共查询到20条相似文献,搜索用时 15 毫秒
1.
Shu-Yu Hsu 《Geometriae Dedicata》2013,162(1):375-388
We give a simple proof for the rotational symmetry of ancient solutions of Ricci flow on surfaces. As a consequence we obtain a simple proof of some results of Daskalopoulos, Hamilton and Sesum on the a priori estimates for the ancient solutions of Ricci flow on surfaces. We also give a simple proof for the solution to be a Rosenau solution under some mild conditions on the solutions of Ricci flow on surfaces. 相似文献
2.
We consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface. The boundaries of the surfaces are assumed to remain fixed. G. Huisken has shown that if the boundary of the domain over which this flow is considered satisfies the “mean curvature” condition of H. Jenkins and J. Serrin (that is, the boundary of the domain is convex “in the mean”) then the corresponding initial boundary value problem with Dirichlet boundary data and smooth initial data admits a smooth solution for all time. In this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of Jenkins-Serrin. In this case, even if the flow starts with smooth initial data and homogeneous Dirichlet boundary data, singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition. We prove, however, existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time. From that moment on such solutions assume the boundary values in the classical sense. We also give sufficient conditions that guarantee the existence of classical solutions for all time t ≧ 0. In addition, we establish estimates of the rate at which solutions tend to zero as t → ∞. 相似文献
3.
Yu. N. Sadkov 《Computational Mathematics and Modeling》1997,8(2):161-164
We consider the flow of supersonic homogeneous gas past a supersonic spherical source. This problem provides a gas-dynamic
model of the interaction of interstellar wind with solar wind, and is thus also of independent interest. It is solved using
an explicit through divergence scheme of third-order approximation. The analysis focuses on formation and stability of the
structure of discontinuity surfaces and convergence to the stationary solution. The results are compared qualitatively and
quantitatively with solutions obtained by other methods.
Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 125–128. 相似文献
4.
V. G. Vyskrebtsov 《Journal of Mathematical Sciences》2001,104(5):1456-1463
We investigate self-similar solutions of the Navier–Stokes equations for the axisymmetric flow of a viscous incompressible fluid. The original equations are transformed by the Slezkin method. On the basis of analysis of physical properties of the flow and the Slezkin general equation, we show that, in parallel with the known solutions of this equation, there exist several other solutions with physical meaning. We consider the simplest case of irrotational flows for which current lines may be circles, ellipses, parabolas, and hyperbolas. Unlike the Landau and Squire solutions, these flows are interpreted as nonjet flows of fluid flowing into and out of a homogeneous porous axially symmetric body. 相似文献
5.
Maria Buzano 《Annals of Global Analysis and Geometry》2014,45(1):25-45
We consider the Ricci flow equation for invariant metrics on compact and connected homogeneous spaces whose isotropy representation decomposes into two irreducible inequivalent summands. By studying the corresponding dynamical system, we completely describe the behaviour of the homogeneous Ricci flow on this kind of spaces. Moreover, we investigate the existence of ancient solutions and relate this to the existence and non-existence of invariant Einstein metrics. 相似文献
6.
Jeffrey Streets 《Advances in Mathematics》2010,223(2):454-3542
We study the behavior of the Ricci Yang-Mills flow for U(1) bundles on surfaces. By exploiting a coupling of the Liouville and Yang-Mills energies we show that existence for the flow reduces to a bound on the isoperimetric constant or the L4 norm of the bundle curvature. We furthermore completely describe the behavior of long time solutions of this flow on surfaces. Finally, in Appendix A we classify all gradient solitons of this flow on surfaces. 相似文献
7.
John Lott 《Mathematische Annalen》2007,339(3):627-666
We show that three-dimensional homogeneous Ricci flow solutions that admit finite-volume quotients have long-time limits given
by expanding solitons. We show that the same is true for a large class of four-dimensional homogeneous solutions. We give
an extension of Hamilton’s compactness theorem that does not assume a lower injectivity radius bound, in terms of Riemannian
groupoids. Using this, we show that the long-time behavior of type-III Ricci flow solutions is governed by the dynamics of
an -action on a compact space.
This work was supported by NSF grant DMS-0306242 相似文献
8.
A. V. Efimov 《Proceedings of the Steklov Institute of Mathematics》2012,279(1):93-109
We present a systematic approach to solving the problem of affine homogeneity of real hypersurfaces in the three-dimensional complex space. This question is an important part of the general problem of holomorphic classification of homogeneous real hypersurfaces in three-dimensional complex spaces. In contrast to the two-dimensional case, the whole problem (just as its affine part) has not yet been fully studied, although there exist a large number of examples of homogeneous manifolds. We study only the class of tubular type surfaces, which is defined by conditions imposed on the 2-jet of their canonical equations and generalizes the class of tube manifolds. We discuss the procedure of describing all matrix Lie algebras corresponding to the homogeneous manifolds under consideration. In the class that we study, we distinguish four cases depending on the third-order Taylor coefficients of the canonical equations; in three of these cases, the Lie algebras and the corresponding affine homogeneous surfaces are completely described. The key point of the proposed approach is the solution of a large system of quadratic equations that corresponds to each of the homogeneous surfaces. 相似文献
9.
Robert A. Van Gorder 《Applied mathematics and computation》2010,216(7):2177-8079
We frame a hierarchy of nonlinear boundary value problems which are shown to admit exponentially decaying exact solutions. We are able to convert the question of the existence and uniqueness of a particular solution to this nonlinear boundary value problem into a question of whether a certain polynomial has positive real roots. Furthermore, if such a polynomial has at least two distinct positive roots, then the nonlinear boundary value problem will have multiple solutions. In certain special cases, these boundary value problems arise in the self-similar solutions for the flow of certain fluids over stretching or shrinking sheets; examples given include the flow of first and second grade fluids over such surfaces. 相似文献
10.
We investigate the motion of homogeneous, spheroidal particles immersed in an incompressible, viscous fluid. We assume the particles to be more dense than the surrounding fluid and small enough that inertia is negligible with respect to viscous forces. We give exact solutions for the motion of the particle’s center of mass for steady, linear flows, either irrotational or without strain. For a weakly strained, two-dimensional, rotational flow we give an asymptotic approximation to the solutions, and we compare it with numerical solutions. In the presence of vorticity we find that the spheroid moves along three-dimensional, non-planar paths. With pure strain the three-dimensionality of the paths is transient. If a two-dimensional rotational flow is perturbed by strain, then the generic path of a spheroid is an open curve, even if all the streamlines of the flow are closed. We conclude by speculating about the significance of these findings for the ecology of phytoplankton. 相似文献
11.
In this note we investigate some basic properties of the level sets of functions which are homogeneous with respect to nonisotropic
dilations. In particular we obtain a formula for the volume of the level sets in terms of the area on the level surfaces.
We relate the results to some well known mean value formulas for solutions of PDE’s. 相似文献
12.
E. V. Altukhov 《Journal of Mathematical Sciences》1998,90(1):1801-1805
By the method of homogeneous solutions we study the stressed state of a transversally isotropic cylinder under local heating
of the lateral surfaces. We describe the results of numerical studies as functions of the elastic properties of the material,
the geometric characteristics, and the variation of the heating zone on one of the lateral surfaces of the cylinder. Five
figures, 3 tables. Bibliography: 3 titles.
Translated fromTeoreticheskaya i Prikladnaya Mekhanika, Vol. 27, 1997, pages 3–10. 相似文献
13.
Recenty, convergent and nonnegativity preserving numerical schemes for the thin film equation have been developed and applied successfully to model the flow of thin films of Newtonian liquids on homogeneous surfaces. It is the aim of this paper to discuss two related, but more involved physical settings. The first topic is flow on chemically structured surfaces which is of great interest in a number of microfluidic applications. We will formulate a convergent, nonnegativity preserving scheme and compare the simulations with recent physical experiments. The second part deals with power‐law‐fluids. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
14.
15.
Shu-Cheng Chang 《Mathematische Annalen》2000,318(2):315-340
Abstract. In this paper, we prove some compactness theorems and collapse phenomenon on compact K?hler surfaces with stable tangent
bundle. We then apply the results to the Calabi flow. More precisely, we prove, under suitable curvature conditions, the longtime
existence and asymptotic convergence for solutions of the Calabi flow on compact K?hler surfaces admitting no nonzero holomorphic
tangent vector fields and with stable tangent bundle. We also give some examples where the Calabi flow blows up.
Received January 7, 1999 / Revised February 2, 2000 / Published online July 20, 2000 相似文献
16.
《Differential Geometry and its Applications》1999,10(1):1-26
The space of lines in R3 can be viewed as a four dimensional homogeneous space of the group of Euclidean motions, E(3). Line congruences arise in the classical method of transforming one surface to another by lines. These transformations are particularly interesting if some geometric property of the original surface is preserved. Line congruences, then, are two parameter families of lines and can be studied as surfaces in the space of lines. In this paper, we use the method of moving frames to study line congruences. We calculate the first order invariants of line congruences for which there are two real focal surfaces, and give the geometric meaning of these invariants. We look specifically at the case where the two first order invariants are constant and give a simple proof of Bäcklund's Theorem which relates to the transformation of one pseudospherical surface, a surface of constant negative Gaussian curvature, to another. These transformations are of interest since pseudospherical surfaces correspond to solutions to the sine-Gordon equation. We also give a proof of Bianchi's permutability theorem for pseudospherical surfaces in this context. Finally, we use the results of these theorems to generate some pseudospherical surfaces. All of these concepts and results are understood in terms of the structure equations of the line congruence. 相似文献
17.
Hamiltonian Stationary Tori in the Complex Projective Plane 总被引:3,自引:0,他引:3
Hamiltonian stationary Lagrangian surfaces are Lagrangian surfacesin a four-dimensional Kähler manifold which are criticalpoints of the area functional for Hamiltonian infinitesimaldeformations. In this paper we analyze these surfaces in thecomplex projective plane: in a previous work we showed thatthey correspond locally to solutions to an integrable system,formulated as a zero curvature on a (twisted) loop group. Herewe give an alternative formulation, using non-twisted loop groupsand, as an application, we show in detail why Hamiltonian stationaryLagrangian tori are finite type solutions, and eventually describethe simplest of them: the homogeneous ones. 2000 MathematicsSubject Classification 53C55 (primary), 53C42, 53C25, 58E12(secondary). 相似文献
18.
Xuan Hien Nguyen 《偏微分方程通讯》2013,38(3):257-280
In this paper, we describe the construction of new examples of self-translating surfaces under the mean curvature flow. We find the new surfaces by desingularizing the intersection of a grim reaper and a plane to obtain approximate solutions, then we solve a perturbation problem to find the exact solutions. Our work is inspired from Kapoulea' construction of minimal surfaces but differs from it by our more abstract and direct study of the linear operator, via Fredholm operators. 相似文献
19.
We study a system of the reaction–diffusion type, where diffusion coefficients depend in an arbitrary way on spatial variables and concentrations, while reactions are expressed as homogeneous functions whose coefficients depend in a special way on spatial variables. We prove that the system has a family of exact solutions that are expressed through solutions to a system of ordinary differential equations (ODE) with homogeneous functions in right-hand sides. For a special case of theODE systemwe construct a general solution represented by Jacobi higher transcendental functions. We also prove that these periodic solutions are analytic functions that can be expressed near each point on the period by convergent power series. 相似文献
20.
M Asunci n Beitia Juan M. Gracia Inmaculada de Hoyos 《Linear and Multilinear Algebra》1992,31(1):93-118
In this paper we study a homogeneous linear matrix equation related to the block similarity of rectangular matrices. We obtain the dimension of the vector space of its solutions and we describe these solutions. We give a characterization of the block similarity by rank tests. We extend Roth's criterion to the corresponding non homogeneous equation. 相似文献