Leray Solutions Research Articles

A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation

We consider the Navier-Stokes system with Oseen and rotational terms describing the stationary flow of a viscous incompressible fluid around a rigid body moving at a constant velocity and rotating at a constant angular velocity. In a previous paper, we proved a representation formula for Leray solutions of this system. Here the representation formula is used as starting point for splitting the velocity into a leading term and a remainder, and for establishing pointwise decay estimates of the remainder and its gradient.

Exponential Decay of the Vorticity in the Steady-State Flow of a Viscous Liquid Past a Rotating Body

Consider the flow of a Navier–Stokes liquid past a body rotating with a prescribed constant angular velocity, \({\omega}\), and assume that the motion is steady with respect to a body-fixed frame. In this paper we show that the vorticity field associated to every “weak” solution corresponding to data of arbitrary “size” (Leray Solution) must decay exponentially fast outside the wake region at sufficiently large distances from the body. Our result improves and generalizes in a non-trivial way famous results by Clark (Indiana Univ Math J 20:633–654, 1971) and Babenko and Vasil’ev (J Appl Math Mech 37:651–665, 1973) obtained in the case \({\omega=0}\).

A regularity criterion for the Navier–Stokes equations based on the gradient of one velocity component

We show that if u is a Leray solution to the Navier–Stokes equations in the full three-dimensional space with an initial condition from W0,σ1,2, T>0 and u∈Lt(0,T;Ls), where 2/t+3/s=59/30 for s∈(2,30/13] and 2/t+3/s=7/4+1/(2s) for s∈(30/13,3) then u is regular on (0,T). We prove our result as a special case of a more general method which might possibly bring a further improvement.

A regularity criterion of Serrin-type for the Navier–Stokes equations involving the gradient of one velocity component

Abstract In the present paper we prove that a weak Leray solution to the Navier–Stokes equations in ℝ 3 × ( 0 , T ] $\mathbb {R}^3\times (0,T]$ is regular provided the gradient of one component belongs to the Prodi–Serrin class ∇ u 3 ∈ L 4 ( 0 , T ; L 2 ) $\nabla u^3 \in L^4(0,T; L^2)$ .

3D viscous incompressible fluid around one thin obstacle

In this article, we consider Leray solutions of the Navier-Stokes equations in the exterior of one obstacle in 3D and we study the asymptotic behavior of these solutions when the obstacle shrinks to a curve or to a surface. In particular, we will prove that a solid curve has no effect on the motion of a viscous fluid, so it is a removable singularity for these equations.

A Derivation of the Magnetohydrodynamic System from Navier–Stokes–Maxwell Systems

We provide a full and rigorous derivation of the standard viscous magnetohydrodynamic system (MHD) as the asymptotic limit of Navier–Stokes–Maxwell systems when the speed of light is infinitely large. We work in the physical setting provided by the natural energy bounds and therefore mainly consider Leray solutions of fluid dynamical systems. Our methods are based on a direct analysis of frequencies and we are able to establish the weak stability of a crucial nonlinear term (the Lorentz force), neither assuming any strong compactness of the components nor applying standard compensated compactness methods (which actually fail in this case).

A note on helicity conservation in Leray models of incompressible flow

We prove conservation of a regularized helicity HL:=∫Ωu⋅wdx for the Leray model (and its variants) of turbulent flow, where w is the solution of a Leray-regularized vorticity equation. The usual definition of helicity is H=∫Ωu⋅(∇×u)dx, which is considered by Navier–Stokes flows, but is not a conserved quantity of the Leray model. However, if u is a Leray solution, then the difference between H and HL is that HL uses a regularized vorticity and H uses the curl of a regularized velocity. The results are extended to show that the standard Crank–Nicolson finite element method for Leray models conserves both discrete energy and discrete regularized helicity.

Stability and instability of Navier boundary layers

We study the inviscid limit problem for the incompressible Navier-Stokes equation on a half-plane with a Navier boundary condition depending on the viscosity. On one hand, we prove the $L^{2}$ convergence of Leray solutions to the solution of the Euler equation. On the other hand, we show the nonlinear instability of some WKB expansions in the stronger $L^{\infty}$ and $\dot{H}^{s}$ ($s>1$) norms. These results are not contradictory, and in the periodic setting, we provide an example for which both phenomena occur simultaneously.

Linearized stationary incompressible flow around rotating and translating bodies -- Leray solutions

We consider Leray solutions of the Oseen system with rotational terms, in an exterior domain.Such solutions are characterized by square-integrability of the gradient of the velocityand local square-integrability of the pressure. In a previous paper, we had shown a pointwisedecay result for a slightly stronger type of solution. Here this result is extended toLeray solutions. We thus present a second access to this result, besides the onein G. P. Galdi, M. Kyed, Arch. Rat. Mech. Anal., 200 (2011), 21-58.

Convergence results for a class of spectrally hyperviscous models of 3-D turbulent flow

We consider the spectrally hyperviscous Navier–Stokes equations (SHNSE) which add hyperviscosity to the NSE but only to the higher frequencies past a cutoff wavenumber m0. In Guermond and Prudhomme (2003) [18], subsequence convergence of SHNSE Galerkin solutions to dissipative solutions of the NSE was achieved in a specific spectral-vanishing-viscosity setting. Our goal is to obtain similar results in a more general setting and to obtain convergence to the stronger class of Leray solutions. In particular we obtain subsequence convergence of SHNSE strong solutions to Leray solutions of the NSE by fixing the hyperviscosity coefficient μ while the spectral hyperviscosity cutoff m0 goes to infinity. This formulation presents new technical challenges, and we discuss how its motivation can be derived from computational experiments, e.g. those in Borue and Orszag (1996, 1998) [3,4]. We also obtain weak subsequence convergence to Leray weak solutions under the general assumption that the hyperviscous coefficient μ goes to zero with no constraints imposed on the spectral cutoff. In both of our main results the Aubin Compactness Theorem provides the underlying framework for the convergence to Leray solutions.

On the uniqueness of weak solutions of the 3D MHD equations in the Orlicz–Morrey space

In this article, we show that if a Leray solution (u, b) to the 3D magneto-hydrodynamic equation belongs to with r ∈ (0, 1) and P > 1, then (u, b) is the unique Leray solution on (0, T).

From Boltzmann’s Equation to the Incompressible Navier–Stokes–Fourier System with Long-Range Interactions

We establish a rigorous demonstration of the hydrodynamic convergence of the Boltzmann equation towards a Navier–Stokes–Fourier system under the presence of long-range interactions. This convergence is obtained by letting the Knudsen number tend to zero and has been known to hold, at least formally, for decades. It is only more recently that a fully rigorous mathematical derivation of this hydrodynamic limit was discovered. However, these results failed to encompass almost all physically relevant collision kernels due to a cutoff assumption, which requires that the cross sections be integrable. Indeed, as soon as long-range intermolecular forces are present, non-integrable collision kernels have to be considered because of the enormous number of grazing collisions in the gas. In this long-range setting, the Boltzmann operator becomes a singular integral operator and the known rigorous proofs of hydrodynamic convergence simply do not carry over to that case. In fact, the DiPerna–Lions renormalized solutions do not even make sense in this situation and the relevant global solutions to the Boltzmann equation are the so-called renormalized solutions with a defect measure developed by Alexandre and Villani. Our work overcomes the new mathematical difficulties coming from the consideration of long-range interactions by proving the hydrodynamic convergence of the Alexandre–Villani solutions towards the Leray solutions.

Long time decay to the Leray solution of the two-dimensional Navier-Stokes equations

Long time decay to the Leray solution of the two-dimensional Navier-Stokes equations

Smoothing estimates of 2d incompressible Navier–Stokes equations in bounded domains with applications

Motivated by the study on the uniqueness problem of the coupled model, in this paper, we revisit 2d incompressible Navier–Stokes equations in bounded domains. In fact, we establish some new smoothing estimates to the Leray solution based on the spectral analysis of Stokes operator. To understand well these estimates, on one hand, we establish some new Brezis–Waigner type inequalities in general domain and in any dimension and disclose the connection between both of them. On the other hand, we show that these new estimates can be applied to prove the existence and uniqueness of the weak solutions for two coupled models: Boussinesq system with partial viscosity (no dissipation for the temperature) and Fluid/Particle system, in two dimension and in bounded domains.

Asymptotic structure of a Leray solution to the Navier–Stokes flow around a rotating body

Consider a body, B, rotating with constant angular velocity, !, and fully submerged in a Navier-Stokes liquid that lls the whole space exterior to B. We analyze the ow of the liquid that is steady with respect to a frame attached to B. Our main theorem shows that the velocity eld, v, of any weak solution, (v;p), in the sense of Leray, has an asymptotic expansion with a suitable Landau solution as leading term, and a remainder decaying point-wise like 1

Inviscid limit for vortex patches in a bounded domain

In this paper, we consider the inviscid limit of the incompressible Navier–Stokes equations in a smooth, bounded and simply connected domain Ω ⊂ R d , d = 2 , 3 . We prove that for a vortex patch initial data, the weak Leray solutions of the incompressible Navier–Stokes equations with Navier boundary conditions will converge (locally in time for d = 3 and globally in time for d = 2 ) to a vortex patch solution of the incompressible Euler equation as the viscosity vanishes. In view of the results obtained in Abidi and Danchin (2004) [5] and Masmoudi (2007) [3] which dealt with the case of the whole space, we derive an almost optimal convergence rate ( ν t ) 3 4 − ε for any small ε > 0 in L 2 .

GLOBAL EXISTENCE RESULTS FOR THE ANISOTROPIC BOUSSINESQ SYSTEM IN DIMENSION TWO

Models with a vanishing anisotropic viscosity in the vertical direction are of relevance for the study of turbulent flows in geophysics. This motivates us to study the two-dimensional Boussinesq system with horizontal viscosity in only one equation. In this paper, we focus on the global existence issue for possibly large initial data. We first examine the case where the Navier–Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transport–diffusion equation with diffusion in the horizontal direction only. For both systems, we construct global weak solutions à la Leray and strong unique solutions for more regular data. Our results rest on the fact that the diffusion acts perpendicularly to the buoyancy force.

On symmetric Leray solutions of the stationary Navier–Stokes equations

A classical result of Amick (Acta Math 161:71–130, 1988) on the nontriviality of the symmetric Leray solutions of the steady-state Navier–Stokes equations in the plane is extended to Lipschitz domains. This results is compared with the famous Stokes paradox of linearized hydrodynamics and applied to a mixed problem of some interest in the applications.

Steady-State Navier–Stokes Flows Past a Rotating Body: Leray Solutions are Physically Reasonable

A rigid body, \({\fancyscript{B}}\), moves in a Navier–Stokes liquid, \({\fancyscript{L}}\), filling the whole space outside \({\fancyscript{B}}\). We assume that, when referred to a frame attached to \({\fancyscript{B}}\), the nonzero velocity of the center of mass, ξ, and the angular velocity, ω, of \({\fancyscript{B}}\) are constant and that the flow of \({\fancyscript{L}}\) is steady. Our main theorem implies that every “weak” steady-state solution in the sense of Leray is, in fact, physically reasonable in the sense of Finn, for data of arbitrary “size”. Such a theorem improves and generalizes an analogous famous result of Babenko (Math USSR Sb 20:1–25, 1973), obtained in the case ω = 0.

Extension d'une classe d'unicité pour les équations de Navier–Stokes

Recently, Q. Chen, C. Miao and Z. Zhang (2009) [4] have proved that weak Leray solutions of the Navier–Stokes are unique in the class L21+r([0,T],B∞r,∞(R3)) with r∈]−12,1]. In this paper, we establish that this criterion remains true for r∈]−1,−12].