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.     

Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions

We tackle the issue of the inviscid limit of the incompressible Navier–Stokes equations when the Navier slip-with-friction conditions are prescribed on impermeable boundaries. We justify an asymptotic expansion which involves a weak amplitude boundary layer, with the same thickness as in Prandtl’s theory and a linear behavior. This analysis holds for general regular domains, in both dimensions two and three.     

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.     

Navier－Stokes方程与湍流研讨会

Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程与湍流研讨会（１９９５年１２月１８—２１日,北京）在国家科委攀登计划“非线性科学”项目首席专家谷超豪院士的倡议下,由郑哲敏院土、了夏畦院士主持,中科院非线性连续介质力学开放研究实验室承办的Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程与...      

Extensions to the Macroscopic Navier–Stokes Equation

A development is provided showing that for any phase, by not neglecting the macroscopic terms of the deviation from the intensive momentum and of the dispersive momentum, we obtain a macroscopic secondary momentum balance equation coupled with a macroscopic dominant momentum balance equation that is valid at a larger spatial scale. The macroscopic secondary momentum balance equation is in the form of a wave equation that propagates the deviation from the intensive momentum while concurrently, in the case of a Newtonian fluid and under certain assumptions, the macroscopic dominant momentum balance equation may be approximated by Darcys equation to address drag dominant flow. We then develop extensions to the dominant macroscopic Navier–Stokes (NS) equation for saturated porous matrices, to account for the pressure gradient at the microscopic solid-fluid interfaces. At the microscopic interfaces we introduce the exchange of inertia between the phases, accounting for the relative fluid square velocities and the rate of these velocities, interpreted as Forchheimer terms. Conditions are provided to approximate the extended dominant NS equation by Forchheimer quadratic momentum law or by Darcys linear momentum law. We also show that the dominant NS equation can conform into a nonlinear wave equation. The one-dimensional numerical solution of this nonlinear wave equation demonstrates good qualitative agreement with experiments for the case of a highly deformable elasto-plastic matrix.     

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.     

Navier–Stokes solutions for steady parallel-sided pendent rivulets

We investigate exact solutions of the Navier–Stokes equations for steady rectilinear pendent rivulets running under inclined surfaces. First we show how to find exact solutions for sessile or hanging rivulets for any profile of the substrate (transversally to the direction of flow) and with no restrictions on the contact angles. The free surface is a cylindrical meniscus whose shape is determined by the static equilibrium between gravity and surface tension, by the shape of the solid surface, and by the contact angles on both contact lines. Given this, the velocity field can be obtained by integrating numerically a Poisson equation. We then perform a systematic study of rivulets hanging below an inclined plane, computing some of their global properties, and discussing their stability.     

A Variational Approach for the Navier–Stokes System

We introduce a variational approach to treat the regularity of the Navier–Stokes equations both in dimensions 2 and 3. Though the method allows the full treatment in dimension 2, we seek to precisely stress where it breaks down for dimension 3. The basic feature of the procedure is to look directly for strong solutions, by minimizing a suitable error functional that measures the departure of feasible fields from being a solution of the problem. By considering the divergence-free property as part of feasibility, we are able to avoid the explicit analysis of the pressure. Two main points in our analysis are:

Isolated Singularity for the Stationary Navier—Stokes System

In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in B_R \{0} B_R \setminus \{0\} , n 3 3 n \geq 3 and |u(x)| = o (|x|^{-(n - 1)/2}) |u(x)| = o\,(|x|^{-(n - 1)/2}) as |x| ? 0 |x| \to 0 or u ? L^{2n/(n - 1)}(B_R) u \in L^{2n/(n - 1)}(B_R) , then (u, p) is a distribution solution and if in addition, u ? L^b(B_R) u \in L^{\beta}(B_R) for some b > n \beta > n then ( u, p) is smooth in BR.     

On the Strong Solvability of the Navier—Stokes Equations

In this paper we study the strong solvability of the Navier—Stokes equations for rough initial data. We prove that there exists essentially only one maximal strong solution and that various concepts of generalized solutions coincide. We also apply our results to Leray—Hopf weak solutions to get improvements over some known uniqueness and smoothness theorems. We deal with rather general domains including, in particular, those having compact boundaries.     

Serrin-Type Blowup Criterion for Full Compressible Navier–Stokes System

The authors establish a Serrin-type blowup criterion for the Cauchy problem of the three-dimensional full compressible Navier–Stokes system, which states that a strong or smooth solution exists globally, provided that the velocity satisfies Serrin’s condition and that the temporal integral of the maximum norm of the divergence of the velocity is bounded. In particular, this criterion extends the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier–Stokes equations to the three-dimensional full compressible system and is just the same as that of the barotropic case.     

Asymptotic Stability of Landau Solutions to Navier–Stokes System

It is known that the three-dimensional Navier–Stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions that are axisymmetric and homogeneous of degree −1. We show that these solutions are asymptotically stable under any L ²-perturbation.     

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.     

Ill-posedness of the Hydrostatic Euler and Navier–Stokes Equations

We prove that the linearization of the hydrostatic Euler equations at certain parallel shear flows is ill-posed. The result also extends to the hydrostatic Navier–Stokes equations with a small viscosity.     

On Kato’s Method for Navier–Stokes Equations

We investigate Kato’s method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier–Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in \mathbbR³{\mathbb{R}}^{3} and irregular domains in \mathbbRⁿ{\mathbb{R}}^{n}.     

The Instationary Navier–Stokes Equations in Weighted Bessel-Potential Spaces

We investigate the solvability of the instationary Navier–Stokes equations with fully inhomogeneous data in a bounded domain W ì \mathbbRⁿ \Omega \subset {{\mathbb{R}}^{n}} . The class of solutions is contained in L^r(0, T; H^{b, q}_w (W))L^{r}(0, T; H^{\beta, q}_{w} (\Omega)), where H^{b, q}_w (W){H^{\beta, q}_{w}} (\Omega) is a Bessel-Potential space with a Muckenhoupt weight w. In this context we derive solvability for small data, where this smallness can be realized by the restriction to a short time interval. Depending on the order of this Bessel-Potential space we are dealing with strong solutions, weak solutions, or with very weak solutions.     

Spectral Stability of Noncharacteristic Isentropic Navier–Stokes Boundary Layers

Building on the work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the shock wave case, we study stability of compressive, or shock-like, boundary layers of the isentropic compressible Navier–Stokes equations with γ-law pressure by a combination of asymptotic ODE estimates and numerical Evans function computations. Our analytical results include convergence of the Evans function in the shock and large-amplitude limits and stability in the large-amplitude limit, the first rigorous stability result for other than the nearly constant case, for all . Together with these analytical results, our numerical investigations indicate stability for γ ϵ [1, 3] for all compressive boundary-layers, independent of amplitude, save for inflow layers in the characteristic limit (not treated). Expansive inflow boundary-layers have been shown to be stable for all amplitudes by Matsumura and Nishihara using energy estimates. Besides the parameter of amplitude appearing in the shock case, the boundary-layer case features an additional parameter measuring displacement of the background profile, which greatly complicates the resulting case structure. Inflow boundary layers turn out to have quite delicate stability in both large-displacement (shock) and large-amplitude limits, necessitating the additional use of a mod-two stability index studied earlier by Serre and Zumbrun in order to decide stability.