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

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 Beir?o da Veiga et al. in J Math Anal Appl 377:216–227, 2011) if  W = \mathbb R³₊,{\,\Omega = \mathbb R^3_+,} convergence holds in C([0,T]; W^k,p(\mathbb R³₊)){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.     

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.     

Stability for the 3D Navier–Stokes Equations with Nonzero far Field Velocity on Exterior Domains

Concerning to the non-stationary Navier–Stokes flow with a nonzero constant velocity at infinity, just a few results have been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary solutions for the Navier–Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata (J Math Fluid Mech 7:339–367, 2005), in L ^p spaces for p ≥ 3. In this article, we first extend their result to the case \frac32 < p{\frac{3}{2} < p} by modifying the method in Bae and Jin (J Math Fluid Mech 10:423–433, 2008) that was used to obtain weighted estimates for the Navier–Stokes flow with the zero velocity at infinity. Then, by using our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier–Stokes flow with a nonzero velocity at infinity.     

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.     

Another Proof of Stability for Global Weak Solutions of 2D Degenerated Shallow Water Models

We consider non-linear viscous shallow water models with varying topography, extra friction terms and capillary effects, in a two-dimensional framework. Water-depth dependent laminar and turbulent friction coefficients issued from an asymptotic analysis of the three-dimensional free-surface Navier–Stokes equations are considered here. A new proof of stability for global weak solutions is given in periodic domain Ω = T², adapting the method introduced by J. Simon in [15] for the non-homogeneous Navier–Stokes equations. Existence results for such solutions can be obtained from this stability analysis.     

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

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.     

Optimal Decay Rate of the Compressible Navier–Stokes–Poisson System in {\mathbb {R}^3}

The compressible Navier–Stokes–Poisson (NSP) system is considered in ${\mathbb {R}^3}The compressible Navier–Stokes–Poisson (NSP) system is considered in \mathbb R³{\mathbb {R}^3} in the present paper, and the influences of the electric field of the internal electrostatic potential force governed by the self-consistent Poisson equation on the qualitative behaviors of solutions is analyzed. It is observed that the rotating effect of electric field affects the dispersion of fluids and reduces the time decay rate of solutions. Indeed, we show that the density of the NSP system converges to its equilibrium state at the same L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 respectively as the compressible Navier–Stokes system, but the momentum of the NSP system decays at the L ²-rate (1+t)^{-\frac 14}{(1+t)^{-\frac {1}{4}}} or L ^∞-rate (1 + t)⁻¹ respectively, which is slower than the L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 for compressible Navier–Stokes system [Duan et al., in Math Models Methods Appl Sci 17:737–758, 2007; Liu and Wang, in Comm Math Phys 196:145–173, 1998; Matsumura and Nishida, in J Math Kyoto Univ 20:67–104, 1980] and the L ^∞-rate (1 + t)^−p with p ? (1, 3/2){p \in (1, 3/2)} for irrotational Euler–Poisson system [Guo, in Comm Math Phys 195:249–265, 1998]. These convergence rates are shown to be optimal for the compressible NSP system.     

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.     

The Fundamental Solution of the Linearized Navier–Stokes Equations for Spinning Bodies in Three Spatial Dimensions – Time Dependent Case

Explicit formulae for the fundamental solution of the linearized time dependent Navier–Stokes equations in three spatial dimensions are obtained. The linear equations considered in this paper include those used to model rigid bodies that are translating and rotating at a constant velocity. Estimates extending those obtained by Solonnikov in [23] for the fundamental solution of the time dependent Stokes equations, corresponding to zero translational and angular velocity, are established. Existence and uniqueness of solutions of these linearized problems is obtained for a class of functions that includes the classical Lebesgue spaces L^p(R³), 1 < p < ∞. Finally, the asymptotic behavior and semigroup properties of the fundamental solution are established.     

Analyticity and Decay Estimates of the Navier–Stokes Equations in Critical Besov Spaces

In this paper, we establish analyticity of the Navier–Stokes equations with small data in critical Besov spaces . The main method is Gevrey estimates, the choice of which is motivated by the work of Foias and Temam (Contemp Math 208:151–180, 1997). We show that mild solutions are Gevrey regular, that is, the energy bound holds in , globally in time for p < ∞. We extend these results for the intricate limiting case p = ∞ in a suitably designed E _∞ space. As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov spaces. Finally, we provide a regularity criterion in Besov spaces.     

On the Impossibility, in Some Cases, of the Leray�CHopf Condition for Energy Estimates

Current proofs of time independent energy bounds for solutions of the time dependent Navier–Stokes equations, and of bounds for the Dirichlet norms of steady solutions, are dependent upon the construction of an extension of the prescribed boundary values into the domain that satisfies the inequality (1.1) below, for a value of κ less than the kinematic viscosity. It is known from the papers of Leray (J Math Pure Appl 12:1–82, 1993), Hopf (Math Ann 117:764–775, 1941) and Finn (Acta Math 105:197–244, 1961) that such a construction is always possible if the net flux of the boundary values across each individual component of the boundary is zero. On the other hand, the nonexistence of such an extension, for small values of κ, has been shown by Takeshita (Pac J Math 157:151–158, 1993) for any two or three-dimensional annular domain, when the boundary values have a net inflow toward the origin across each component of the boundary. Here, we prove a similar result for boundary values that have a net outflow away from the origin across each component of the boundary. The proof utilizes a class of test functions that can detect and measure deformation. It appears likely that much of our reasoning can be applied to other multiply connected domains.     

Boundary Regularity of Shear Thickening Flows

This article is concerned with the global regularity of weak solutions to systems describing the flow of shear thickening fluids under the homogeneous Dirichlet boundary condition. The extra stress tensor is given by a power law ansatz with shear exponent p≥ 2. We show that, if the data of the problem are smooth enough, the solution u of the steady generalized Stokes problem belongs to W^{1,(np+2-p)/(n-2)}(W){W^{1,(np+2-p)/(n-2)}(\Omega)} . We use the method of tangential translations and reconstruct the regularity in the normal direction from the system, together with anisotropic embedding theorem. Corresponding results for the steady and unsteady generalized Navier–Stokes problem are also formulated.     

Incompressible Fluids in Thin Domains with Navier Friction Boundary Conditions (I)

This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier–Stokes equations in a three-dimensional domain Ω_ɛ, whose thickness is of order O(ɛ) as ɛ → 0, having non-trivial topography. The velocity field is subject to the Navier friction boundary conditions on the bottom and top boundaries of Ω_ɛ, and to the periodicity condition on its sides. Assume that the friction coefficients are of order O(ɛ^3/4) as ɛ → 0. It is shown that if the initial data, respectively, the body force, belongs to a large set of H¹(Ω_ɛ), respectively, L²(Ω_ɛ), then the strong solution of the Navier–Stokes equations exists for all time. Our proofs rely on the study of the dependence of the Stokes operator on ɛ, and the non-linear estimate in which the contributions of the boundary integrals are non-trivial.     

A Navier–Stokes Approximation of the 3D Euler Equation with the Zero Flux on the Boundary

Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T ₀ > 0, ν ₀ > 0 and a unique continuous family of strong solutions u ^ν (0 ≤ ν < ν ₀) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T ₀). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary conditions imposed on curl u ^ν and curl ² u ^ν .      

Weak in Space, Log in Time Improvement of the Lady?enskaja�CProdi�CSerrin Criteria

In this article we present a Ladyženskaja–Prodi–Serrin Criteria for regularity of solutions for the Navier–Stokes equation in three dimensions which incorporates weak L ^p norms in the space variables and log improvement in the time variable.     

Steady Flow of a Navier–Stokes Liquid Past an Elastic Body

We perform a mathematical analysis of the steady flow of a viscous liquid, L{\mathcal{L}} , past a three-dimensional elastic body, B{\mathcal{B}} . We assume that L{\mathcal{L}} fills the whole space exterior to B{\mathcal{B}} , and that its motion is governed by the Navier–Stokes equations corresponding to non-zero velocity at infinity, v _∞. As for B{\mathcal{B}} , we suppose that it is a St. Venant–Kirchhoff material, held in equilibrium either by keeping an interior portion of it attached to a rigid body or by means of appropriate control body force and surface traction. We treat the problem as a coupled steady state fluid-structure problem with the surface of B{\mathcal{B}} as a free boundary. Our main goal is to show existence and uniqueness for the coupled system liquid-body, for sufficiently small |v _∞|. This goal is reached by a fixed point approach based upon a suitable reformulation of the Navier–Stokes equation in the reference configuration, along with appropriate a priori estimates of solutions to the corresponding Oseen linearization and to the elasticity equations.     

A New Class of Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous Data

We investigate a class of weak solutions, the so-called very weak solutions, to stationary and nonstationary Navier–Stokes equations in a bounded domain . This notion was introduced by Amann [3], [4] for the nonstationary case with nonhomogeneous boundary data leading to a very large solution class of low regularity. Here we are mainly interested in the investigation of the “largest possible” class of solutions u for the more general problem with arbitrary divergence k = div u, boundary data g = u|_∂Ω and an external force f, as weak as possible, but maintaining uniqueness. In principle, we will follow Amann’s approach.     

Artificial Boundary Conditions of Pressure Type for Viscous Flows in a System of Pipes

In a three-dimensional domain Ω with J cylindrical outlets to infinity the problem is treated how solutions to the stationary Stokes and Navier–Stokes system with pressure conditions at infinity can be approximated by solutions on bounded subdomains. The optimal artificial boundary conditions turn out to have singular coefficients. Existence, uniqueness and asymptotically precise estimates for the truncation error are proved for the linear problem and for the nonlinear problem with small data. The results include also estimates for the so called “do-nothing” condition.     

Point Singularities of 3D Stationary Navier–Stokes Flows

This article characterizes the singularities of very weak solutions of 3D stationary Navier–Stokes equations in a punctured ball which are sufficiently small in weak L ³.