Concentration sets of the Landau–Lifshitz system and quasi-mean curvature flows

The aim of this work is to analyze the concentration set of the stationary weak solutions to the Landau-Lifshitz system of the ferromagnetic spin chain from R ^m into the unit sphere S ² of R ³ . Suppose that u _k → u weakly in W ^1,2(R ^m × R ₊, S ²) and that Σ ^t is the concentration set for fixed t. In the present paper we first prove that Σ ^t is a -rectifiable set for almost all t ∈ R ₊. And then we verify that Σ ^t moves by the quasi-mean curvature under some assumptions, which is a new codimension 2 curvature flow. Finally we analyze the behavior of the solution at the singular point and get the blow up formulas. The main barrier to Landau–Lifshtiz system is that there is no energy monotonicity inequality. After the seminal work to on the study of the concentration set of minimizing energy harmonic maps by Leon Simon, there are several papers dealing with the stationary harmonic maps and its heat flows, and so on. This investigation is inspired by the study on the heat flow of harmonic maps and it largely depends on our result of the partial regularity.     

On the singular set of stationary harmonic maps

LetM andN be compact riemannian manifolds, andu a stationary harmonic map fromM toN. We prove thatH ⁿ⁻² (Σ)=0, wheren=dimM and Σ is the singular set ofu. This is a generalization of a result of C. Evans [7], where this is proved in the special caseN is a sphere. We also prove that, ifu is a weakly harmonic map inW ^1,n (M, N), thenu is smooth. This extends results of F. Hélein for the casen=2, or the caseN is a sphere ([9], [10]).     

The blow-up rate for a system of heat equations with neumann boundary conditions

This paper deals with the blow-up properties of solutions to a system of heat equations u _t=Δu, v _t=Δv in B _R×(0, T) with the Neumann boundary conditions εu/εη=e ^v, εv/εη=e ^u on S _R×[0, T). The exact blow-up rates are established. It is also proved that the blow-up will occur only on the boundary. This work is supported by the National Natural Science Foundation of China     

Asymptotiques d’un systeme dynamique conservatif associe a une equation elliptique non lineaire

LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar ^P-B. We prove that any non-negative solutionu ofu _tt+Δ_gu=f(u) onM x ℝ⁺ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ _g Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C ²(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that for every λ the map (u(0,·),u _t(0,·))→(u(t,·), u _t(t,·)) defines a dynamical system onW ^1/2(M)⊂C(M)×L ²(M) which possesses a compact maximal attractor.      

On Asymptotic Behavior for Singularities of the Powers of Mean Curvature Flow

Let M^n be a smooth, compact manifold without boundary, and F0 ： M^n→ R^n＋1 a smooth immersion which is convex. The one-parameter families F（·, t） ： M^n× [0, T） → R^n＋1 of hypersurfaces Mt^n= F（·,t）（M^n） satisfy an initial value problem dF/dt （·,t） = -H^k（· ,t）v（· ,t）, F（· ,0） = F0（· ）, where H is the mean curvature and u（·,t） is the outer unit normal at F（·, t）, such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton.     

Nonstationary weak limit of a stationary harmonic map sequence

Let M and N be two compact Riemannian manifolds. Let u_k be a sequence of stationary harmonic maps from M to N with bounded energies. We may assume that it converges weakly to a weakly harmonic map u in H^1,2 (M, N) as k → ∞. In this paper, we construct an example to show that the limit map u may not be stationary. © 2002 Wiley Periodicals, Inc.     

Finite Time Blow-up for a Non-linear Parabolic Equation with a Gradient Term and Applications

We give new finite time blow-up results for the non-linear parabolic equations u_t−Δu = u^p and u_t−Δu+μ∣∇u∣^q = u^p. We first establish an a priori bound in L^p+1 for the positive non-decreasing global solutions. As a consequence, we prove in particular that for the second equation on ℝ^N, with q = 2p/(p+1) and small μ>0, blow-up can occur for any N≥1, p>1, (N−2)p<N+2 and without energy restriction on the initial data. Incidentally, we present a simple model in population dynamics involving this equation.     

Generalizations of coatomic modules

For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N+X=M with M/X singular, we have X=M. Let ℘ be the class of all singular simple modules. Then δ(M)=Σ{ L≤ M| L is a δ-small submodule of M} = Re _jm(℘)=∩{ N⊂ M: M/N∈℘. We call M δ-coatomic module whenever N≤ M and M/N=δ(M/N) then M/N=0. And R is called right (left) δ-coatomic ring if the right (left) R-module R _R(_RR) is δ-coatomic. In this note, we study δ-coatomic modules and ring. We prove M=⊕ _i=1 ⁿ M_i is δ-coatomic if and only if each M _i (i=1,…, n) is δ-coatomic.     

Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension

Let f:Σ₁↦Σ₂ be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in Σ₁×Σ₂ by the mean curvature flow. Under suitable conditions on the curvature of Σ₁ and Σ₂ and the differential of the initial map, we show that the flow exists smoothly for all time. At each instant t, the flow remains the graph of a map f _t and f _t converges to a constant map as t approaches infinity. This also provides a regularity estimate for Lipschitz initial data. Oblatum 30-I-2001 & 24-X-2001?Published online: 1 February 2002     

Evolution equations with a free boundary condition

In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary) with a free boundary condition. That is, the following initial boundary value problem ∂₁,u −Δu = Γ(u)(∇u, ∇u) [tT T_{u u}N, on M × [0, ∞), u(t, x) ∈ Σ, for x ∈ ∂M, t > 0, ∂u/t6n(t, x) ⊥_u T_u(t,x) Σ, for x ∈ ∂M, t > 0, u(o,x) = u_o(x), on M, where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M. Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties, even for the case N = ℝⁿ, in which the nonlinear equation reduces to ∂_tu-Δu = 0. We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical conditions on N and Σ which are weaker than K_N <-0 and Σ is totally geodesic in N.     

Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$

We prove the existence of a global heat flow u : Ω ×  \mathbbR⁺ ? \mathbbR^N {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR⁺ {\mathbb{R}^{+}}) ⊂  \mathbbRⁿ {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbR^N {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbR^N {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.     

带非局部源的退化半线性抛物方程的解的爆破性质

This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u _t − (x ^a u _x ) _x =∫ ₀ ^a f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely determined for the special case: f(u)=u ^p , p>1.     

Unique continuation for parabolic operators

It is shown that if a functionu satisfies a backward parabolic inequality in an open set Ω∉R ⁿ⁺¹ and vanishes to infinite order at a point (x ₀·t ₀) in Ω, thenu(x, t ₀)=0 for allx in the connected component ofx ₀ in Ω⌢(R ⁿ ×{t ₀}).     

The wave equation for Dunkl operators

Let k = (k_α)_αε, be a positive-real valued multiplicity function related to a root system , and Δ_k be the Dunkl-Laplacian operator. For (x, t) ε ^N, × , denote by u_k(x, t) the solution to the deformed wave equation Δ_ku_k,(x, t) = δ_ttu_k(x, t), where the initial data belong to the Schwartz space on ^N. We prove that for k 0 and N l, the wave equation satisfies a weak Huygens' principle, while a strict Huygens' principle holds if and only if (N − 3)/2 + Σ_αε+k_α ε . Here ⁺ is a subsystem of positive roots. As a particular case, if the initial data are supported in a closed ball of radius R > 0 about the origin, the strict Huygens principle implies that the support of u_k(x, t) is contained in the conical shell {(x, t), ε ^N × | |t| − R x |t| + R}. Our approach uses the representation theory of the group SL(2, ), and Paley-Wiener theory for the Dunkl transform. Also, we show that the (t-independent) energy functional of u_k is, for large |t|, partitioned into equal potential and kinetic parts.     

A lower bound on the energy of travelling waves of fixed speed for the Gross-Pitaevskii equation

In this paper we consider the Gross-Pitaevskii equation iu _t = Δu + u(1 − |u|²), where u is a complex-valued function defined on \Bbb R^N×\Bbb R{\Bbb R}^N\times{\Bbb R} , N ≥ 2, and in particular the travelling waves, i.e., the solutions of the form u(x, t) = ν(x ₁ − ct, x ₂, …, x _N ), where c ? \Bbb Rc\in{\Bbb R} is the speed. We prove for c fixed the existence of a lower bound on the energy of any non-constant travelling wave. This bound provides a non-existence result for non-constant travelling waves of fixed speed having small energy.     

Initial trace for a porous medium equation: I. The strong absorption case

We study the equation (E): u_t−Δu^m+u^q=0, (m, q>0) in Δ×ℝ₊, in a regular bounded open set Ω, or the whole space. We first prove that when 0<m<q, distributional solutions of (E) have an initial trace which is a Borel measure, then we study existence and uniqueness results with measure initial data. Entrata in Redazione il 12 giugno 1999. Ricevuta versione finale il 5 febbraio 2000.     

Sharp estimation of the almost-sure Lyapunov exponent for the Anderson model in continuous space

In this article we study the exponential behavior of the continuous stochastic Anderson model, i.e. the solution of the stochastic partial differential equation u(t,x)=1+∫₀^tκΔ_xu (s,x) ds+∫₀^t W(ds,x) u (s,x), when the spatial parameter x is continuous, specifically x∈R, and W is a Gaussian field on R₊×R that is Brownian in time, but whose spatial distribution is widely unrestricted. We give a partial existence result of the Lyapunov exponent defined as lim_t→∞t⁻¹ log u(t,x). Furthermore, we find upper and lower bounds for lim sup_t→∞t⁻¹ log u(t,x) and lim inf_t→∞t⁻¹ log u(t,x) respectively, as functions of the diffusion constant κ which depend on the regularity of W in x. Our bounds are sharper, work for a wider range of regularity scales, and are significantly easier to prove than all previously known results. When the uniform modulus of continuity of the process W is in the logarithmic scale, our bounds are optimal. This author's research partially supported by NSF grant no. : 0204999     

Harmonic maps from manifolds of <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>-Riemannian metrics

For a bounded domain Ω ⊂ R ⁿ endowed with L ^∞-metric g, and a C ⁵-Riemannian manifold (N, h) ⊂ R ^k without boundary, let u ∈ W ^1,2(Ω, N) be a weakly harmonic map, we prove that (1) u ∈ C ^α (Ω, N) for n = 2, and (2) for n ≥ 3, if, in additions, g ∈ VMO(Ω) and u satisfies the quasi-monotonicity inequality (1.5), then there exists a closed set Σ ⊂ Ω, with H ^n-2(Σ) = 0, such that for some α ∈ (0, 1). C. Y. Wang Partially supported by NSF.     

Sard’s theorem for mappings between Fréchet manifolds

We prove an infinite-dimensional version of Sard’s theorem for Fréchet manifolds. Let M (respectively, N) be a bounded Fréchet manifold with compatible metric d _M (respectively, d _N ) modeled on Fréchet spaces E (respectively, F) with standard metrics. Let f : M → N be an MC ^k -Lipschitz–Fredholm map with k > max{Ind f, 0}: Then the set of regular values of f is residual in N.