Some spatial decay estimates in time-dependent stokes slow flows

This paper uses a time-weighted cross-sectional area measure in order to establish spatial decay estimates for the time-dependent Stokes slow flow of an incompressible viscous fluid in a semi- infinite cylindrical pipe of smooth cross section. The decay rate predicted in this paper depends only on the constant kinematic viscosity and the smallest positive eigenvalue of the free membrane problem as well as two geometric characteristics of the cross section of the pipe.     

Local well-posedness to the cauchy problem of the 3-D compressible navier-stokes equations with density-dependent viscosity

In this article, we prove the local existence and uniqueness of the classical solution to the Cauchy problem of the 3-D compressible Navier-Stokes equations with large initial data and vacuum, if the shear viscosity μ is a positive constant and the bulk viscosity λ(ρ) = ρ^β with β ≥ 0. Note that the initial data can be arbitrarily large to contain vacuum states.     

On decay estimates of the 3D nematic liquid crystal flows in critical Besov spaces

In this paper, we develop the energy argument in homogeneous Besov space framework to study the large time behavior of global‐in‐time strong solutions to the Cauchy problem of the three‐dimensional incompressible nematic liquid crystal flows with low regularity assumptions on initial data. More precisely, if the small initial data with 1 < p < ∞ and further assume that with 1 < q≤p and , then the global‐in‐time strong solution (u,d) to the nematic liquid crystal flows admits the following temporal decay rate: Here, is a constant unit vector. The highlight of our argument is to show that the ‐norms (with ) of solution are preserved along time evolution. Copyright © 2016 John Wiley & Sons, Ltd.     

On the spatial decay of solutions of parabolic equations

Zusammenfassung Es wird ein von Edelstein erzieltes Ergebnis über den exponentiellen Temperaturabfall eines Körpers an von Wärmezufuhrquellen entfernten Stellen auf eine erweiterte Klasse von Randwertproblemen verallgemeinert.

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This work supported by the U.S. Department of the Navy Contract NOw-62-0604-c.     

On the spatial decay of solutions of the heat equation

Zusammenfassung Es werden Energieungleichungen aufgestellt, welche die exponentielle räumliche Abnahme der Lösungen der Wärmegleichung ausdrücken.

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This research is based on work supported in part by the United States Office of Naval Research under Contract Nonr 220(56). Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Continuous dependence on the time and spatial geometry for the navier-stokes equations

In this paper we derive explicit a priori inequalities which imply stability of the solution of the initial-boundary value problem for the Navier-Stokes equations under perturbations of the initial time geometry and of the spatial geometry. These inequalities bound the solution perturbation In L₂ in terms of some well defined measure of the perturbation in geometry. We establish continuous dependence on spatial geometry in both two and three dimensions and continuous dependence on initial geometry in two dimensions. In the latter problem the three dimensional case will be somewhat more complicated due to the different form of the Sobolev inequality.     

On partial regularity results for the navier-stokes equations

Looking at the regularity results of Scheffer, respectively, Caffarelli, Kohn and Nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the Navier-Stokes equations.     

Multi-level spectral galerkin method for the navier-stokes problem I : spatial discretization

A multi-level spectral Galerkin method for the two-dimensional non-stationary Navier-Stokes equations is presented. The method proposed here is a multiscale method in which the fully nonlinear Navier-Stokes equations are solved only on a low-dimensional space subsequent approximations are generated on a succession of higher-dimensional spaces j=2, . . . ,J, by solving a linearized Navier-Stokes problem around the solution on the previous level. Error estimates depending on the kinematic viscosity 0<ν<1 are also presented for the J-level spectral Galerkin method. The optimal accuracy is achieved when We demonstrate theoretically that the J-level spectral Galerkin method is much more efficient than the standard one-level spectral Galerkin method on the highest-dimensional space . The work of this author was supported in part by the NSF of China 10371095, City University of Hong Kong Research Project 7001093 Hong Kong and the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1084/02P)     

On a new 3D model for incompressible euler and navier-stokes equations

In this paper, we investigate some new properties of the incompressible Euler and Navier-Stokes equations by studying a 3D model for axisymmetric 3D incompressible Euler and Navier-Stokes equations with swirl. The 3D model is derived by reformulating the axisymmetric 3D incompressible Euler and Navier-Stokes equations and then neglecting the convection term of the resulting equations. Some properties of this 3D model are reviewed. Finally, some potential features of the incompressible Euler and Navier-Stokes equations such as the stabilizing effect of the convection are presented.     

On the existence and stability of 2-D perturbed steady subsonic circulatory flows

In this paper, under the generalized conservation condition of mass flux in a unbounded domain, we are concerned with the global existence and stability of a perturbed subsonic circulatory flow for the two-dimensional steady Euler equation, which is assumed to be isentropic and irrotational. Such a problem can be reduced into a second order quasi-linear elliptic equation on the stream function in an exterior domain with a Dirichlet boundary value condition on the circular body and a stability condition at i...     

On the navier-stokes equation with the additional conditionu 1 1 =u 3=0

We study the Navier-Stokes equation with the additional conditionu ₁ ¹ =u ³=0. In certain cases, solutions are represented in a closed form. In other cases, the investigated system reduces to simpler systems of partial differential equations. We study the symmetry properties of these systems and construct classes of their particular solutions.     

On the stationary,compressible and incompressible navier-stokes equations

Summary In this paper we study the system (1.1), (1.3) which describes the stationary motion of a given amount of a compressible heat conducting, viscous fluid in a bounded domain of Rⁿ, n2, and we consider the incompressible limit of the solutions of that system of equations (for barotropic flows) as the Mach number becomes small.     

On the temporal decay of solutions to the two‐dimensional nematic liquid crystal flows

We consider the temporal decay estimates for weak solutions to the two‐dimensional nematic liquid crystal flows, and we show that the energy norm of a global weak solution has non‐uniform decay under suitable conditions on the initial data. We also show the exact rate of the decay (uniform decay) of the energy norm of the global weak solution.     

On interior regularity criteria for weak solutions of the navier-stokes equations

We are concerned with the behavior of weak solutions of the Navier-Stokes equations near possible singularities. We shall show that if a weak solution is in some Lebesgue space or small in some Lorentz space locally, it does not blowup there. Our basic idea is to estimate integral formulas for vorticity which satisfies parabolic equations.     

On the decay of higher order derivatives of solutions to Ladyzhenskaya model for incompressible viscous flows

This article concerns large time behavior of Ladyzhenskaya model for incompressible viscous flows in R~3.Based on linear L~P-L~q estimates,the auxiliary decay properties of the solutions and generalized Gronwall type arguments,some optimal upper and lower bounds for the decay of higher order derivatives of solutions are derived without assuming any decay properties of solutions and using Fourier splitting technology.