Uniform stability at permanently acting disturbances of solutions of Navier-Stokes equations for compressible isothermic fluids

We prove that the uniform stability at permanently acting disturbances of a given solution of the Navier-Stokes equations for viscous compressible isothermic fluid is a consequence of the uniform exponential stability of the zero solution of so-called linearized equations.The research was supported by the grant No. 201/93/2177 of Grant Agency of Czech Republic.     

Existence and uniqueness of the solution to a free boundary-value problem with thermocapillary convection in an unbounded domain

A free boundary-value problem for a nonisothermal flow of two heavy viscous incompressible capillary liquids descibing a slot coating process is mathematically studied. The problem is stationary and two-dimensional and the Boussinesq approximation for the Navier-Stokes equations is used. The existence and uniqueness of a weak solution is proved in weighted Holder spaces for small data.     

Existence results for viscous polytropic fluids with vacuum

We consider the full Navier-Stokes equations for viscous polytropic fluids with nonnegative thermal conductivity. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover our results hold for both bounded and unbounded domains.     

Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R³ and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a different idea from [X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. (2008), in press] to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.     

Solvability of the Navier-Stokes equations on manifolds with boundary

This paper deals with the solvability of the Navier-Stokes equations on manifolds with boundary. In particular, we concentrate on the inhomogeneous slip boundary condition. Our formulation of the equations takes into account a curvature term which results from a proper derivation of the Navier-Stokes equations. This term has not been considered in prior work. During the work on this version, the author received technical support through a fellowship of the DFG     

Global solutions for micro-macro models of polymeric fluids

We provide a new proof for the global well-posedness of systems coupling fluids and polymers in two space dimensions. Compared to the well-known existing method based on the losing a priori estimates, our method is more direct and much simpler. The co-rotational FENE dumbbell model and the coupling Smoluchowski and Navier-Stokes equations are studied as examples to illustrate our main ideas.     

Existence of solutions for a class of singular quasilinear elliptic resonance problems

The paper concerns a resonance problem for a class of singular quasilinear elliptic equations in weighted Sobolev spaces. The equation set studied is one of the most useful sets of Navier-Stokes equations; these describe the motion of viscous fluid substances such as liquids, gases and so on. By using Galerkin-type techniques, the Brouwer fixed point theorem, and a new weighted compact Sobolev-type embedding theorem established by Shapiro, we show the existence of a nontrivial solution.     

Shallow water equations: viscous solutions and inviscid limit

We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C ² test-functions, are confined in a compact set in H ^?1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.     

On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations

The dispersive effect of the Coriolis force for the stationary and non-stationary Navier-Stokes equations is investigated. Existence of a unique solution is shown for arbitrary large external force provided the Coriolis force is large enough. In addition to the stationary case, counterparts of several classical results for the non-stationary Navier-Stokes problem have been proven. The analysis is carried out in a new framework of the Fourier-Besov spaces.     

A study on the global regularity for a model of the 3D axisymmetric Navier-Stokes equations

This paper investigates the global regularity issue concerning a model equation proposed by Hou and Lei (2008) [9] to understand the stabilizing effects of the nonlinear terms in the 3D axisymmetric Navier-Stokes and Euler equations. We establish the global regularity of a generalized version of their model with a fractional Laplacian when the fractional power satisfies an explicit condition. This condition is exactly the same as in the case of the 3D generalized Navier-Stokes equations and is due to the balance between a more regular nonlinearity and a less effective (five-dimensional) Laplacian.     

Vanishing viscosity of isentropic Navier-Stokes equations for interacting shocks

We study the vanishing viscosity of the Navier-Stokes equations for interacting shocks. Given an entropy solution to p-system which consists of two different families of shocks interacting at some positive time,we show that such entropy solution is the vanishing viscosity limit of a family of global smooth solutions to the isentropic Navier-Stokes equations. The key point of the proofs is to derive the estimates separately before and after the interaction time and connect the incoming and outgoing viscous shock profiles.     

Stationary viscous flows at zero surface tension

Stationary solutions of free boundary problems for the Navier-Stokes equations are considered with and without surface tension. The linearized problem in a half-space is studied. Exact solutions of the Poiseuille type are obtained. Properties of the free boundary smoothness in 3-D and 2-D cases are discussed.     

Behaviour of weak solutions of compressible Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor

The aim of this paper is to study the behaviour of a weak solution to Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor for time going to infinity. In an analogous way as in [18], we construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is mentioned as well.     

A blow-up criterion for compressible viscous heat-conductive flows

We study an initial boundary value problem for the Navier-Stokes equations of compressible viscous heat-conductive fluids in a 2-D periodic domain or the unit square domain. We establish a blow-up criterion for the local strong solutions in terms of the gradient of the velocity only, which coincides with the famous Beale-Kato-Majda criterion for ideal incompressible flows.     

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.     

On a new boundary-value problem for the Navier-Stokes system with discontinuity in the type of boundary conditions depending on time

An initial boundary-value problem for the nonlinear system of Navier-Stokes equations with timedependent discontinuity in the type of boundary conditions is proposed to describe the dynamics of a hurricane over solid ground. The tools for the solution of the stated problem are displayed in the case of both smooth and nonsmooth evolution of a hurricane.This paper continues the authors' work [1].     

Symmetries and conservation laws of Navier-Stokes equations

All the symmetries and conservation laws of Navier-Stokes equations are calculated.     

On the existence of periodic weak solutions of Navier-Stokes equations in regions with periodically moving boundaries

We prove the existence of periodic weak solutions to the Navier-Stokes equations in regions with moving boundaries using the elliptic regularization.     

A novel variational method for 3D viscous flow in flow channel of turbomachines based on differential geometry

ABSTRACT

This paper presents a novel variational method for treating three-dimensional rotational Navier-Stokes equations in flow channel of turbomachines. The proposed method establishes a new semi-geodesic coordinate system on the central surface of blades. From the perspective of differential geometry, the system under concern is split into a set of membrane operator equations on two-dimensional manifolds and bending operator equations along hub circle. The third variable of the new coordinate system is approximated by the central difference scheme. We derive a new formulation of two-dimensional Navier-Stokes equations with three components on the manifolds in the variational sense. The well-posedness of the proposed variational formulation is rigorously justified.     

Crossing of eigenvalues: Measure type estimates

This paper deals with the behavior of eigenvalues for some non-homogeneous elliptic operators. More precisely, we present measure type estimates evaluating neighborhoods of the so-called resonant set. Various problems like the non-homogeneous incompressible limit of the compressible Navier-Stokes equations lead to such studies. This will be the purpose of a forthcoming paper.