共查询到20条相似文献,搜索用时 32 毫秒
1.
Sven Winklmann 《Calculus of Variations and Partial Differential Equations》2003,16(4):439-447
We define a generalized notion of mean curvature for regular hypersurfaces in . This enables us to introduce a new class of geometric curvature flows for which we prove enclosure theorems, using methods
of Dierkes [D] and Hildebrandt [H]. In particular, we obtain “neck-pinching” results that generalize previous observations
by Ecker [E] concerning the classical mean curvature flow.
Received: 8 October 2001 / Accepted: 1 March 2002 / Published online: 23 May 2002 相似文献
2.
In this paper, we study the asymptotic behavior of solutions to the simplified Ginzburg–Landau model for superconductivity. We prove that, asymptotically, vortex-filaments evolves according to the mean curvature flow in the sense of weak formulation. This can be seen as a first attempt to understand the nature of the motion of vortex filaments in three dimensions with magnetic field. On the other hand, this paper revisits the pioneering work of Bethuel–Orlandi–Smets [F. Bethuel, G. Orlandi, D. Smets, Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature, Ann. of Math. 163 (2006) 37–163] in a slightly relaxed setting. 相似文献
3.
Santiago Cano-Casanova Julián López-Gómez Kazuhiro Takimoto 《Journal of Differential Equations》2012,252(1):323-343
This paper studies a quasilinear perturbation, through the mean curvature flow operator, of the classical linear heat equation. The mean curvature has the effect of maintaining bounded all classical positive steady-states of the model. 相似文献
4.
Well‐posedness of a parabolic free boundary problem driven by diffusion and surface tension 下载免费PDF全文
Martijn M. Zaal 《Mathematical Methods in the Applied Sciences》2015,38(2):380-392
Well‐posedness and regularity results are shown for a class of free boundary problems consisting of diffusion on a free domain where the boundary movement depends on its mean curvature of the boundary and the diffusion on the boundary, and initial conditions are radially symmetric. Short‐time existence and uniqueness of solutions in a suitable Sobolev space are shown using a fixed‐point argument. Higher regularity is a posteriori. Finally, it is shown that solutions exist globally in time and converge to equilibrium if the boundary movement depends on the mean curvature of the boundary and diffusion in a specific way. A mathematical model describing the swelling of a cell due to osmosis is treated as an example. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
5.
Philip Broadbridge Joanna M. Goard 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2004,55(3):534-538
An exact solution is given for the evolution of an
initially v-shaped surface by a fully nonlinear diffusion
equation. This is the unique generalized solution that is
continuous but not twice differentiable. Since the profile
velocity decreases faster than the reciprocal of the profile
curvature, the point of infinite curvature persists for a finite
positive time. 相似文献
6.
We introduce a new sharp interface model for the flow of two immiscible, viscous, incompressible fluids. In contrast to classical models for two-phase flows we prescribe an evolution law for the interfaces that takes diffusional effects into account. This leads to a coupled system of Navier–Stokes and Mullins–Sekerka type parts that coincides with the asymptotic limit of a diffuse interface model. We prove the long-time existence of weak solutions, which is an open problem for the classical two-phase model. We show that the phase interfaces have in almost all points a generalized mean curvature. 相似文献
7.
Hongjing Pan 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(4):1234-1260
We investigate various properties of time maps for one-dimensional prescribed mean curvature equations. Using these properties, we obtain some exact multiplicity results of positive solutions and sign-changing solutions. As it turned out, these quasilinear problems show many different phenomena from semilinear problems. Our methods are based on a detailed analysis of time maps. 相似文献
8.
We show that almost all level sets of the unique viscosity solution for general anisotropic mean curvature flow satisfy a
weak form of the flow equation. This generalizes the case of isotropic mean curvature flow studied by Evans and Spruck, in
which a relation between the viscosity solution and Brakke's varifold mean curvature flow is established.
Received June 4, 1998 / Accepted February 26, 1999 相似文献
9.
Yu-Chu Lin 《Journal of Differential Equations》2009,247(9):2620-2636
Motivated by a recent curvature flow introduced by Professor S.-T. Yau [S.-T. Yau, Private communication on his “Curvature Difference Flow”, 2007], we use a simple curvature flow to evolve a convex closed curve to another one (under the assumption that both curves have the same length). We show that, under the evolution, the length is preserved and if the curvature is bounded above during the evolution, then an initial convex closed curve can be evolved to another given one. 相似文献
10.
11.
Bogdan-Vasile Matioc 《Archiv der Mathematik》2007,89(4):365-372
The motion of surfaces by their mean curvature has been studied by several authors from different points of view. K. A. Brake
studied this problem from the geometric measure theory point of view, the parametric problem was studied by G. Huisken [5].
Nonparametric mean curavture flow with boundary conditions was studied in [6] and [7]. Rotationally symmetric mean curvature
flows have been treated by G. Dziuk, B. Kawohl [3], but also by S. Altschuler, S. B. Angenent and Y. Giga [2].
In this paper we consider the case in which the initial surface has rotational symmetry and we shall generalize the results
in [3] in the sense that we shall give more general boundary conditions which enforce the formation of a singularity in finite
time. The proofs rely entirely on parabolic maximum principles.
Received: 6 September 2006 相似文献
12.
In this paper, we prove the existence of classical solutions to the Dirichlet problem of a class of quasi-linear elliptic equations on an unbounded cone and a U-type domain in Rn(n?2). This problem comes from the study of mean curvature flow or its generalization, the flow by powers of mean curvature. Our approach is a modified version of the classical Perron method, where the solutions to the minimal surface equation are used as sub-solutions and a family auxiliary functions are constructed as super-solutions. 相似文献
13.
This paper deals with positive solutions of degenerate and strongly coupled quasi-linear parabolic system not in divergence form: ut=vp(u+au), vt=uq (v+bv) with null Dirichlet boundary condition and positive initial condition, where p, q, a and b are all positive constants, and p, q 1. The local existence of positive classical solution is proved. Moreover, it will be proved that: (i) When min {a, b} 1 then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm (we can not prove the uniqueness result in general); (ii) When min {a, b} > 1, there is no global positive classical solution (we can not still prove the uniqueness result), if in addition the initial datum (u0v0) satisfies u0 + au0 0, v0+bv0 0 in , then the positive classical solution is unique and blows up in finite time, where 1 is the first eigenvalue of – in with homogeneous Dirichlet boundary condition.This project was supported by PRC grant NSFC 19831060 and 333 Project of JiangSu Province. 相似文献
14.
We study a two-phase free boundary problem in which the speed of the free boundary depends also on its curvature. It is assumed
that the free boundary is Lipschitz and it is proved that the solution as well as the free boundary are classical. 相似文献
15.
Yu. R. Romanovsky 《Acta Appl Math》1989,15(1-2):149-160
Two aims are pursued in this article. The first one is methodological and consists of demonstrating the symmetry calculation techniques based on the commutation relation on the example of the quasi-linear heat equation. The second one consists of investigating the following open question: is the algebra of higher symmetries of the linear heat equation exhausted by those which may be obtained by means of the classical symmetries and well-known recursion operators? A positive solution of this question is given. 相似文献
16.
The aim of this work is to ascertain the characterization of the existence of coexistence states for a class of cooperative systems supported by the study of an associated non-local equation through classical variational methods. Thanks to those results, we are able to obtain the blow-up behavior of the solutions in the whole domain for certain values of the main continuation parameter. 相似文献
17.
Andrew Stone 《Calculus of Variations and Partial Differential Equations》1994,2(4):443-480
We study singularity formation in the mean curvature flow of smooth, compact, embedded hypersurfaces of non-negative mean curvature in
n+1, primarily in the boundaryless setting. We concentrate on the so-called Type I case, studied by Huisken in [Hu 90], and extend and refine his results. In particular, we show that a certain restriction on the singular points covered by his analysis may be removed, and also establish results relating to the uniqueness of limit rescalings about singular points, and to the existence of slow-forming singularities of the flow.The main new ingredient introduced, to address these issues, is a certain density function, analogous to the usual density function in the study of harmonic maps in the stationary setting. The definition of this function is based on Huisken's important monotonicity formula for mean curvature flow. 相似文献
18.
A diffusive Lotka–Volterra competition model with nonlocal intraspecific and interspecific competition between species is formulated and analyzed. The nonlocal competition strength is assumed to be determined by a diffusion kernel function to model the movement pattern of the biological species. It is shown that when there is no nonlocal intraspecific competition, the dynamics properties of nonlocal diffusive competition problem are similar to those of classical diffusive Lotka–Volterra competition model regardless of the strength of nonlocal interspecific competition. Global stability of nonnegative constant equilibria are proved using Lyapunov or upper–lower solution methods. On the other hand, strong nonlocal intraspecific competition increases the system spatiotemporal dynamic complexity. For the weak competition case, the nonlocal diffusive competition model may possess nonconstant positive equilibria for some suitably large nonlocal intraspecific competition coefficients. 相似文献
19.
C.V. Pao 《Journal of Differential Equations》2010,248(5):1175-540
Coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems, including systems of parabolic and ordinary differential equations are investigated. The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i=1,…,N, and the boundary condition is ui=0. Using the method of upper and lower solutions, we show that a unique global classical time-dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a scalar polynomial growth problem, a coupled system of polynomial growth problem, and a two component competition model in ecology. 相似文献
20.
Rafael López 《Journal of Differential Equations》2003,194(1):185-197
In this work, we give a priori height and gradient estimates for solutions of the prescribed constant Gauss curvature equation in Euclidean space. We shall consider convex radial graphs with positive constant mean curvature. The estimates are established by considering in such a graph, the Riemannian metric given by the second fundamental form of the immersion. 相似文献