Evolutionary Navier Stokes Equations Research Articles

On the long-time behaviour of solutions to unforced evolution Navier–Stokes equations under Navier boundary conditions

We study the asymptotic behaviour of the solutions to Navier–Stokes unforced equations under Navier boundary conditions in a wide class of merely Lipschitz domains of physical interest. The paper draws its main motivation from celebrated results by Foias and Saut (1984) under Dirichlet conditions; here the choice of the boundary conditions requires carefully considering the geometry of the domain Ω, due to the possible lack of the Poincaré inequality in presence of symmetries. In non-axially symmetric domains we show the validity of the Foias–Saut result about the limit at infinity of the Dirichlet quotient, in axially symmetric domains we provide two invariants of the flow which completely characterize the motion and we prove that the Foias–Saut result holds for initial data belonging to one of the invariants.

The evolution Navier–Stokes equations in a cube under Navier boundary conditions: rarefaction and uniqueness of global solutions

We study the evolution Navier–Stokes equations in a cube under Navier boundary conditions. For the related stationary Stokes problem, we determine explicitly all the eigenvectors, eigenvalues and the corresponding Weyl asymptotic. We introduce the notion of rarefaction, namely families of eigenvectors that weakly interact with each other through the nonlinearity. By combining the spectral analysis with rarefaction, we expand the solutions in Fourier series, making explicit some of their properties. We then suggest several new points of view in order to explain the striking difference in uniqueness results between 2D and 3D. First, we construct examples of solutions for which the nonlinearity plays a minor role, both in 2D and 3D. Second, we show that, if a solution is rarefied, then its energy is decreasing: hence, rarefaction may be seen as an almost two dimensional assumption. Finally, by exploiting the explicit form of the eigenvectors we provide a numerical explanation of the difficulty in using energy methods for general solutions of the 3D equations.

A survey on some vanishing viscosity limit results

Abstract We present a survey concerning the convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier-Stokes equations to solutions of the Euler equations. After considering the Cauchy problem, particular attention is given to the convergence under Navier slip-type boundary conditions. We show that, in the presence of flat boundaries (typically, the half-space case), convergence holds, uniformly in time, with respect to the initial data’s norm. In spite of this result (and of a similar result for arbitrary two-dimensional domains), strong inviscid limit results are proved to be false in general domains, in correspondence to a very large family of smooth initial data. In Section 6, we present a result in this direction.

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 consider a full discretization in which the classical Crank–Nicolson method (θ-method with theta =1/2) is used to discretize the time variable. In contrast, 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 toward 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 alternate proofs of the existence of “physically relevant” solutions in three space dimensions.

Analysis of an exactly mass conserving space-time hybridized discontinuous Galerkin method for the time-dependent Navier–Stokes equations

We introduce and analyze a space-time hybridized discontinuous Galerkin method for the evolutionary Navier–Stokes equations. Key features of the numerical scheme include pointwise mass conservation, energy stability, and pressure robustness. We prove that there exists a solution to the resulting nonlinear algebraic system in two and three spatial dimensions, and that this solution is unique in two spatial dimensions under a small data assumption. A priori error estimates are derived for the velocity in a mesh-dependent energy norm.

Convergence to weak solutions of a space-time hybridized discontinuous Galerkin method for the incompressible Navier–Stokes equations

We prove that a space-time hybridized discontinuous Galerkin method for the evolutionary Navier–Stokes equations converges to a weak solution as the time step and mesh size tend to zero. Moreover, we show that this weak solution satisfies the energy inequality. To perform our analysis, we make use of discrete functional analysis tools and a discrete version of the Aubin–Lions–Simon theorem.

Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains

<p style='text-indent:20px;'>For the evolution Navier-Stokes equations in bounded 3D domains, it is well-known that the uniqueness of a solution is related to the existence of a regular solution. They may be obtained under suitable assumptions on the data and smoothness assumptions on the domain (at least <inline-formula><tex-math id="M1">\begin{document}$ C^{2,1} $\end{document}</tex-math></inline-formula>). With a symmetrization technique, we prove these results in the case of Navier boundary conditions in a wide class of merely <i>Lipschitz domains</i> of physical interest, that we call <i>sectors</i>.</p>

Well-posedness of evolutionary Navier-Stokes equations with forces of low regularity on two-dimensional domains

Existence and uniqueness of solutions to the Navier-Stokes equations in dimension two with forces in the space Lq((0, T); W−1,p(Ω)) for p and q in appropriate parameter ranges are proven. The case of spatially measured-valued forces is included. For the associated Stokes equation the well-posedness results are verified in arbitrary dimensions for any 1 < p, q < ∞.

Weak Solutions and Optimal Control of Hemivariational Evolutionary Navier-Stokes Equations under Rauch Condition

In this paper, we consider the evolutionary Navier-Stokes equations subject to the nonslip boundary condition together with a Clarke subdifferential relation between the dynamic pressure and the normal component of the velocity. Under the Rauch condition, we use the Galerkin approximation method and a weak precompactness criterion to ensure the convergence to a desired solution. Moreover, a control problem associated with such system of equations is studied with the help of a stability result with respect to the external forces. At the end of this paper, a more general condition due to Z. Naniewicz, namely the directional growth condition, is considered and all the results are reexamined.

Higher-order discontinuous Galerkin time discretizations for the evolutionary Navier–Stokes equations

Abstract Discontinuous Galerkin methods of higher order are applied as temporal discretizations for the transient Navier–Stokes equations. The spatial discretization based on inf–sup stable pairs of finite element spaces is stabilized using a one-level local projection stabilization method. Optimal error bounds for the velocity with constants independent of the viscosity parameter are obtained for both the semidiscrete case and the fully discrete case. Numerical results support the theoretical predictions.

A posteriori error estimations for mixed finite element approximations to the Navier–Stokes equations based on Newton-type linearization

In this paper we derive a posteriori error estimates for inf–sup stable mixed finite element approximations to the evolutionary Navier–Stokes equations. We reduce the problem of getting a posteriori error estimations of a non-linear evolutionary problem to that of getting a posteriori estimations of a linear steady problem. The main idea is based on the fact that the solutions of some Newton-type linearized Navier–Stokes equations around the plain Galerkin approximations approach the solution of the original Navier–Stokes equations with a bigger rate of convergence than the plain Galerkin method. As a consequence, the difference between the Galerkin approximations and the solution of the linearized problem is an estimator of the Galerkin error. Moreover, since the Galerkin approximation of the evolutionary Navier–Stokes equations is also the Galerkin approximation of the linearized equations, any a posteriori estimator of the error in the Newton-type linearized Navier–Stokes equations is also an a posteriori error estimator of the full Navier–Stokes equations.

Fully Discrete Approximations to the Time-Dependent Navier–Stokes Equations with a Projection Method in Time and Grad-Div Stabilization

This paper studies fully discrete approximations to the evolutionary Navier–Stokes equations by means of inf-sup stable $$H^1$$ -conforming mixed finite elements with a grad-div type stabilization and the Euler incremental projection method in time. We get error bounds where the constants do not depend on negative powers of the viscosity. We get the optimal rate of convergence in time of the projection method. For the spatial error we get a bound $$O(h^k)$$ for the $$L^2$$ error of the velocity, k being the degree of the polynomials in the velocity approximation. We prove numerically that this bound is sharp for this method.

On a two components condition for regularity of the 3D Navier–Stokes equations under physical slip boundary conditions on non-flat boundaries

This work concerns the sufficient condition for the regularity of solutions to the evolution Navier–Stokes equations known in the literature as Prodi–Serrin condition. H.-O. Bae and H. J. Choe proved in 1997 that, in the whole space $$\mathbb {R}^3,$$ it is sufficient that two components of the velocity satisfy the above condition in order to guarantee the regularity of solutions. In 2017, H. Beirao da Veiga extended this result (Beirao da Veiga, J Math Anal Appl 453:212–220, 2017) to the half-space case $$\mathbb {R}^n_+$$ under slip boundary conditions by assuming that the velocity components parallel to the boundary enjoy the above condition. It remained open whether the flat boundary geometry is essential. Below, we prove that, under physical slip boundary conditions imposed in cylindrical boundaries, the result still holds.

Error analysis of non inf-sup stable discretizations of the time-dependent Navier–Stokes equations with local projection stabilization

Abstract This paper studies non inf-sup stable finite element approximations to the evolutionary Navier–Stokes equations. Several local projection stabilization (LPS) methods corresponding to different stabilization terms are analyzed, thereby separately studying the effects of the different stabilization terms. Error estimates are derived in which the constants are independent of inverse powers of the viscosity. For one of the methods, using velocity and pressure finite elements of degree $l$, it will be proved that the velocity error in $L^\infty (0,T;L^2(\varOmega ))$ decays with rate $l+1/2$ in the case that $\nu \le h$, with $\nu$ being the dimensionless viscosity and $h$ being the mesh width. In the analysis of another method it was observed that the convective term can be bounded in an optimal way with the LPS stabilization of the pressure gradient. Numerical studies confirm the analytical results.

On the Truth, and Limits, of a Full Equivalence $${\mathbf{p \cong \,v^2 }}$$ p ≅ v 2 in the Regularity Theory of the Navier–Stokes Equations: A Point of View

The motivation at the origin of this note is the well known sufficient condition for regularity of solutions to the evolution Navier–Stokes equations, sometimes referred to in the literature as Ladyzhenskaya–Prodi–Serrin’s condition. Such a condition requires that the velocity field \(\,v\,\), alone, satisfies sufficiently strong integrability requirements in space–time. On the other hand, a relation like \(\,{p \cong \,|v|^2}\,\), with p pressure field, is loosely suggested by the Navier–Stokes equations themselves. In three papers published nearly 20 years ago we have considered this problem. The results obtained there immediately suggest new interesting questions. In this paper, we propose, and solve, some of them, while many other related problems remain still open.

Error estimates for the approximation of the velocity tracking problem with Bang-Bang controls

The velocity tracking problem for the evolutionary Navier–Stokes equations in 2d is studied. The controls are of distributed type but the cost functional does not involve the usual quadratic term for the control. As a consequence the resulting controls can be of bang-bang type. First and second order necessary and sufficient conditions are proved. A fully-discrete scheme based on discontinuous (in time) Galerkin approach combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, τ and h respectively, satisfy τ ≤ C h 2 , then L 2 error estimates are proved for the difference between the states corresponding to locally optimal controls and their discrete approximations.

Two-Grid Mixed Finite-Element Approximations to the Navier–Stokes Equations Based on a Newton-Type Step

A two-grid scheme to approximate the evolutionary Navier–Stokes equations is introduced and analyzed. A standard mixed finite element approximation is first obtained over a coarse mesh of size H at any positive time $$T>0$$ . Then, the approximation is postprocessed by means of solving a steady problem based on one step of a Newton iteration over a finer mesh of size $$h<H$$ . The method increases the rate of convergence of the standard Galerkin method in one unit in terms of H and equals the rate of convergence of the standard Galerkin method over the fine mesh h. However, the computational cost is essentially the cost of approaching the Navier–Stokes equations with the plain Galerkin method over the coarse mesh of size H since the cost of solving one single steady problem is negligible compared with the cost of computing the Galerkin approximation over the full time interval (0, T]. For the analysis we take into account the loss of regularity at initial time of the solution of the Navier–Stokes equations in the absence of nonlocal compatibility conditions. Some numerical experiments are shown.

Error Analysis of Projection Methods for Non inf-sup Stable Mixed Finite Elements: The Navier–Stokes Equations

We obtain error bounds for a modified Chorin-Teman (Euler non-incremental) method for non inf-sup stable mixed finite elements applied to the evolutionary Navier-Stokes equations. The analysis of the classical Euler non-incremental method is obtained as a particular case. We prove that the modified Euler non-incremental scheme has an inherent stabilization that allows the use of non inf-sup stable mixed finite elements without any kind of extra added stabilization. We show that it is also true in the case of the classical Chorin-Temam method. The relation of the methods with the so called pressure stabilized Petrov Galerkin method (PSPG) is established. We do not assume non-local compatibility conditions for the solution.

On the extension to slip boundary conditions of a Bae and Choe regularity criterion for the Navier–Stokes equations. The half-space case

This note concerns the sufficient condition for regularity of solutions to the evolution Navier–Stokes equations known in the literature as Prodi–Serrin's condition. H.-O. Bae and H.J. Choe proved in a 1997 paper that, in the whole space R3, it is merely sufficient that two components of the velocity satisfy the above condition. Below, we extend the result to the half-space case R+n under slip boundary conditions. We show that it is sufficient that the velocity component parallel to the boundary enjoys the above condition. Flat boundary geometry is not essential, as shown in a forthcoming paper in cylindrical domains, prepared in collaboration with J. Bemelmans and J. Brand.

Regularity Criteria in Weak &lt;i&gt;L&lt;/i&gt;&lt;sup&gt;3&lt;/sup&gt; for 3D Incompressible Navier-Stokes Equations

We study the regularity of a distributional solution (u, p) of the 3D incompressible evolution Navier-Stokes equations. Let Br denote concentric balls in R3 with radius r. We will show that if p ∈ Lm(0,1; L1(B2)), m > 2, and if u is sufficiently small in L∞(0,1; L3,∞(B2)), without any assumption on its gradient, then u is bounded in B1 × (1/10,1). It is an endpoint case of the usual Serrin-type regularity criteria, and extends the steady-state result of Kim-Kozono to the time dependent setting. In the appendix we also show some nonendpoint borderline regularity criteria.