Lq$L^q$-weak solutions to the time-dependent 3D Oseen system: Decay estimates

This article deals with $L^q$-weak solutions to the 3D time-dependent Oseen system. This type of solution is defined in terms of the velocity only. It is shown that the velocity may be represented by a sum of integrals none of which involves the pressure and without a surface integral of the spatial gradient of the velocity. On the basis of this representation formula, an estimate of the spatial decay of the velocity and its spatial gradient is derived. No boundary conditions have to be imposed for these results.     

First‐order system least squares for the Oseen equations

Following earlier work for Stokes equations, a least squares functional is developed for two‐ and three‐dimensional Oseen equations. By introducing a velocity flux variable and associated curl and trace equations, ellipticity is established in an appropriate product norm. The form of Oseen equations examined here is obtained by linearizing the incompressible Navier–Stokes equations. An algorithm is presented for approximately solving steady‐state, incompressible Navier–Stokes equations with a nested iteration‐Newton‐FOSLS‐AMG iterative scheme, which involves solving a sequence of Oseen equations. Some numerical results for Kovasznay flow are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

Thorough analysis of the Oseen system in 2D exterior domains

We construct L_p‐estimates for the inhomogeneous Oseen system studied in a two‐dimensional exterior domain Ω with inhomogeneous slip boundary conditions. The kernel of the paper is a result for the half space ?. Analysis of this model system shows us a parabolic character of the studied problem, resulting as an appearance of the wake region behind the obstacle. Main tools are given by the Fourier analysis to obtain the maximal regularity estimates. The results imply the solvability for the Navier–Stokes system for small velocity at infinity. Copyright © 2009 John Wiley & Sons, Ltd.     

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate L^q‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D^1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D^1,r is the homogeneous Sobolev space.     

The method of fundamental solutions for the Oseen steady‐state viscous flow past obstacles of known or unknown shapes

In this paper, the steady‐state Oseen viscous flow equations past a known or unknown obstacle are solved numerically using the method of fundamental solutions (MFS), which is free of meshes, singularities, and numerical integrations. The direct problem is linear and well‐posed, whereas the inverse problem is nonlinear and ill‐posed. For the direct problem, the MFS computations of the fluid flow characteristics (velocity, pressure, drag, and lift coefficients) are in very good agreement with the previously published results obtained using other methods for the Oseen flow past circular and elliptic cylinders, as well as past two circular cylinders. In the inverse obstacle problem the boundary data and the internal measurement of the fluid velocity are minimized using the MATLAB^© optimization toolbox lsqnonlin routine. Regularization was found necessary in the case the measured data are contaminated with noise. Numerical results show accurate and stable reconstructions of various star‐shaped obstacles of circular, bean, or peanut cross‐section.     

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.     

A numerical method for the viscous incompressible Oseen flow in shape reconstruction

This paper is concerned with the shape reconstruction of a bounded domain with a viscous incompressible fluid driven by the Oseen equations. For the approximate solution of the ill-posed and nonlinear problem we propose a regularized Gauss-Newton method. A theoretical foundation for the method is given by establishing the differentiability of the boundary value problem with respect to the boundary in the sense of the domain derivative. The results of several numerical experiments show that our theory is useful for practical purpose, and the proposed algorithm is feasible.     

Global smooth solutions of the 2D MHD equations for a class of large data without magnetic diffusion

In this paper, we establish the global regularity of classical solutions for the two-dimensional MHD equations with only velocity diffusion for a class of large initial data. Both the initial velocity and magnetic field can be arbitrarily large in $H^s$.     

Weak and strong solutions for the incompressible Navier-Stokes equations with damping

In this paper, we show that the Cauchy problem of the Navier-Stokes equations with damping α|u|^β−1u(α>0) has global weak solutions for any β?1, global strong solution for any β?7/2 and that the strong solution is unique for any 7/2?β?5.     

Lump solutions of the 2D Toda equation

In this research, the lump solution, which is rationally localized and decays along the directions of space variables, of a 2D Toda equation is studied. The effective method of constructing the lump solutions of this 2D Toda equation is derived, and the constraint conditions that make the lump solutions analytical and positive are obtained as well. Finally, three classes of lump solutions are constructed, 3D plots, density plots, and contour plots are given to illustrate this proposed method.     

Regularity criterion of weak solutions to the 3D Navier-Stokes equations via two velocity components

This paper is concerned with the regularity criterion of Leray-Hopf weak solutions to the 3D Navier-Stokes equations with respect to Serrin type condition on two velocity filed components. It is shown that the weak solution u=(u₁,u₂,u₃) is regular on (0,T] if there exist two solution components, for example, u₂ and u₃, satisfying the condition

     

Uniqueness for some Leray-Hopf solutions to the Navier-Stokes equations

We exhibit simple sufficient conditions which give weak-strong uniqueness for the 3D Navier-Stokes equations. The main tools are trilinear estimates and energy inequalities. We then apply our result to the framework of Lorentz, Morrey and Besov over Morrey spaces so as to get new weak-strong uniqueness classes and so uniqueness classes for solutions in the Leray-Hopf class. In the last section, we give a uniqueness and regularity result. We obtain new uniqueness classes for solutions in the Leray-Hopf class without energy inequalities but sufficiently regular.     

Periodic solutions in unbounded domains for the Boussinesq system

Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Boussinesq equations in several classes of unbounded domains of Rn. Our analysis is based on the framework of weak-Lp spaces.     

Decaying solutions for discrete boundary value problems on the half line

Some nonlocal boundary value problems, associated to a class of functional difference equations on unbounded domains, are considered by means of a new approach. Their solvability is obtained by using properties of the recessive solution to suitable half-linear difference equations, a half-linearization technique and a fixed point theorem in Frechét spaces. The result is applied to derive the existence of nonoscillatory solutions with initial and final data. Examples and open problems complete the paper.     

Existence and uniqueness of global classical solutions to 3D isentropic compressible Navier-Stokes equations with general initial data

In this paper we study the global existence and uniqueness of classical solutions to the Cauchy problem for 3D isentropic compressible Navier-Stokes equations with general initial data which could contain vacuum.We give the relation between the viscosity coefficients and the initial energy,which implies that the Cauchy problem under consideration has a global classical solution.     

Geometric measure-type regularity criteria for the 3D magnetohydrodynamical system

Several formulations of a local geometric measure-type condition are imposed on super-level sets of mild solutions to the homogeneous incompressible 3D magnetohydrodynamical system with bounded initial data to prevent finite-time singularity formation. Supporting this, results regarding the existence, uniqueness, and real analyticity of mild solutions are established as is a sharp lower bound on the radius of analyticity.     

Positive solutions of fourth order Emden-Fowler type differential equations in the framework of regular variation

The fourth order nonlinear differential equations

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