A stochastic Lagrangian proof of global existence of the Navier–Stokes equations for flows with small Reynolds number

We consider the incompressible Navier–Stokes equations with spatially periodic boundary conditions. If the Reynolds number is small enough we provide an elementary short proof of the existence of global in time Hölder continuous solutions. Our proof uses a stochastic representation formula to obtain a decay estimate for heat flows in Hölder spaces, and a stochastic Lagrangian formulation of the Navier–Stokes equations.     

An elementary approach to the analysis of exact solutions for the Navier-Stokes stagnation flows with slips

We deal with the exact solutions of the Navier-Stokes equations for stagnation flows with slips. The problem becomes the solvability of certain third-order ordinary differential equations (ODEs). Reducing the order of ODEs, we exhibit another elementary proof of the existence and asymptotic behavior of solutions. Numerical investigations are also provided. Received: 14 August 2003     

Navier-Stokes equations in 3D thin domains with Navier friction boundary condition

In this article we study the 3D Navier-Stokes equations with Navier friction boundary condition in thin domains. We prove the global existence of strong solutions to the 3D Navier-Stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small. We generalize the techniques developed to study the 3D Navier-Stokes equations in thin domains, see [G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains I: Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993) 503-568; G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains II: Global regularity of spatially periodic conditions, in: Nonlinear Partial Differential Equations and Their Application, College de France Seminar, vol. XI, Longman, Harlow, 1994, pp. 205-247; R. Temam, M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations 1 (1996) 499-546; R. Temam, M. Ziane, Navier-Stokes equations in thin spherical shells, in: Optimization Methods in Partial Differential Equations, in: Contemp. Math., vol. 209, Amer. Math. Soc., Providence, RI, 1996, pp. 281-314], to the Navier friction boundary condition by introducing a new average operator Mε in the thin direction according to the spectral decomposition of the Stokes operator Aε. Our analysis hinges on the refined investigation of the eigenvalue problem corresponding to the Stokes operator Aε with Navier friction boundary condition.     

A new proof of partial regularity of solutions to Navier-Stokes equations

In this paper we give a new proof of the partial regularity of solutions to the incompressible Navier-Stokes equation in dimension 3 first proved by Caffarelli, Kohn and Nirenberg. The proof relies on a method introduced by De Giorgi for elliptic equations. This work was supported in part by NSF Grant DMS-0607953.     

Vanishing viscosity in the plane for vorticity in borderline spaces of Besov type

The existence and uniqueness of solutions to the Euler equations for initial vorticity in BΓ∩Lp₀∩Lp₁ was proved by Misha Vishik, where BΓ is a borderline Besov space parameterized by the function Γ and 1<p₀<2<p₁. Vishik established short time existence and uniqueness when Γ(n)=O(logn) and global existence and uniqueness when . For initial vorticity in BΓ∩L², we establish the vanishing viscosity limit in L²(R²) of solutions of the Navier-Stokes equations to a solution of the Euler equations in the plane, convergence being uniform over short time when Γ(n)=O(logn) and uniform over any finite time when Γ(n)=O(logκn), 0?κ<1, and we give a bound on the rate of convergence. This allows us to extend the class of initial vorticities for which both global existence and uniqueness of solutions to the Euler equations can be established to include BΓ∩L² when Γ(n)=O(logκn) for 0<κ<1.     

A scaling and energy equality preserving approximation for the 3D Navier-Stokes equations in the finite energy case

We discuss a new model (inspired by the work of Vishik and Fursikov) approximating the 3D Navier-Stokes equations, which preserves the scaling as in the Navier-Stokes equations and thus allows the study of self-similar solutions. Using some energy estimates and Leray’s limiting process, we show the existence of a solution of this model in the finite energy case, and the energy equality and inequality fulfilled by it. This approximation can be shown to converge to the Navier-Stokes equations using a mild approach based on the approximated pressure, and the solution satisfies Scheffer’s local energy inequality, an essential tool for proving Caffarelli, Kohn and Nirenberg’s regularity criterion. We also give a partial result of self-similarity satisfied by the approximated solution in the infinite energy case.     

Asymptotic energy concentration in the phase space of the weak solutions to the Navier-Stokes equations

We study the asymptotic behavior of the energy of weak solutions of Navier-Stokes equations as t→∞. We characterize the space of the initial data which causes a concentration of the kinetic energy in the phase space. Moreover, an explicit convergence rate is obtained.     

Global strong solutions of Navier-Stokes equations with interface boundary in three-dimensional thin domains

In this article, we study the spectrum of the Stokes operator in a 3D two layer domain with interface, obtain the asymptotic estimates on the spectrum of the Stokes operator as thickness ε goes to zero. Based on the spectral decomposition of the Stokes operator, a new average-like operator is introduced and applied to the study of Navier-Stokes equation in the two layer thin domains under interface boundary condition. We prove the global existence of strong solutions to the 3D Navier-Stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small. This article is a continuation of our study on the Stokes operator under Navier friction boundary condition. Due to the viscosity distinction between the two layers, the Stokes operator displays radically different spectral structure from that under Navier friction boundary condition, then causes great difficulty to the analysis.     

Distributed control for a class of non-Newtonian fluids

We consider control problems with a general cost functional where the state equations are the stationary, incompressible Navier-Stokes equations with shear-dependent viscosity. The equations are quasi-linear. The control function is given as the inhomogeneity of the momentum equation. In this paper, we study a general class of viscosity functions which correspond to shear-thinning or shear-thickening behavior. The basic results concerning existence, uniqueness, boundedness, and regularity of the solutions of the state equations are reviewed. The main topic of the paper is the proof of Gâteaux differentiability, which extends known results. It is shown that the derivative is the unique solution to a linearized equation. Moreover, necessary first-order optimality conditions are stated, and the existence of a solution of a class of control problems is shown.     

Global existence of strong solutions of Navier-Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

The aim of this paper is to discuss the global existence and uniqueness of strong solution for a class of the isentropic compressible Navier-Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove two global existence results on strong solutions of the isentropic compressible Navier-Stokes equations. The first result shows only the existence, and the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition.     

Asymptotic behavior for the Navier-Stokes equations with nonzero external forces

We estimate the asymptotic behavior for the Stokes solutions, with external forces first. We found that if there are external forces, then the energy decays slowly even if the forces decay quickly. Then, we also obtain the asymptotic behavior in the temporal-spatial direction for weak solutions of the Navier-Stokes equations. We also provide a simple example of external forces which shows that the Stokes solution does not decay quickly.     

Asymptotic stability of viscous contact discontinuity to an inflow problem for compressible Navier-Stokes equations

This paper is concerned with an initial-boundary value problem for one-dimensional full compressible Navier-Stokes equations with inflow boundary conditions in the half space R₊=(0,+∞). The asymptotic stability of viscous contact discontinuity is established under the conditions that the initial perturbations and the strength of contact discontinuity are suitably small. Compared with the free-boundary and the initial value problems, the inflow problem is more complicated due to the additional boundary effects and the different structure of viscous contact discontinuity. The proofs are given by the elementary energy method.     

On the time discretization for the globally modified three dimensional Navier-Stokes equations

In this work, we analyze the discrete in time 3D system for the globally modified Navier-Stokes equations introduced by Caraballo (2006) [1]. More precisely, we consider the backward implicit Euler scheme, and prove the existence of a sequence of solutions of the resulting equations by implementing the Galerkin method combined with Brouwer’s fixed point approach. Moreover, with the aid of discrete Gronwall’s lemmas we prove that for the time step small enough, and the initial velocity in the domain of the Stokes operator, the solution is H² uniformly stable in time, depends continuously on initial data, and is unique. Finally, we obtain the limiting behavior of the system as the parameter N is big enough.     

Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations

We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. The coupling arises from a drag force exerted by each other. We establish existence of global weak solutions for the system in two and three dimensions. Furthermore, we obtain the existence and uniqueness result of global smooth solutions for dimension two. In case of three dimensions, we also prove that strong solutions exist globally in time for the Vlasov-Stokes system.     

On weighted-norm estimates for nonstationary incompressible Navier-Stokes flows in a 3D exterior domain

We study the time-decay of weighted norms of weak and strong solutions to the Navier-Stokes equations in a 3D exterior domain. Moment estimates for weak solutions and weighted Lq-estimates for strong solutions are deduced, both of which seem to be optimal. The relation is discussed between the space-time decay and the vanishing of the total net force exerted by the fluid to the body. A class of initial data is given so that the total net force associated to the corresponding fluid flows does not vanish.     

On cylindrical symmetric flows through pipe-like domains

We prove existence of cylindrical symmetric solutions to the steady Navier-Stokes equations in bounded pipe-like domains in with the slip boundary conditions. The result is shown for any large flows across the boundary assuming only a geometrical constraint on the shape of the domain which is independent of data. The simply connectedness of the domain is not required. The technique is based on a reformulation of the original problem and delivers us a new type of estimates in the Hölder spaces for this class of the solutions.     

A removable isolated singularity theorem for the stationary Navier-Stokes equations

We show that an isolated singularity at the origin 0 of a smooth solution (u,p) of the stationary Navier-Stokes equations is removable if the velocity u satisfies u∈Lⁿ or |u(x)|=o(|x|^-1) as x→0. Here n?3 denotes the dimension. As a byproduct of the proof, we also obtain a new interior regularity theorem.     

Unique solvability of equations of motion for ferrofluids

We study the differential system governing the flow of an incompressible ferrofluid under the action of a magnetic field. The system is a combination of the Navier-Stokes equations, the angular momentum equation, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of the unique strong solution to the system posed in a bounded domain of R³ and equipped with initial and boundary conditions.     

A stochastic representation for backward incompressible Navier-Stokes equations

By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer’s forward formulations in Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for incompressible Navier-Stokes equations in the whole space. In two dimensions or large viscosity, an alternative proof to the global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero.