Optimal Results for the Nonhomogeneous Initial-Boundary Value Problem for the Two-Dimensional Navier–Stokes Equations

In this work we study the fully nonhomogeneous initial boundary value problem for the two-dimensional time-dependent Navier–Stokes equations in a general open space domain in R² with low regularity assumptions on the initial and the boundary value data. We show that the perturbed Navier–Stokes operator is a diffeomorphism from a suitable function space onto its own dual and as a corollary we get that the Navier–Stokes equations are uniquely solvable in these spaces and that the solution depends smoothly on all involved data. Our source data space and solution space are in complete natural duality and in this sense, without any smallness assumptions on the data, we solve the equations for data with optimally low regularity in both space and time.     

Neural network solution to an inverse problem associated with the eigenvalues of the Stokes operator

A numerical method, based on the design of two artificial neural networks, is presented in order to approximate the viscosity and density features of fluids from the eigenvalues of the Stokes operator. The finite element method is used to solve the direct problem by training a first artificial neural network. A nonlinear map of eigenvalues of the Stokes operator as a function of the viscosity and density of the fluid under study is then obtained. This relationship is later inverted and refined by training a second artificial neural network, solving the aforementioned inverse problem. Numerical examples are presented in order to show the effectiveness and the limitations of this methodology.     

On a Resolvent Estimate of the Stokes Equation on an Infinite Layer Part 2, λ=: 0 Case

This paper is concerned with the standard L_p L_p estimate of solutions to the Stokes resolvent problem on an infinite layer in the case where 5 is close to zero (see eq.(1.1)). Combining this result with that in [1], we find that the Stokes operator on an infinite layer generates an analytic semigroup. As an application, we prove the local stability of some steady flows.     

Strong Unique Continuation for the Navier–Stokes Equation with Non-Analytic Forcing

We establish the strong unique continuation property for differences of solutions to the Navier–Stokes system with Gevrey forcing. For this purpose, we provide Carleman-type inequalities with the same singular weight for the Laplacian and the heat operator.     

Stability of Discrete Stokes Operators in Fractional Sobolev Spaces

Using a general approximation setting having the generic properties of finite-elements, we prove uniform boundedness and stability estimates on the discrete Stokes operator in Sobolev spaces with fractional exponents. As an application, we construct approximations for the time-dependent Stokes equations with a source term in L ^p (0, T; L ^q (Ω)) and prove uniform estimates on the time derivative and discrete Laplacian of the discrete velocity that are similar to those in Sohr and von Wahl [20]. On long leave from LIMSI (CNRS-UPR 3251), BP 133, 91403, Orsay, France.     

Homogenization of Stiff Plates and Two-Dimensional High-Viscosity Stokes Equations

The paper deals with the homogenization of stiff heterogeneous plates. Assuming that the coefficients are equi-bounded in L ¹, we prove that the limit of a sequence of plate equations remains a plate equation which involves a strongly local linear operator acting on the second gradients. This compactness result is based on a div-curl lemma for fourth-order equations. On the other hand, using an intermediate stream function we deduce from the plates case a similar result for high-viscosity Stokes equations in dimension two, so that the nature of the Stokes equation is preserved in the homogenization process. Finally, we show that the L ¹-boundedness assumption cannot be relaxed. Indeed, in the case of the Stokes equation the concentration of one very rigid strip on a line induces the appearance of second gradient terms in the limit problem, which violates the compactness result obtained under the L ¹-boundedness condition.     

Adiabatic shock capturing in perfect gas hypersonic flows

This paper considers the streamline‐upwind Petrov/Galerkin (SUPG) method applied to the compressible Euler and Navier–Stokes equations in conservation‐variable form. The spatial discretization, including a modified approach for interpolating the inviscid flux terms in the SUPG finite element formulation, is briefly reviewed. Of particular interest is the behavior of the shock‐capturing operator, which is required to regularize the scheme in the presence of strong, shock‐induced gradients. A standard shock‐capturing operator that has been widely used in previous studies by several authors is presented and discussed. Specific modifications are then made to this standard operator that is designed to produce a more physically consistent discretization in the presence of strong shock waves. The actual implementation of the term in a finite‐dimensional approximation is also discussed. The behavior of the standard and modified scheme is then compared for several supersonic/hypersonic flows. The modified shock‐capturing operator is found to preserve enthalpy in the inviscid portion of the flowfield substantially better than the standard operator. Published in 2009 by John Wiley & Sons, Ltd.     

Computational Aspects of Pseudospectra in Hydrodynamic Stability Analysis

This paper addresses the analysis of spectrum and pseudospectrum of the linearized Navier–Stokes operator from the numerical point of view. The pseudospectrum plays a crucial role in linear hydrodynamic stability theory and is closely related to the non-normality of the underlying differential operator and the matrices resulting from its discretization. This concept offers an explanation for experimentally observed instability in situations when eigenvalue-based linear stability analysis would predict stability. Hence the reliable numerical computation of the pseudospectrum is of practical importance particularly in situations when the stationary “base flow” is not analytically but only computationally given. The proposed algorithm is based on a finite element discretization of the continuous eigenvalue problem and uses an Arnoldi-type method involving a multigrid component. Its performance is investigated theoretically as well as practically at several two-dimensional test examples such as the linearized Burgers equations and various problems governed by the Navier–Stokes equations for incompressible flow.     

On Single Mode Forcing of the 2D-NSE

We examine how the global attractor of the 2D periodic Navier- Stokes equations projects in the energy–enstrophy-plane when the force is an eigenvector of the Stokes operator. We find sharper bounds than in Dascaliuc et al. (J Dyn Diff Equat 17:643–736, 2005) for a general force. We also find regions of the plane in which the energy and enstrophy must decrease, and calculate the curvature of the projected solution at certain initial data. Finally, we obtain restrictions on solutions that project onto a single point in the plane.     

On stabilized finite element methods for the Stokes problem in the small time step limit

Recent studies indicate that consistently stabilized methods for unsteady incompressible flows, obtained by a method of lines approach may experience difficulty when the time step is small relative to the spatial grid size. Using as a model problem the unsteady Stokes equations, we show that the semi‐discrete pressure operator associated with such methods is not uniformly coercive. We prove that for sufficiently large (relative to the square of the spatial grid size) time steps, implicit time discretizations contribute terms that stabilize this operator. However, we also prove that if the time step is sufficiently small, then the fully discrete problem necessarily leads to unstable pressure approximations. The semi‐discrete pressure operator studied in the paper also arises in pressure‐projection methods, thereby making our results potentially useful in other settings. Copyright © 2006 John Wiley & Sons, Ltd.     

Maximal L p – L q -Estimates for the Stokes Equation: a Short Proof of Solonnikov’s Theorem

Introducing a new localization method involving Bogovskiĭ's operator we give a short and new proof for maximal L^p – L^q-estimates for the solution of the Stokes equation. Moreover, it is shown that, up to constants, the Stokes operator is an R{\mathcal{R}}-sectorial operator in L^p_s(W)L^{p}_{\sigma}(\Omega), 1 < p < ￥1 < p < \infty, of R{\mathcal{R}}-angle 0, for bounded or exterior domains of Ω.     

Navier–Stokes Equations with Navier Boundary Conditions for an Oceanic Model

We consider the Navier–Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided the initial data belong to a “large” set in the Sobolev space H ¹. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the class of all strong solutions. This attractor is proved to be also the global attractor for the Leray–Hopf weak solutions of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear) inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits a short and simple proof of the existence of strong solutions for all time.     

Spectral Properties of the Linearization at the Burgers Vortex in the High Rotation Limit

We study a linearized operator of the equation for the axisymmetric Burgers vortex which gives a stationary solution to the three dimensional Navier–Stokes equations with an axisymmetric background straining flow. It is numerically known that the Burgers vortex obtains better stabilities as the circulation number (or the vortex Reynolds number) is increasing. Although the global stability of the axisymmetric Burgers vortex is already proved rigorously, mathematical understanding of this numerical observation has been lacking. In this paper we study a linearized operator that includes the circulation number as a parameter, and prove that if the operator is restricted on a suitable invariant subspace, then its spectrum moves to the left as the circulation number goes to infinity. As an application, we show that the axisymmetric Burgers vortex with a high rotation has a better stability, in the sense that the non-radially symmetric part of solutions to the associated evolution equation decays faster in time if the circulation number is sufficiently large.     

Dealing with pressure: FEM solution strategies for the pressure in the time‐dependent Navier–Stokes equations

This paper presents a survey of solution methods for calculating the pressure in the time‐dependent Navier–Stokes equations. The primary focus is on the treatment of the pressure‐Poisson equation deriving from index‐1 DAE formulations of the Navier–Stokes equations. Based on extensive operational experience with a variety of solution strategies, the combination of a stabilized pressure‐Poisson operator with an A‐conjugate projection and SSOR preconditioned conjugate gradient method has been found to yield the overall best performance relative to the resolve cost of a high performance direct solver. Copyright © 2002 John Wiley & Sons, Ltd.     

Three-Dimensional Instability of Planar Flows

We study the stability of two-dimensional solutions of the three-dimensional Navier–Stokes equations, in the limit of small viscosity. We are interested in steady flows with locally closed streamlines. We consider the so-called elliptic and centrifugal instabilities, which correspond to the continuous spectrum of the underlying linearized Euler operator. Through the justification of highly oscillating Wentzel–Kramers–Brillouin expansions, we prove the nonlinear instability of such flows. The main difficulty is the control of nonoscillating and nonlocal perturbations issued from quadratic interactions.     

The fundamental solution method for incompressible Navier–Stokes equations

A complete boundary integral formulation for incompressible Navier–Stokes equations with time discretization by operator splitting is developed using the fundamental solutions of the Helmholtz operator equation with different order. The numerical results for the lift and the drag hysteresis associated with a NACA0012 aerofoil oscillating in pitch show good agreement with available experimental data. © 1998 John Wiley & Sons, Ltd.     

Deconvolution‐based nonlinear filtering for incompressible flows at moderately large Reynolds numbers

We consider a Leray model with a deconvolution‐based indicator function for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under‐resolved meshes. For the implementation of the model, we adopt a three‐step algorithm called evolve–filter–relax that requires (i) the solution of a Navier–Stokes problem, (ii) the solution of a Stokes‐like problem to filter the Navier–Stokes velocity field, and (iii) a final relaxation step. We take advantage of a reformulation of the evolve–filter–relax algorithm as an operator‐splitting method to analyze the impact of the filter on the final solution versus a direct simulation of the Navier–Stokes equations. In addition, we provide some direction for tuning the parameters involved in the model based on physical and numerical arguments. Our approach is validated against experimental data for fluid flow in an idealized medical device (consisting of a conical convergent, a narrow throat, and a sudden expansion, as recommended by the U.S. Food and Drug Administration). Numerical results are in good quantitative agreement with the measured axial components of the velocity and pressures for two different flow rates corresponding to turbulent regimes, even for meshes with a mesh size more than 40 times larger than the smallest turbulent scale. After several numerical experiments, we perform a preliminary sensitivity analysis of the computed solution to the parameters involved in the model. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence,uniqueness and asymptotics of steady jets

We consider the three-dimensional flow through an aperture in a plane either with a prescribed flux or pressure drop condition. We discuss the existence and uniqueness of solutions for small data in weighted spaces and derive their complete asymptotic behaviour at infinity. Moreover, we show that each solution with a bounded Dirichlet integral, which has a certain weak additional decay, behaves like O(r ⁻²) as r=|x|→∞ and admits a wide jet region. These investigations are based on the solvability properties of the linear Stokes system in a half space ℝ ₊ ³ . To investigate the Stokes problem in ℝ ₊ ³ , we apply the Mellin transform technique and reduce the Stokes problem to the determination of the spectrum of the corresponding invariant Stokes-Beltrami operator on the hemisphere.     

Spectral galerkin approximation of couette-taylor flow

Axisymmetric Couette-Taylor flow between two concentric rotating cylinders was simulated numerically by the spectral method. First, stream function form of the Navier-Stokes equations which homogeneous boundary condition was given by introducing Couette flow. Second, the analytical expressions of the eigenfunction of the Stokes operator in the cylindrical gap region were given and its orthogonality was proved. The estimates of growth rate of the eigenvalue were presented. Finally, spectral Galerkin approximation of Couette-Taylor flow was discussed by introducing eigenfunctions of Stokes operator as basis of finite dimensional approximate subspaces. The existence, uniquence and convergence of spectral Galerkin approximation of nonsingular solution for the steady-state Navier-Stokes equations are proved. Moreover, the error estimates are given. Numerical result is presented.     

Stokes and Navier–Stokes Equations with Robin Boundary Conditions in a Half-Space

We study the initial-boundary value problem for the Stokes equations with Robin boundary conditions in the half-space It is proved that the associated Stokes operator is sectorial and admits a bounded H^∞-calculus on As an application we prove also a local existence result for the nonlinear initial value problem of the Navier–Stokes equations with Robin boundary conditions.