Nonlinear Degenerate Parabolic Equation with Nonlinear Boundary Condition

In this paper, we obtain the necessary and sufficient conditions on the global existence of all positive （weak） solutions to a nonlinear degenerate parabolic equation with nonlinear boundary condition.     

On Nonlinear Rayleigh-Taylor Instabilities

Some of the mathematical properties of the interface between two incompressible inviscid and immiscible fluids with different densities under the influence of a constant gravity field 9 are investigated. The purpose of this paper is to prove that linearly unstable modes for Rayleigh-Taylor instabilities give birth to nonlinear instabilities for the full nonlinear system. The main ingredient is a general instability theorem in an analytic framework which enables us to go from linear to nonlinear instabilities.     

Nonlinear PDEs with Modulated Dispersion I: Nonlinear Schrödinger Equations

We start a study of various nonlinear PDEs under the effect of a modulation in time of the dispersive term. In particular in this paper we consider the modulated non-linear Schrödinger equation (NLS) in dimension 1 and 2 and the derivative NLS in dimension 1. We introduce a deterministic notion of “irregularity” for the modulation and obtain local and global results similar to those valid without modulation. In some situations, we show how the irregularity of the modulation improves the well–posedness theory of the equations. We develop two different approaches to the analysis of the effects of the modulation. A first approach is based on novel estimates for the regularizing effect of the modulated dispersion on the non-linear term using the theory of controlled paths. A second approach is an extension of a Strichartz estimated first obtained by Debussche and Tsutsumi in the case of the Brownian modulation for the quintic NLS.     

Nonlinear Extrapolation Estimates of π

The classical Archimedean approximation of π uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in R² and it is well-known that by using linear combinations of these basic estimates,modern extrapolation techniques can greatly speed up the approximation process.Similarly,when n vertices are randomly selected on the circle,the semiperimeter and area of the corresponding random inscribed and circumscribing polygons are known to converge t...     

A Nonlinear Degenerate Parabolic Equation

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Nonlinear Least Squares in ℝ N

Recent research and new paradigms in mathematics, engineering, and science assume nonlinear signal models of the form ?=∪ _i∈I V _i consisting of a union of subspaces V _i instead of a single subspace ?=V. These models have been used in sampling and reconstruction of signals with finite rate of innovation, the Generalized Principle Component Analysis and the subspace segmentation problem in computer vision, and problems related to sparsity, compressed sensing, and dictionary design. In this paper, we develop an algorithm that searches for the best nonlinear model of the form ?=∪ _i=1 ^l V _i ?? ^N that is optimally compatible with a set of observations ?={f ₁,…,f _m }?? ^N . When l=1 this becomes the classical least squares optimization. Thus, this problem is a nonlinear version of the least squares problem. We test our algorithm on synthetic data as well as images.     

Nonlinear Doob-Meyer Decomposition for General Filtration

The comparison theorem for generalized backward stochastic differential equations is discussed. Some topics related to equations of this type are also investigated.     

Periodic Solutions for Nonlinear Diferential Equations

ＰｅｒｉｏｄｉｃＳｏｌｕｔｉｏｎｓｆｏｒＮｏｎｌｉｎｅａｒＤｉｆｅｒｅｎｔｉａｌＥｑｕａｔｉｏｎｓＣｏｎｇＦｕｚｈｏｎｇ（从福仲）（ＯｆｆｉｃｅｏｆＭａｔｈｅｍａｔｉｃｓ，８６００３Ｕｎｉｔ，Ｃｈａｎｇｃｈｕｎ，１３００２２）ＭａｏＤｏｎｇｍｉｎｇ（...     

Global Nonlinear Brascamp–Lieb Inequalities

We prove global versions of certain known nonlinear Brascamp–Lieb inequalities under a natural homogeneity assumption. We also establish a conditional theorem allowing one to generally pass from local to global nonlinear Brascamp–Lieb estimates under such a homogeneity assumption.     

Nonlinear Biharmonic Equations with Critical Potential

In this paper, we study two semilinear singular biharmonic equations： one with subcritical exponent and critical potential, another with sub-critical potential and critical exponent. By Pohozaev identity for singular solution, we prove there is no nontrivial solution for equations with critical exponent and critical potential. And by using the concentrate compactness principle and Mountain Pass theorem, respectively, we get two existence results for the two problems. Meanwhile, we have compared the changes of the critical dimensions in singular and non-singular cases, and we get an interesting result.     

Nonlinear Programming——a Review

Nonlinear programming problems are to minimize general nonlinear functions, possibly subject to some nonlinear constraints. In this paper, we review some recent results on nonlinear optimization.     

The Nonlinear Steepest Descent Method to Long-Time Asymptotics of the Coupled Nonlinear Schrödinger Equation

The Riemann–Hilbert problem for the coupled nonlinear Schrödinger equation is formulated on the basis of the corresponding \(3\times 3\) matrix spectral problem. Using the nonlinear steepest descent method, we obtain leading-order asymptotics for the Cauchy problem of the coupled nonlinear Schrödinger equation.     

Nonlinear evolutions with Carathéodory forcing

Let A be an m-accretive operator in a real Banach space X and f : J x X X a function of Carathéodory type, where $ J = [0, a] \subset \mathbb{R} $. This paper investigates the existence of mild solutions of the evolution system satisfying additional time-dependent constraints $ u(t) \in K(t) $ on J for a given tube K(·). Main emphasis is on existence results that are valid under minimal assumptions on f, K and X.AMS Subject Classification: Primary: 47J35, 35K90, Secondary: 34G25, 47H20.     

Nonlinear Doob—Meyer Decomposition with Jumps

Concepts of g-supersolution ,g-martingale,g-supermartingale are introduced,wihch are related to BSDE with Brownian motion and Poisson Point process.A strict comparison theorem,monotonic limit theorem related to this type of BSDE are also discussed.As an application of these results,a nonlinear Doob-Meyer decomposition theorem is obtained.     

Nonlinear Galerkin Optimal Truncated Low－dimensionalDynamicalSystems

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Nonlinear algebra and Bogoliubov’s recursion

We give many examples of applying Bogoliubov’s forest formula to iterative solutions of various nonlinear equations. The same formula describes an extremely wide class of objects, from an ordinary quadratic equation to renormalization in quantum field theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 316–343, February, 2008.     

Blow-up Solutions for Mixed Nonlinear Schrodinger Equations

This paper is concerned with the initial boundary-value problem for the following nonlinear evolution equation: Фt=iαФxx βФ^2Ф^-x γ|Ф|^2Фx ig(|Ф|^2)Ф Under certain conditions on the initial data and the function g(s),we study the existence and nonexistence of global solution for this equation.The blow-up solution and the blow-up time are also investigated.     

Frozen Landweber Iteration for Nonlinear Ill-Posed Problems

In this paper we propose a modification of the Landweber iteration termed frozen Landweberiteration for nonlinear ill-posed problems.A convergence analysis for this iteration is presented.The numericalperformance of this frozen Landweber iteration for a nonlinear Hammerstein integral equation is compared withthat of the Landweber iteration.We obtain a shorter running time of the frozen Landweber iteration based onthe same convergence accuracy.