A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

Differential equations with non-standard growth have been a very active field of investigation in recent years. In this survey we present an overview of the field, as well as several of the most important results. We consider both existence and regularity questions. Finally, we provide a comprehensive list of papers published to date.

Overview of differential equations with non-standard growth

Cambridge University Press. Paperback. Book Condition: New. Paperback. 480 pages. Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This new edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems. This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN. Paperback.

A First Course in the Numerical Analysis of Differential Equations: Stiff equations

This paper presents a variational multiscale method (VMS) for the incompressible Navier--Stokes equations which is defined by a large scale space LH for the velocity deformation tensor and a turbulent viscosity $\nu_T$. The connection of this method to the standard formulation of a VMS is explained. The conditions on LH under which the VMS can be implemented easily and efficiently into an existing finite element code for solving the Navier--Stokes equations are studied. Numerical tests with the Smagorinsky large eddy simulation model for $\nu_T$ are presented.

A Finite Element Variational Multiscale Method for the Navier-Stokes Equations

Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity

We study the existence of the ground state
solution of the problem
$$
\cases
-\Delta u+V(x)u=f'(u) & x\in\mathbb R^N, \\ 
u(x)> 0,
\endcases
$$
under the assumption that
$\lim_{x\to\infty}V(x)=0$.

/pdf/existence-and-non-existence-of-the-ground-state-solution-for-2ronzwrx4i.pdf

Existence and non existence of the ground state solution for the nonlinear Schroedinger equations with $V(\infty)=0$

We study the existence of a solution of the problem 

$$ \left\{ \begin{gathered} - \Delta u + V\left( x \right)u = f'\left( u \right)x \in \mathbb{R}^N , \hfill \\ u\left( x \right) > 0, \hfill \\ \end{gathered} \right. $$

under the assumption that 

$$ \mathop {\lim }\limits_{\left| x \right| \to \infty } V\left( x \right) = 0 $$

where V > 0 and there is no ground state solution.

Existence of Solutions for the Nonlinear Schrödinger Equation with V (∞) = 0

Given a multiple-valued function f, we deal with the problem of selecting its single-valued branches. This problem can be stated in a rather abstract setting considering a metric space E and a finite group G of isometries of E. Given a function f, which takes values in the equivalence classes of E/G, the problem consists of finding a map g with the same domain as f and taking values in E, such that at every point t the equivalence class of g(t) coincides with f(t). If the domain of f is an interval, we show the existence of a function g with these properties which, moreover, has the same modulus of continuity of f. In the particular case where E is the product of Q copies of ℝ
 n
 and G is the group of permutations of Q elements, it is possible to introduce a notion of differentiability for multiple-valued functions. In this case, we prove that the function g can be constructed in such a way to preserve C
 k,α regularity. Some related problems are also discussed.

/pdf/regular-selections-for-multiple-valued-functions-1siunbigti.pdf

Regular selections for multiple-valued functions

On the C1,γ(Ω¯)∩W2,2(Ω) regularity for a class of electro-rheological fluids

We consider a stationary Navier–Stokes system with shear dependent viscosity, under Dirichlet boundary conditions. We prove Holder continuity, up to the boundary, for the gradient of the velocity field together with the L
 2-summability of the weak second derivatives. The results hold under suitable smallness assumptions on the force term and without any restriction on the range of $$p \in (1, 2)$$
 .

Carlo Romano Grisanti

Papers

Existence and non existence of the ground state solution for the nonlinear Schroedinger equations with $V(\infty)=0$

Existence of Solutions for the Nonlinear Schrödinger Equation with V (∞) = 0

Regular selections for multiple-valued functions

On the C1,γ(Ω¯)∩W2,2(Ω) regularity for a class of electro-rheological fluids

On the Existence, Uniqueness and \(C^{1,\gamma} (\bar{\Omega}) \cap W^{2,2}(\Omega)\) Regularity for a Class of Shear-Thinning Fluids