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.     

2D Navier–Stokes Equations in a Time Dependent Domain with Neumann Type Boundary Conditions

In this paper the two-dimensional Navier–Stokes system for incompressible fluid coupled with a parabolic equation through the Neumann type boundary condition for the second component of the velocity is considered. Navier–Stokes equations are defined on a given time dependent domain. We prove the existence of a weak solution for this system. In addition, we prove the continuous dependence of solutions on the data for a regularized version of this system. For a special case of this regularized system also a problem with an unknown interface is solved.     

Vanishing Viscous Limits for 3D Navier–Stokes Equations with a Navier-Slip Boundary Condition

In this paper, we investigate the vanishing viscosity limit for solutions to the Navier–Stokes equations with a Navier slip boundary condition on general compact and smooth domains in R ³. We first obtain the higher order regularity estimates for the solutions to Prandtl’s equation boundary layers. Furthermore, we prove that the strong solution to Navier–Stokes equations converges to the Eulerian one in C([0, T]; H ¹(Ω)) and ${L^\infty((0,T) \times \Omega)}$ , where T is independent of the viscosity, provided that initial velocity is regular enough. Furthermore, rates of convergence are obtained also.     

Random Kick-Forced 3D Navier–Stokes Equations in a Thin Domain

We consider the Navier–Stokes equations in the thin 3D domain , where is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick force. We establish that, firstly, when ε ≪ 1, the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as ε → 0) to a unique stationary measure for the Navier–Stokes equation on . Thus, the 2D Navier–Stokes equations on surfaces describe asymptotic in time, and limiting in ε, statistical properties of 3D solutions in thin 3D domains.     

Global Well-Posedness of 2D Compressible Navier–Stokes Equations with Large Data and Vacuum

In this paper, we study the global well-posedness of the 2D compressible Navier–Stokes equations with large initial data and vacuum. It is proved that if the shear viscosity μ is a positive constant and the bulk viscosity λ is the power function of the density, that is, λ(ρ) = ρ ^β with β > 3, then the 2D compressible Navier–Stokes equations with the periodic boundary conditions on the torus \({\mathbb{T}^2}\) admit a unique global classical solution (ρ, u) which may contain vacuums in an open set of \({\mathbb{T}^2}\) . Note that the initial data can be arbitrarily large to contain vacuum states.     

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.     

Stationary Stokes, Oseen and Navier–Stokes Equations with Singular Data

The concept of very weak solution introduced by Giga (Math Z 178:287–329, 1981) for the Stokes equations has hardly been studied in recent years for either the Navier–Stokes equations or the Navier–Stokes type equations. We treat the stationary Stokes, Oseen and Navier–Stokes systems in the case of a bounded open set, connected of class C^1,1{\mathcal{C}^{1,1}} of \mathbbR³{\mathbb{R}^3}. Taking up once again the duality method introduced by Lions and Magenes (Problèmes aus limites non-homogènes et applications, vols. 1 & 2, Dunod, Paris, 1968) and Giga (Math Z 178:287–329, 1981) for open sets of class C^￥{\mathcal{C}^{\infty}} [see also chapter 4 of Necas (Les méthodes directes en théorie des équations elliptiques. (French) Masson et Cie, éd., Paris; Academia, éditeurs, Prague, 1967), which considers the Hilbertian case p = 2 for general elliptic operators], we give a simpler proof of the existence of a very weak solution for stationary Oseen and Navier–Stokes equations when data are not regular enough, based on density arguments and a functional framework adequate for defining more rigourously the traces of non-regular vector fields. In the stationary Navier–Stokes case, the results will be valid for external forces not necessarily small, which lets us extend the uniqueness class of solutions for these equations. Considering more regular data, regularity results in fractional Sobolev spaces will also be discussed for the three systems. All these results can be extended to other dimensions.     

About the Regularized Navier–Stokes Equations

The first goal of this paper is to study the large time behavior of solutions to the Cauchy problem for the 3-dimensional incompressible Navier–Stokes system. The Marcinkiewicz space L^3, is used to prove some asymptotic stability results for solutions with infinite energy. Next, this approach is applied to the analysis of two classical regularized Navier–Stokes systems. The first one was introduced by J. Leray and consists in mollifying the nonlinearity. The second one was proposed by J.-L. Lions, who added the artificial hyper-viscosity (–)^{/ 2}, > 2 to the model. It is shown in the present paper that, in the whole space, solutions to those modified models converge as t toward solutions of the original Navier–Stokes system.     

Stochastic Three-Dimensional Rotating Navier–Stokes Equations: Averaging, Convergence and Regularity

We consider stochastic three-dimensional rotating Navier?CStokes equations and prove averaging theorems for stochastic problems in the case of strong rotation. Regularity results are established by bootstrapping from global regularity of the limit stochastic equations and convergence theorems.     

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.     

Time-Periodic Solutions to the Full Navier–Stokes–Fourier System

We consider the full Navier–Stokes–Fourier system describing the motion of a compressible viscous and heat conducting fluid driven by a time-periodic external force. We show the existence of at least one weak time periodic solution to the problem under the basic hypothesis that the system is allowed to dissipate the thermal energy through the boundary. Such a condition is in fact necessary, as energetically closed fluid systems do not possess non-trivial (changing in time) periodic solutions as a direct consequence of the Second law of thermodynamics.     

Incompressible 3D Navier–Stokes Equations as a Limit of a Nonlinear Parabolic System

In this paper, we consider the Cauchy problem for a nonlinear parabolic system ${u^\epsilon_t - \Delta u^\epsilon + u^\epsilon \cdot \nabla u^\epsilon + \frac{1}{2}u^\epsilon\, {\rm div}\, u^\epsilon - \frac{1}{\epsilon}\nabla\, {\rm div}\, u^\epsilon = 0}$ in ${\mathbb {R}^3 \times (0,\infty)}$ with initial data in Lebesgue spaces ${L^2(\mathbb {R}^3)}$ or ${L^3(\mathbb {R}^3)}$ . We analyze the convergence of its solutions to a solution of the incompressible Navier?CStokes system as ${\epsilon \to 0}$ .     

On the Two-Phase Navier–Stokes Equations with Boussinesq–Scriven Surface Fluid

Two-phase flows with interface modeled as a Boussinesq–Scriven surface fluid are analysed concerning their fundamental mathematical properties. This extended form of the common sharp-interface model for two-phase flows includes both surface tension and surface viscosity. For this system of partial differential equations with free interface it is shown that the energy serves as a strict Ljapunov functional, where the equilibria of the model without boundary contact consist of zero velocity and spheres for the dispersed phase. The linearizations of the problem are derived formally, showing that equilibria are linearly stable, but nonzero velocities may lead to problems which linearly are not well-posed. This phenomenon does not occur in absence of surface viscosity. The present paper aims at initiating a rigorous mathematical study of two-phase flows with surface viscosity.     

Compressible Navier–Stokes Equations on Thin Domains

We consider the barotropic Navier–Stokes system describing the motion of a compressible viscous fluid confined to a straight layer \({\Omega_\varepsilon = \omega\times (0, \varepsilon)}\) , where ω is a particular 2-D domain (a periodic cell, bounded domain or the whole 2-D space). We show that the weak solutions in the 3D domain converge to a (strong) solutions of the 2-D Navier–Stokes system on ω as \({\varepsilon \to 0}\) on the maximal life time of the strong solution.     

On Nonexistence for Stationary Solutions to the Navier–Stokes Equations with a Linear Strain

We consider stationary solutions to the three-dimensional Navier–Stokes equations for viscous incompressible flows in the presence of a linear strain. For certain class of strains we prove a Liouville type theorem under suitable decay conditions on vorticity fields.