Geometry of PDE's. IV: Navier-Stokes equation and integral bordism groups

Following our previous results on this subject [R.P. Agarwal, A. Prástaro, Geometry of PDE's. III(I): Webs on PDE's and integral bordism groups. The general theory, Adv. Math. Sci. Appl. 17 (2007) 239-266; R.P. Agarwal, A. Prástaro, Geometry of PDE's. III(II): Webs on PDE's and integral bordism groups. Applications to Riemannian geometry PDE's, Adv. Math. Sci. Appl. 17 (2007) 267-285; A. Prástaro, Geometry of PDE's and Mechanics, World Scientific, Singapore, 1996; A. Prástaro, Quantum and integral (co)bordism in partial differential equations, Acta Appl. Math. (5) (3) (1998) 243-302; A. Prástaro, (Co)bordism groups in PDE's, Acta Appl. Math. 59 (2) (1999) 111-201; A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing Co, Singapore, 2004, 500 pp.; A. Prástaro, Geometry of PDE's. I: Integral bordism groups in PDE's, J. Math. Anal. Appl. 319 (2006) 547-566; A. Prástaro, Geometry of PDE's. II: Variational PDE's and integral bordism groups, J. Math. Anal. Appl. 321 (2006) 930-948; A. Prástaro, Th.M. Rassias, Ulam stability in geometry of PDE's, Nonlinear Funct. Anal. Appl. 8 (2) (2003) 259-278; I. Stakgold, Boundary Value Problems of Mathematical Physics, I, The MacMillan Company, New York, 1967; I. Stakgold, Boundary Value Problems of Mathematical Physics, II, Collier-MacMillan, Canada, Ltd, Toronto, Ontario, 1968], integral bordism groups of the Navier-Stokes equation are calculated for smooth, singular and weak solutions, respectively. Then a characterization of global solutions is made on this ground. Enough conditions to assure existence of global smooth solutions are given and related to nullity of integral characteristic numbers of the boundaries. Stability of global solutions are related to some characteristic numbers of the space-like Cauchy data. Global solutions of variational problems constrained by (NS) are classified by means of suitable integral bordism groups too.     

Long Time Stability in Perturbations of Completely Resonant PDE's

In this paper we will present some results concerning long time stability in nonlinear perturbations of resonant linear PDE's with discrete spectrum. In particular we will prove that if the perturbation satisfies a suitable nondegeneracy condition then there exists a periodic like trajectory, i.e. a closed curve in the phase space with the property that solutions starting close to it remain close to it for very long times. Secondly, in the special case where the average of the main part of the perturbation is integrable we will prove that if the energy is initially essentially concentrated on finitely many modes, then along the corresponding solutions all the actions are approximatively constant for very long times. Applications to nonlinear wave and Schrödinger equations on a segment will also be given.     

On the singular PDE's geometry and boundary value problems

A geometric formulation of singular partial differential equations (PDEs) is considered. Surgery techniques and integral bordism groups are utilized, following previous works by Prástaro on PDEs, in order to build global solutions crossing also singular points and to study their stability properties.     

An inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities

The augmented Lagrangian method is attractive in constraint optimizations. When it is applied to a class of constrained variational inequalities, the sub-problem in each iteration is a nonlinear complementarity problem (NCP). By introducing a logarithmic-quadratic proximal term, the sub-NCP becomes a system of nonlinear equations, which we call the LQP system. Solving a system of nonlinear equations is easier than the related NCP, because the solution of the NCP has combinatorial properties. In this paper, we present an inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities, in which the LQP system is solved approximately under a rather relaxed inexactness criterion. The generated sequence is Fejér monotone and the global convergence is proved. Finally, some numerical test results for traffic equilibrium problems are presented to demonstrate the efficiency of the method.      

New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

In a previous paper we gave a new formulation and derived the Euler equations and other necessary conditions to solve strong, pathwise, stochastic variational problems with trajectories driven by Brownian motion. Thus, unlike current methods which minimize the control over deterministic functionals (the expected value), we find the control which gives the critical point solution of random functionals of a Brownian path and then, if we choose, find the expected value.This increase in information is balanced by the fact that our methods are anticipative while current methods are not. However, our methods are more directly connected to the theory and meaningful examples of deterministic variational theory and provide better means of solution for free and constrained problems. In addition, examples indicate that there are methods to obtain nonanticipative solutions from our equations although the anticipative optimal cost function has smaller expected value.In this paper we give new, efficient numerical methods to find the solution of these problems in the quadratic case. Of interest is that our numerical solution has a maximal, a priori, pointwise error of O(h3/2) where h is the node size. We believe our results are unique for any theory of stochastic control and that our methods of proof involve new and sophisticated ideas for strong solutions which extend previous deterministic results by the first author where the error was O(h²).We note that, although our solutions are given in terms of stochastic differential equations, we are not using the now standard numerical methods for stochastic differential equations. Instead we find an approximation to the critical point solution of the variational problem using relations derived from setting to zero the directional derivative of the cost functional in the direction of simple test functions.Our results are even more significant than they first appear because we can reformulate stochastic control problems or constrained calculus of variations problems in the unconstrained, stochastic calculus of variations formulation of this paper. This will allow us to find efficient and accurate numerical solutions for general constrained, stochastic optimization problems. This is not yet being done, even in the deterministic case, except by the first author.     

Law of large numbers and central limit theorem for randomly forced PDE's

We consider a class of dissipative PDE's perturbed by an external random force. Under the condition that the distribution of perturbation is sufficiently non-degenerate, a strong law of large numbers (SLLN) and a central limit theorem (CLT) for solutions are established and the corresponding rates of convergence are estimated. It is also shown that the estimates obtained are close to being optimal. The proofs are based on the property of exponential mixing for the problem in question and some abstract SLLN and CLT for mixing-type Markov processes.     

From formal numerical solutions of elliptic PDE's to the true ones

We propose a discretization scheme for a numerical solution of elliptic PDE's, based on local representation of functions, by their Taylor polynomials (jets). This scheme utilizes jet calculus to provide a very high order of accuracy for a relatively small number of unknowns involved.

     

Bubbling phenomena for fourth-order four-dimensional PDEs with exponential growth

We are concerned in this paper with the bubbling phenomenon for nonlinear fourth-order four-dimensional PDE's. The operators in the equations are perturbations of the bi-Laplacian. The nonlinearity is of exponential growth. Such equations arise naturally in statistical physics and geometry. As a consequence of our theorem we get a priori bounds for solutions of our equations.

     

Sharp condition of global existence for second-order derivative nonlinear Schrödinger equations in two space dimensions

This paper discusses a class of second-order derivative nonlinear Schrödinger equations which are used to describe the upper-hybrid oscillation propagation. By establishing a variational problem, applying the potential well argument and the concavity method, we prove that there exists a sharp condition for global existence and blow-up of the solutions to the nonlinear Schrödinger equation. In addition, we also answer the question: how small are the initial data, the global solutions exist?     

Sharp Threshold for Blowup and Global Existence in Nonlinear Schrödinger Equations Under a Harmonic Potential

ABSTRACT

We present a variational approach to study the nonlinear Schrödinger equations under a harmonic potential. By constructing a type of cross constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp threshold for global existence and blowup of the solution. The stability of the standing waves is also discussed.     

Blow-up for Stochastic Reaction-Diffusion Equations with Jumps

In this paper, we focus on stochastic reaction-diffusion equations with jumps. By a new auxiliary function, we investigate non-negative property of the local strong (variational) solutions, which applies to stochastic reaction-diffusion equations with highly nonlinear noise terms. As a byproduct, we study the problem of non-existence of global strong solutions by imposing appropriate conditions on the drift terms, which can cover many more models than the existing literature. Moreover, we also investigate the subject of Lévy-type noise-induced explosion by bringing some plausible assumptions to bear on the noise terms, which, however, need not guarantee local strong (variational) solutions to enjoy the non-negative property. Meanwhile, several examples are presented to illustrate the theory established.     

A self-dual variational approach to stochastic partial differential equations

Unlike many of their deterministic counterparts, stochastic partial differential equations are not amenable to the methods of calculus of variations à la Euler–Lagrange. In this paper, we show how self-dual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields. We construct solutions as minima of suitable non-negative and self-dual energy functionals on Itô spaces of stochastic processes. We show how a stochastic version of Bolza's duality leads to solutions for equations with additive noise. We then use a Hamiltonian formulation to construct solutions for non-linear equations with non-additive noise such as the stochastic Navier–Stokes equations in dimension two.     

一类具非线性二阶导数项的Schrōdinger方程整体解存在的最佳条件

本文讨论一类具非线性二阶导数项的Schrodinger方程,它在物理上描述了上混合振荡传播．根据基态的变分特征,运用势井方法和凹方法,我们获得了其初值问题整体解存在的一个最佳条件,另外还证明了当初值多小时,初值问题的整体解存在．     

Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's

We present a general result of transverse nonlinear instability of 1d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1-d model and the transverse perturbation “have the same sign”. Our result applies to the generalized KP-I equation, the Nonlinear Schrödinger equation, the generalized Boussinesq system and the Zakharov–Kuznetsov equation and we hope that it may be useful in other contexts.     

Weak solutions to a nonlinear variational wave equation and some related problems

In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L ^p Young measure theory and related compactness results, in the first section. Then we use the L ^p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.     

一类具非线性二阶导数项的Schr dinger方程整体解存在的最佳条件

本文讨论一类具非线性二阶导数项的Schr(?)dinger方程,它在物理上描述了上混合振荡传播．根据基态的变分特征,运用势井方法和凹方法,我们获得了其初值问题整体解存在的一个最佳条件,另外还证明了当初值多小时,初值问题的整体解存在．     

求解盒子形区域上变分不等式的逐次逼近类Broyden算法

In this paper, a successive approximation Broyden-like method is presented for the box constrained variational inequality problems based on its equivalent nonsmooth equations. The global convergence of the algorithm is obtained under suitable conditions. Numerical results are also reported.     

The self-similar solutions of the one-dimensional semilinear Tricomi-type equations

In this article we investigate the issue of existence of global in time solutions of semilinear Tricomi-type equations. We give conditions that relate the nonlinearity, the speed of propagation, and the order of singularities of initial data. These conditions guarantee existence of global in time solutions. In particular, we prove existence of solutions invariant under dilation by solving the Cauchy problem with initial data which are homogeneous functions.     

Bifurcation calculus by the extended functional method

We justify variational principles of a new type corresponding to bifurcations of solutions for families of equations given in variational form. To illustrate the method, we consider elliptic equations with sign-indefinite nonlinearities and prove the existence of pairwise creation-annihilation bifurcations of their positive solutions. The corresponding bifurcation points are expressed via explicitly specified variational principles.     

Instability of standing waves for a class of nonlinear Schrödinger equations

This paper discusses a class of nonlinear Schrödinger equations with different power nonlinearities. We first establish the existence of standing wave associated with the ground states by variational calculus. Then by the potential well argument and the concavity method, we get a sharp condition for blowup and global existence to the solutions of the Cauchy problem and answer such a problem: how small are the initial data, the global solutions exist? At last we prove the instability of standing wave by combing those results.