ to the Navier-Stokes equations in a domain $\Omega \subset {\mathbb{R}}^{3}$. We refine the criterion for the local regularity of this solution at the point $(\mathbf{f}{x}_{0},{t}_{0})$, which uses the ${L}^{3}$-norm of $\mathbf{v}$ and the ${L}^{3/2}$-norm of $p$ in a shrinking backward parabolic neighbourhood of $({\mathbf{x}}_{0},{t}_{0})$. The refinement consists in the fact that only the values of $\mathbf{v}$, respectively $p$, in the exterior of a space-time paraboloid with vertex at $({\mathbf{x}}_{0},{t}_{0})$, respectively in a ”small” subset of this exterior, are considered. The consequence is that...

The paper is dedicated to the global well-posedness of the barotropic compressible Navier-Stokes-Poisson system in the whole space ${\mathbb{R}}^{N}$ with N ≥ 3. The global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces. The initial velocity has the same critical regularity index as for the incompressible homogeneous Navier-Stokes equations. The proof relies on a uniform estimate for a mixed hyperbolic/parabolic linear system with a convection term.

We recover the Navier–Stokes equation as the incompressible limit of a stochastic lattice gas in which particles are allowed to jump over a mesoscopic scale. The result holds in any dimension assuming the existence of a smooth solution of the Navier–Stokes equation in a fixed time interval. The proof does not use nongradient methods or the multi-scale analysis due to the long range jumps.

In this work we study a fully discrete mixed scheme, based on continuous finite elements in space and a linear semi-implicit first-order integration in time, approximating an Ericksen–Leslie nematic liquid crystal model by means of a Ginzburg–Landau penalized problem. Conditional stability of this scheme is proved via a discrete version of the energy law satisfied by the continuous problem, and conditional convergence towards generalized Young measure-valued solutions to the Ericksen–Leslie problem...

This article is dedicated to localization of the principal eigenvalue (PE) of the Stokes operator acting on solenoidal vector fields that vanish outside a large random domain modeling the pore space in a cubic block of porous material with disordered micro-structure. Its main result is an asymptotically deterministic lower bound for the PE of the sum of a low compressibility approximation to the Stokes operator and a small scaled random potential term, which is applied to produce a similar bound...