Nonlinear artificial boundary conditions with pointwise error estimates for the exterior three dimensional Navier–Stokes problem

On a three–dimensional exterior domain Ω we consider the Dirichlet problem for the stationary Navier–Stokes system. We construct an approximation problem on the domain Ω_R, which is the intersection of Ω with a sufficiently large ball, while we create nonlinear, but local artificial boundary conditions on the truncation boundary. We prove existence and uniqueness of the solutions to the approximating problem together with asymptotically precise pointwise error estimates as R tends to infinity.     

Regularity of minimizers of semilinear elliptic problems up to dimension 4

We consider the class of semistable solutions to semilinear equations ?Δu = f(u) in a bounded smooth domain Ω of \input amssym $\Bbb R^n$ (with Ω convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions n ≤ 4, we establish an a priori L^∞‐bound that holds for every positive semistable solution and every nonlinearity f. This estimate leads to the boundedness of all extremal solutions when n = 4 and Ω is convex. This result was previously known only in dimensions n ≤ 3 by a result of G. Nedev. In dimensions 5 ≤ n ≤ 9 the boundedness of all extremal solutions remains an open question. It is only known to hold in the radial case Ω = B_R by a result of A. Capella and the author. © 2010 Wiley Periodicals, Inc.     

A priori error estimates for upwind finite volume schemes for two-dimensional linear convection diffusion problems

It is still an open problem to prove a priori error estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domain Ω in R² and we can prove such kind of an a priori error estimate. In the part of the estimate, which refers to the discretization of the convective term, we gain h^1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique.     

Positive splutions of semilinear elliptic systems

we investigata th existence of positive solutions of a Dirichlet problem for the system in a bounded convex domain Ω of Rⁿ with smooth boundary. In particular a priori bounds are obtained in the same spirit as in De Figueiredo – Lions – Nussbaum.     

Singular solutions of the Poisson equation at edges of three‐dimensional domains and their treatment with a predictor–corrector finite element method

Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ? ?³ with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H¹(Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions.     

Local and Global Existence of Solutions to Initial Value Problems of Modified Nonlinear Kawahara Equations

This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third ＋α the second partial dervative of u to x ,the second the third ＋β the third partial dervative of u to x ,the second the thire ＋γ the fifth partial dervative of u to x = 0,（x,t）∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo（x） ∈ H^s（R） with s ≥ 1/4, and a global solution exists if s ≥ 2.     

Global spherically symmetric classical solution to the Navier-Stokes-Maxwell system with large initial data and vacuum

We study the initial boundary value problem to the system of the compressible Navier-Stokes equations coupled with the Maxwell equations through the Lorentz force in a bounded annulus Ω of R3. And a result on the existence and uniqueness of global spherically symmetric classical solutions is obtained. Here the initial data could be large and initial vacuum is allowed.     

Global Existence and Convergence Rates of Smooth Solutions for the 3-D Compressible Magnetohydrodynamic Equations Without Heat Conductivity

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H³-norm of the initial perturbation around a constant states is small enough and its L¹-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.     

Remarks on an S-shaped bifurcation curve

We study the non-negative solutions of the boundary value problem $\text{?Δu = λ [}\text{exp}\text{αu}\text{(α + u)}\text{]; x ? Ω. u = 0; x ? ?Ω,}\text{where}\text{α > 0, λ ? 0, Ω ? R}{}^{\mathrm{n}}$ is bounded with smooth boundary ?Ω.This problem arises in the theory of combustion. We study the estimates on the supremum norm of the solutions and estimates on the critical values of λ.     

On the existence of steady flows of a Navier–Stokes liquid around a moving rigid body

We prove the existence of a strong solution to the three‐dimensional steady Navier–Stokes equations in the exterior of an obstacle undergoing a rigid motion. Unlike the classical exterior problem for the Navier–Stokes equations, that only takes into account the translational motion of the obstacle, is this case, the obstacle can also rotate. Assuming the total flux of the velocity field through the boundary to be sufficiently small, we first construct approximating solutions in bounded regions Ω_R = Ω∩ {x ∈ ?³:∣x∣< R} invading the liquid domain Ω. A set of estimates independent of R are shown to hold for the approximating solutions which allows to obtain a strong solution by taking the limit R→∞. Copyright © 2004 John Wiley & Sons, Ltd.     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

Identifying a spherically symmetric conductivity in a nonlinear parabolic equation

In this article we determine, i.e. we prove existence and uniqueness of a radially symmetric operator conductivity in a nonlinear heat equation of the form (1.1) related to a spherical corona Ω ? R ³ under a suitable additional information.     

The problem of recovering the permittivity coefficient from the modulus of the scattered electromagnetic field

Under consideration is the stationary system of equations of electrodynamics relating to a nonmagnetic nonconducting medium. We study the problem of recovering the permittivity coefficient ε from given vectors of electric or magnetic intensities of the electromagnetic field. It is assumed that the field is generated by a point impulsive dipole located at some point y. It is also assumed that the permittivity differs from a given constant ε₀ only inside some compact domain Ω ? R³ with smooth boundary S. To recover ε inside Ω, we use the information on a solution to the corresponding direct problem for the system of equations of electrodynamics on the whole boundary of Ω for all frequencies from some fixed frequency ω ₀ on and for all y ∈ S. The asymptotics of a solution to the direct problem for large frequencies is studied and it is demonstrated that this information allows us to reduce the initial problem to the well-known inverse kinematic problem of recovering the refraction index inside Ω with given travel times of electromagnetic waves between two arbitrary points on the boundary of Ω. This allows us to state uniqueness theorem for solutions to the problem in question and opens up a way of its constructive solution.     

欧拉 - 玻尔兹曼方程整体光滑解

本文证明了R^d 中具有某一类小初值的等熵欧拉 - 玻尔兹曼方程整体光滑解的存在性.本文首先构造了等熵欧拉 - 玻尔兹曼方程的局部解, 并证明了局部解的适定性. 此外,文中还构造了关于原方程的随时间 t 增加、具有良好的衰减性质的整体光滑背景解. 同时, 当方程的辐射项系数满足一定条件时, 本文建立了关于源项的估计.通过将背景解的衰减与源项的估计结合起来, 文中证明了存在整数 s>d/2 + 1 ,使得背景解与原方程解的 H^s(R^d)x L²(R₊ x S^d-1;H^s(R^d))范数之差始终是有界的, 从而保证了原方程整体光滑解的存在性.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

On an A Priori Estimate for Solutions of an Elliptic Equation

In the investigation of the spectral theory of non-selfadjoint elliptic boundary value problems involving an indefinite weight function, there arises the problem of obtaining L^p a priori estimates for solutions about points of discontinuity of the weight function. Here we deal with this problem for the case where the weight function vanishes on a set of positive measure.     

Geometric Characterizations for Variational Minimization Solutions of the 3-Body Problem

Abstract In this paper, we prove that for any given positive masses the variational minimization solutions of the 3-body problem in R ³ or R ² are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular restricted 3-body problem in R ³ or R ² are also planar equilateral triangle circular solutions. *Partially supported by the NNSF and MCME of China, the Qiu Shi Sci. and Tech. Foundation, and Edu. Comm. of Tianjin City. Associate Member of the ICTP. **Partially supported by the NNSF of China     

Global existence of solutions to the compressible Navier-Stokes equation around parallel flows

The initial boundary value problem for the compressible Navier-Stokes equation is considered in an infinite layer of Rn. It is proved that if n?3, then strong solutions to the compressible Navier-Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations, provided that the Reynolds and Mach numbers are sufficiently small. The proof is given by a variant of the Matsumura-Nishida energy method based on a decomposition of solutions associated with a spectral property of the linearized operator.     

Spectral properties for layer potentials associated to the Stokes equation in Lipschitz domains

In this paper, we consider the spectral properties of the double layer potentials K and \({\tilde{K}}\) related to the traction boundary value problem and the slip boundary value problem, respectively, of the Stokes equations in a bounded Lipschitz domain Ω in R ⁿ . We show the invertibility of λI ? K and \({\lambda I - \tilde{K}}\) in L ²(?Ω) for \({\lambda \in {\bf R}{\setminus} [-\frac 12, \frac12]}\). As an application, we study the transmission problems of the Stokes equations.