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

We prove local existence and uniqueness for the two-dimensional Prandtl system in weighted Sobolev spaces under Oleinik's monotonicity assumption. In particular we do not use the Crocco transform or any change of variables. Our proof is based on a new nonlinear energy estimate for the Prandtl system. This new energy estimate is based on a cancellation property that is valid under the monotonicity assumption. To construct the solution, we use a regularization of the system that preserves this nonlinear structure. This new nonlinear structure may give some insight into the convergence properties from the Navier-Stokes system to the Euler system when the viscosity goes to 0. © 2015 Wiley Periodicals, Inc.

Local-in-Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods

We prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the Navier–Stokes equation with the Navier boundary condition has a strong solution. This solution is uniformly bounded in a conormal Sobolev space and has only one normal derivative bounded in L∞. This allows us to obtain the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument.

Uniform regularity for the Navier-Stokes equation with Navier boundary condition

Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on the data and on the boundary of the domain is assumed. The case of arbitrary bounded convex domains is also included.

Global Boundedness of the Gradient for a Class of Nonlinear Elliptic Systems

We consider the evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, and study the convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We obtain quite sharp results in the 2-D and 3-D cases. However, in the 3-D case, we need to assume that the boundary is flat.

Sharp Inviscid Limit Results under Navier Type Boundary Conditions. An L p Theory

This note is concerned with some interpolation inequalities of Gagliardo-Nirenberg type. In [4, 13] it is proved that if V%R is a bounded domain having the cone property or if V4R , then any function u belonging to L q (V) with derivatives D a u , a multi-index, belonging to L p (V), q , pF1, satisfies the following inequality: for m4NaN and j40, R , m21 ND j uNL r (V) GC1 ND m uNL p (V) a NuNL q (V) 12a 1C2 NuNL q (V) () , (1.1)

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An Interpolation Inequality in Exterior Domains.

We consider the 3-D evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, see (1.2), and study the problem of the strong convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We prove here, in the flat boundary case, convergence in Sobolev spaces W
 k, p
 (Ω), for arbitrarily large k and p (for previous results see Xiao and Xin in Comm Pure Appl Math 60:1027–1055, 2007 and Beirao da Veiga and Crispo in J Math Fluid Mech, 2009, doi:
 10.1007/s00021-009-0295-4
 ). However this problem is still open for non-flat, arbitrarily smooth, boundaries. The main obstacle consists in some boundary integrals, which vanish on flat portions of the boundary. However, if we drop the convective terms (Stokes problem), the inviscid, strong limit result holds, as shown below. The cause of this different behavior is quite subtle. As a by-product, we set up a very elementary approach to the regularity theory, in L
 p
 -spaces, for solutions to the Navier–Stokes equations under slip type boundary conditions.

Concerning the W k,p -Inviscid Limit for 3-D Flows Under a Slip Boundary Condition

We show that, in general, the solutions to the initial-boundary value problem for the Navier-Stokes equations under a widely adopted Navier-type slip boundary condition do not converge, as the viscosity goes to zero, to the solution of the Euler equations under the classical zero-flux boundary condition, and same smooth initial data, in any arbitrarily small neighborhood of the initial time. Convergence does not hold with respect to any space-topology which is sufficiently strong as to imply that the solution to the Euler equations inherits the complete slip type boundary condition. In our counter-example Ω is a sphere, and the initial data may be infinitely differentiable. The crucial point here is that the boundary is not flat. In fact (see Beirao da Veiga et al. in J Math Anal Appl 377:216–227, 2011) if $${\,\Omega = \mathbb R^3_+,}$$
 convergence holds in $${C([0,T]; W^{k,p}(\mathbb R^3_+))}$$
 , for arbitrarily large k and p. For this reason, the negative answer given here was not expected.

The 3-D Inviscid Limit Result Under Slip Boundary Conditions. A Negative Answer

We consider the Dirichlet boundary value problem for nonlinear N -systems of partial differential equations with p -growth, 1 p ≤ 2 , in the n -dimensional case. For clearness, we confine ourselves to a particularly representative case, the well known p -Laplacian system. We are interested in regularity results, up to the boundary, for the second order derivatives of the solution. We prove W 2 , q -global regularity results, for arbitrarily large values of q . In turn, the regularity achieved implies the Holder continuity of the gradient of the solution. It is worth noting that we cover the singular case μ = 0 . See Theorem 2.1 .

Francesca Crispo

Papers

Sharp Inviscid Limit Results under Navier Type Boundary Conditions. An L p Theory

An Interpolation Inequality in Exterior Domains.

Concerning the W k,p -Inviscid Limit for 3-D Flows Under a Slip Boundary Condition

The 3-D Inviscid Limit Result Under Slip Boundary Conditions. A Negative Answer

On the global W2,q regularity for nonlinear N-systems of the p-Laplacian type in n space variables