The Cauchy problem for the Schrödinger-KdV system

In this paper we prove that in the general case (i.e. β not necessarily vanishing) the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed in , and if β=0 then it is locally well-posed in with . These results improve the corresponding results of Corcho and Linares (2007) [5]. Idea of the proof is to establish some bilinear and trilinear estimates in the space Gs×Fs, where Gs and Fs are dyadic Bourgain-type spaces related to the Schrödinger operator and the Airy operator , respectively, but with a modification on Fs in low frequency part of functions with a weaker structure related to the maximal function estimate of the Airy operator.     

Neighborhood radius estimation for variable-neighborhood random fields

We consider random fields defined by finite-region conditional probabilities depending on a neighborhood of the region which changes with the boundary conditions. To predict the symbols within any finite region, it is necessary to inspect a random number of neighborhood symbols which might change according to the value of them. In analogy with the one-dimensional setting we call these neighborhood symbols the context associated to the region at hand. This framework is a natural extension, to d-dimensional fields, of the notion of variable length Markov chains introduced by Rissanen [24] in his classical paper. We define an algorithm to estimate the radius of the smallest ball containing the context based on a realization of the field. We prove the consistency of this estimator. Our proofs are constructive and yield explicit upper bounds for the probability of wrong estimation of the radius of the context.     

Asymptotic stability for a free boundary problem arising in a tumor model

We consider a tumor model in which all cells are proliferating at a rate μ and their density is proportional to the nutrient concentration. The model consists of a coupled system of an elliptic equation and a parabolic equation, with the tumor boundary as a free boundary. It is known that for an appropriate choice of parameters, there exists a unique spherically symmetric stationary solution with radius RS which is independent of μ. It was recently proved that there is a function μ_∗(RS) such that the spherical stationary solution is linearly stable if μ<μ_∗(RS) and linearly unstable if μ>μ_∗(RS). In this paper we prove that the spherical stationary solution is nonlinearly stable (or, asymptotically stable) if μ<μ_∗(RS).     

An arctic circle theorem for Groves

In earlier work, Jockusch, Propp, and Shor proved a theorem describing the limiting shape of the boundary between the uniformly tiled corners of a random tiling of an Aztec diamond and the more unpredictable ‘temperate zone’ in the interior of the region. The so-called arctic circle theorem made precise a phenomenon observed in random tilings of large Aztec diamonds.Here we examine a related combinatorial model called groves. Created by Carroll and Speyer as combinatorial interpretations for Laurent polynomials given by the cube recurrence, groves have observable frozen regions which we describe precisely via asymptotic analysis of a generating function. Our approach also provides another way to prove the arctic circle theorem for Aztec diamonds.     

Green function estimates for relativistic stable processes in half-space-like open sets

In this paper, we establish sharp two-sided estimates for the Green functions of relativistic stable processes (i.e. Green functions for non-local operators m−(m^2/α−Δ)^α/2) in half-space-like C^1,1 open sets. The estimates are uniform in m∈(0,M] for each fixed M∈(0,∞). When m↓0, our estimates reduce to the sharp Green function estimates for −(−Δ)^α/2 in such kind of open sets that were obtained recently in Chen and Tokle [12]. As a tool for proving our Green function estimates, we show that a boundary Harnack principle for X^m, which is uniform for all m∈(0,∞), holds for a large class of non-smooth open sets.     

Embeddings of homology equivalent manifolds with boundary

We prove a theorem on equivariant maps implying the following two corollaries:(1) Let N and M be compact orientable n-manifolds with boundaries such that M⊂N, the inclusion M→N induces an isomorphism in integral cohomology, both M and N have (n−d−1)-dimensional spines and . Then the restriction-induced map Embm(N)→Embm(M) is bijective. Here Embm(X) is the set of embeddings X→Rm up to isotopy (in the PL or smooth category).(2) For a 3-manifold N with boundary whose integral homology groups are trivial and such that N?D³ (or for its special 2-spine N) there exists an equivariant map , although N does not embed into R³.The second corollary completes the answer to the following question: for which pairs (m,n) for each n-polyhedron N the existence of an equivariant map implies embeddability of N into Rm? An answer was known for each pair (m,n) except (3,3) and (3,2).     

Long range scattering for the modified Schrödinger map in two space dimensions

We study the asymptotic behaviour in time of solutions and the theory of scattering for the modified Schrödinger map in two space dimensions. We solve the Cauchy problem with large finite initial time, up to infinity in time, and we determine the asymptotic behaviour in time of the solutions thereby obtained. As a by product, we obtain global existence for small data in Hk∩FHk with k>1. We also solve the Cauchy problem with infinite initial time, namely we construct solutions defined in a neighborhood of infinity in time, with prescribed asymptotic behaviour of the previous type.     

Kirchhoff systems with dynamic boundary conditions

We are interested in the study of the global non-existence of solutions of hyperbolic nonlinear problems, governed by the p-Kirchhoff operator, under dynamic boundary conditions, when p>p_n with p_n<2. The systems involve nonlinear external forces and may be affected by a perturbation of the type |u|^p−2u. Several models already treated in the literature are covered in special subcases, and concrete examples are provided for the source term f and the external nonlinear boundary damping Q.     

Multilinear estimates for periodic KdV equations, and applications

We prove an endpoint multilinear estimate for the X^s,b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm.     

Homogenization of transport equations: A simple PDE approach to the Kubo formula

We are interested in the behavior with respect to the small parameter ?>0 of solutions ρ? of the conservative transport(-diffusion) equation t_∂ρ?+∇x(ρ?u?)=ηΔxρ?, with η?0, driven by a large random velocity field: |u?|=O(1/?). Assuming that the velocity does not have long-time memory we justify the convergence of the expectation Eρ? to the solution of a diffusion equation. This question has been widely investigated; here we present a simple proof which only relies on PDE tools.     

Asymptotics of the principal eigenvalue and expected hitting time for positive recurrent elliptic operators in a domain with a small puncture

Let X(t) be a positive recurrent diffusion process corresponding to an operator L on a domain D⊆R^d with oblique reflection at ∂D if D≠R^d. For each x∈D, we define a volume-preserving norm that depends on the diffusion matrix a(x). We calculate the asymptotic behavior as ε→0 of the expected hitting time of the ε-ball centered at x and of the principal eigenvalue for L in the exterior domain formed by deleting the ball, with the oblique derivative boundary condition at ∂D and the Dirichlet boundary condition on the boundary of the ball. This operator is non-self-adjoint in general. The behavior is described in terms of the invariant probability density at x and Det(a(x)). In the case of normally reflected Brownian motion, the results become isoperimetric-type equalities.     

The Wiener disorder problem with finite horizon

The Wiener disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ‘disorder’ when the drift of an observed Wiener process changes from one value to another. In this paper we present a solution of the Wiener disorder problem when the horizon is finite. The method of proof is based on reducing the initial problem to a parabolic free-boundary problem where the continuation region is determined by a continuous curved boundary. By means of the change-of-variable formula containing the local time of a diffusion process on curves we show that the optimal boundary can be characterized as a unique solution of the nonlinear integral equation.     

Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

We consider the entire graph S of a continuous real function over RN−1 with N?3. Let Ω be a domain in RN with S as a boundary. Consider in Ω the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in Ω. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Ampère-type equation.     

Dynamical behaviors for 1D compressible Navier-Stokes equations with density-dependent viscosity

The dynamical behaviors of vacuum states for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficient are considered. It is first shown that a unique strong solution to the free boundary value problem exists globally in time, the free boundary expands outwards at an algebraic rate in time, and the density is strictly positive in any finite time but decays pointwise to zero time-asymptotically. Then, it is proved that there exists a unique global weak solution to the initial boundary value problem when the initial data contains discontinuously a piece of continuous vacuum and is regular away from the vacuum. The solution is piecewise regular and contains a piece of continuous vacuum before the time T_∗>0, which is compressed at an algebraic rate and vanishes at the time T_∗, meanwhile the weak solution becomes either a strong solution or a piecewise strong one and tends to the equilibrium state exponentially.     

Stochastic processes with proportional increments and the last-arrival problem

The notion of stochastic processes with proportional increments is introduced. This notion is of general interest as indicated by its relationship with several stochastic processes, as counting processes, Lévy processes, and others, as well as martingales related with these processes. The focus of this article is on the motivation to introduce processes with proportional increments, as instigated by certain characteristics of stopping problems under weak information. We also study some general properties of such processes. These lead to new insights into the mechanism and characterization of Pascal processes. This again will motivate the introduction of more general f-increment processes as well as the analysis of their link with martingales. As a major application we solve the no-information version of the last-arrival problem   which was an open problem. Further applications deal with the impact of proportional increments on modelling investment problems, with a new proof of the 1/e$1/e$-law of best choice, and with other optimal stopping problems.     

Decomposability for stable processes

We characterize all possible independent symmetric α-stable (SαS) components of an SαS process, 0<α<2. In particular, we focus on stationary SαS processes and their independent stationary SαS components. We also develop a parallel characterization theory for max-stable processes.     

On the low regularity of the fifth order Kadomtsev-Petviashvili I equation

We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as , while α∈R. We introduce an interpolated energy space Es to consider the well-posedness of the initial value problem (IVP) of the fifth order KP-I equation. We obtain the local well-posedness of IVP of the fifth order KP-I equation in Es for 0<s?1. To obtain the local well-posedness, we present a bilinear estimate in the Bourgain space in the framework of the interpolated energy space. It crucially depends on the dyadic decomposed Strichartz estimate, the fifth order dispersive smoothing effect and maximal estimate.     

On the low regularity of the modified Korteweg-de Vries equation with a dissipative term

We consider a dissipative version of the modified Korteweg-de Vries equation ut+uxxx−uxx+x(u³)=0. We prove global well-posedness results on the associated Cauchy problem in the Sobolev spaces Hs(R) for s>−1/4 while for s<−1/2 we prove some ill-posedness issues.