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1.
In this paper we introduce a class of nonlinear vector fields on infinite dimensional manifolds such that the corresponding
evolution equations can be solved with the same method one uses to solve ordinary differential equations with constant coeficients.
Mostly, these equations are nonlinear partial differential equations. It is shown that these flows are characterized by a
generalization of the ‘method of variation of constants’ which is widely used for second order problems to find general solutions
out of particular ones. Invariant densities are constructed for these flows in a natural way. These invariant densities are
providing an essential tool for solving initial value and boundary value problems for the equations under consideration. Many
applications are presented 相似文献
2.
On the Inviscid Limit Problem of the Vorticity Equations for Viscous Incompressible Flows in the Half‐Plane 下载免费PDF全文
Yasunori Maekawa 《纯数学与应用数学通讯》2014,67(7):1045-1128
We consider the Navier‐Stokes equations for viscous incompressible flows in the half‐plane under the no‐slip boundary condition. By using the vorticity formulation we prove the local‐in‐time convergence of the Navier‐Stokes flows to the Euler flows outside a boundary layer and to the Prandtl flows in the boundary layer in the inviscid limit when the initial vorticity is located away from the boundary. © 2014 Wiley Periodicals, Inc. 相似文献
3.
Boundary integral equations are an important class of methods for acoustic and electromagnetic scattering from periodic arrays
of obstacles. For piecewise homogeneous materials, they discretize the interface alone and can achieve high order accuracy
in complicated geometries. They also satisfy the radiation condition for the scattered field, avoiding the need for artificial
boundary conditions on a truncated computational domain. By using the quasi-periodic Green’s function, appropriate boundary conditions are automatically satisfied on the boundary of the unit cell. There are
two drawbacks to this approach: (i) the quasi-periodic Green’s function diverges for parameter families known as Wood’s anomalies,
even though the scattering problem remains well-posed, and (ii) the lattice sum representation of the quasi-periodic Green’s
function converges in a disc, becoming unwieldy when obstacles have high aspect ratio. 相似文献
4.
The vanishing viscosity limit is considered for the viscous lake equations with Navier friction boundary conditions. We prove that the inviscid limit satisfies the inviscid lake equations, and the results include flows generated by Lp initial vorticity with 1<p?∞. 相似文献
5.
Local well‐posedness for volume‐preserving mean curvature and Willmore flows with line tension 下载免费PDF全文
We show the short‐time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by the volume‐preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non‐local nature of the resulting second order, nonlinear PDE. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a curvilinear cordinate system due to Vogel to write the flows as graphs over a fixed reference hypersurface. 相似文献
6.
7.
Low-Reynolds recirculating cavity flows are traditionally generated from lid-driven boundary motion at a solid-fluid interface or result from shear flow over an opening. Such flows are typically described by the equations of creeping motion, where viscous forces are dominant. We illustrate using Particle Image Velocimetry (PIV) an original family of boundary-driven cavity flows occurring, in contrast to classic configurations, at a liquid-gas interface: thermally-induced Marangoni flows in a thin liquid shell generate forced, steady-state recirculating flows inside the cavity. Forcing relies on viscous mechanisms at the boundary but resulting flow patterns are, however, inviscid. Here, the inviscid equations of fluid motion are not used as an approximation, but rather come as a result from the solution of the creeping motion equations in the region inside the sphere. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
8.
John Urbas 《Proceedings of the American Mathematical Society》2000,128(3):853-855
We give a simple proof of a result of Xinan Ma concerning a necessary condition for the solvability of the two-dimensional Monge-Ampère equation subject to the contact angle or capillarity boundary condition. Our technique works for more general Monge-Ampère equations in any dimension, and also extends to some other boundary conditions.
9.
We consider approximation by partial time steps of a smooth solution of the Navier-Stokes equations in a smooth domain in two or three space dimensions with no-slip boundary condition. For small k > 0, we alternate the solution for time k of the inviscid Euler equations, with tangential boundary condition, and the solution of the linear Stokes equations for time k, with the no-slip condition imposed. We show that this approximation remains bounded in H2,p and is accurate to order k in Lp for p > ∞. The principal difficulty is that the initial state for each Stokes step has tangential velocity at the boundary generated during the Euler step, and thus does not satisfy the boundary condition for the Stokes step. The validity of such a fractional step method or splitting is an underlying principle for some computational methods. © 1994 John Wiley & Sons, Inc. 相似文献
10.
Nassif Ghoussoub 《Frontiers of Mathematics in China》2008,3(2):167-193
Hamiltonian systems with various time boundary conditions are formulated as absolute minima of newly devised non-negative
action functionals obtained by a generalization of Bogomolnyi’s trick of ‘dcompleting squares’. Reminiscent of the selfdual
Yang-Mills equations, they are not derived from the fact that they are critical points (i.e., from the corresponding Euler-Lagrange
equations) but from being zeroes of the corresponding non-negative Lagrangians. A general method for resolving such variational
problems is also described and applied to the construction of periodic solutions for Hamiltonian systems, but also to study
certain Lagrangian intersections.
相似文献
11.
Yuri A. Melnikov 《Central European Journal of Mathematics》2010,8(1):53-72
Convenient for immediate computer implementation equivalents of Green’s functions are obtained for boundary-contact value
problems posed for two-dimensional Laplace and Klein-Gordon equations on some regions filled in with piecewise homogeneous
isotropic conductive materials. Dirichlet, Neumann and Robin conditions are allowed on the outer boundary of a simply-connected
region, while conditions of ideal contact are assumed on interface lines. The objective in this study is to widen the range
of effective applicability for the Green’s function version of the boundary integral equation method making the latter usable
for equations with piecewise-constant coefficients. 相似文献
12.
Xiaoming WANG 《数学年刊B辑(英文版)》2010,31(5):781-792
The author surveys a few examples of boundary layers for which the Prandtl boundary layer theory can be rigorously validated. All of them are associated with the incompressible Navier-Stokes equations for Newtonian fluids equipped with various Dirichlet boundary conditions (specified velocity). These examples include a family of (nonlinear 3D) plane parallel flows, a family of (nonlinear) parallel pipe flows, as well as flows with uniform injection and suction at the boundary. We also identify a key ingredient in establishing the validity of the Prandtl type theory, i.e., a spectral constraint on the approximate solution to the Navier-Stokes system constructed by combining the inviscid solution and the solution to the Prandtl type system. This is an additional difficulty besides the wellknown issue related to the well-posedness of the Prandtl type system. It seems that the main obstruction to the verification of the spectral constraint condition is the possible separation of boundary layers. A common theme of these examples is the inhibition of separation of boundary layers either via suppressing the velocity normal to the boundary or by injection and suction at the boundary so that the spectral constraint can be verified. A meta theorem is then presented which covers all the cases considered here. 相似文献
13.
Joseph W. Jerome 《Numerische Mathematik》2008,109(1):121-142
We consider nonlinear elliptic systems, with mixed boundary conditions, on a convex polyhedral domain Ω ⊂ R
N
. These are nonlinear divergence form generalizations of Δu = f(·, u), where f is outward pointing on the trapping region boundary. The motivation is that of applications to steady-state reaction/diffusion
systems. Also included are reaction/diffusion/convection systems which satisfy the Einstein relations, for which the Cole-Hopf
transformation is possible. For maximum generality, the theory is not tied to any specific application. We are able to demonstrate
a trapping principle for the piecewise linear Galerkin approximation, defined via a lumped integration hypothesis on integrals
involving f, by use of variational inequalities. Results of this type have previously been obtained for parabolic systems by Estep, Larson,
and Williams, and for nonlinear elliptic equations by Karátson and Korotov. Recent minimum and maximum principles have been
obtained by Jüngel and Unterreiter for nonlinear elliptic equations. We make use of special properties of the element stiffness
matrices, induced by a geometric constraint upon the simplicial decomposition. This constraint is known as the non-obtuseness
condition. It states that the inward normals, associated with an arbitrary pair of an element’s faces, determine an angle
with nonpositive cosine. Drăgănescu, Dupont, and Scott have constructed an example for which the discrete maximum principle
fails if this condition is omitted. We also assume vertex communication in each element in the form of an irreducibility hypothesis
on the off-diagonal elements of the stiffness matrix. There is a companion convergence result, which yields an existence theorem
for the solution. This entails a consistency hypothesis for interpolation on the boundary, and depends on the Tabata construction
of simple function approximation, based on barycentric regions.
This work was supported by the National Science Foundation under grant DMS-0311263. 相似文献
14.
The adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard
explicit residual-based a posteriori error estimator. Each cycle of the adaptive loop consists of the steps ‘SOLVE’, ‘ESTIMATE’,
‘MARK’, and ‘REFINE’. The techniques from the unrestricted variational problem are modified for the convergence analysis to
overcome the lack of Galerkin orthogonality. We establish R-linear convergence of the part of the energy above its minimal
value, if there is appropriate control of the data oscillations. Surprisingly, the adaptive mesh-refinement algorithm is the
same as in the unconstrained case of a linear PDE—in fact, there is no modification near the discrete free boundary necessary
for R-linear convergence. The arguments are presented for a model obstacle problem with an affine obstacle χ and homogeneous
Dirichlet boundary conditions. The proof of the discrete local efficiency is more involved than in the unconstrained case.
Numerical results are given to illustrate the performance of the error estimator. 相似文献
15.
The proposal of this note is to derive the equations of boundary layers in the small viscosity limit for the two-dimensional incompressible Navier–Stokes equations defined in a curved bounded domain with the non-slip boundary condition. By using curvilinear coordinate system in a neighborhood of boundary, and the multi-scale analysis we deduce that the leading profiles of boundary layers of the incompressible flows in a bounded domain still satisfy the classical Prandtl equations when the viscosity goes to zero, which are the same as for the flows defined in the half space. 相似文献
16.
Takemi Yanagimoto Masaaki Sibuya 《Annals of the Institute of Statistical Mathematics》1972,24(1):423-434
Summary Definitions of different strengths are given to the notion of ‘a positively biased random variable’. This notion is related
to that of ‘a stochastically larger component of a two-dimensional random vector’, which was introduced previously by the
authors. Properties of common rank tests of symmetry about zero against our specification of alternatives are studied in detail.
The positive biasedness is extended to ‘positively more biased’. Test of symmetry of a two-dimensional random vector is also
referred to. 相似文献
17.
M. R. Pinheiro 《Optimization Letters》2009,3(1):1-6
In this revisional article, we criticize (strongly) the use made by Medar et al., and those whose work they base themselves
on, of the name ‘convexity’ in definitions which intend to relate to convex functions, or cones, or sets, but actually seem
to be incompatible with the most basic consequences of having the name ‘convexity’ associated to them. We then believe to
have fixed the ‘denominations’ associated with Medar’s (et al.) work, up to a point of having it all matching the existing
literature in the field [which precedes their work (by long)]. We also expand his work scope by introducing s
1-convexity concepts to his group of definitions, which encompasses only convex and its proper extension, s
2-convex, so far. This article is a long version of our previous review of Medar’s work, published by FJMS (Pinheiro, M.R.:
S-convexity revisited. FJMS, 26/3, 2007). 相似文献
18.
V. I. Zhuk 《Journal of Applied Mathematics and Mechanics》1999,63(6):893-908
Large-scale structures with an inviscid, non-linear subdomain (deck) on the bottom of a boundary layer in the case of subsonic and transonic free stream velocities are considered. A class of locally inviscid perturbations with an internal line of discontinuity of the tangential velocity, which leads to the appearance of a free term on the right-hand side of the Benjamin-Ono equations, is investigated. The shape of the above-mentioned line is sought and it is determined from the solution of a system of one-dimensional non-stationary equations in which, apart from the Benjamin-Ono equation, a kinematic condition and an equation for the inviscid deck close to the wall also occur. An example of a periodic, non-linear solution is constructed and amplitude constraints which ensure its realization are formulated. 相似文献
19.
Michal Beneš 《Archiv der Mathematik》2009,93(3):287-297
We study regularity of viscous incompressible fluid flows in a 2D channel with “do nothing” outflow boundary condition on
the output for the steady Stokes and Navier–Stokes equations. 相似文献
20.
Ellipticity of a manifold with edges and boundary is connected to boundary and edge conditions that complete corresponding
operators to Fredholm operators between weighted Sobolev spaces. We study a new parameter-dependent calculus of elliptic operators,
where the interior symbols have specific properties on the boundary. We construct elliptic operators with a prescribed number
of edge conditions and obtain isomorphisms in the scale of edge Sobolev spaces.
Supported by the Chinese-German Cooperation Program ‘Partial Differential Equations’, NSFC of China and DFG of Germany. 相似文献