Almost global wellposedness of the 2-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period [0,T/ε] for initial data of the form ε Ψ, where T depends only on Ψ. In this paper, we show that for such data there exists a unique solution for a time period [0,e ^T/ε ]. This is achieved by better understandings of the nature of the nonlinearity of the full water wave equation. Financial support provided in part by NSF grant DMS-0400643.     

Filtration Consistent Nonlinear Expectations and Evaluations of Contingent Claims

We will study the following problem.Let X_t,t∈[0,T],be an R~d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of X.Assume that,at each fixedtime t≤T,the information available to an agent(an individual,a firm,or even a market)is the trajectory ofX before t.Thus at time T,the random value of Y(ω) will become known to this agent.The question is:howwill this agent evaluate Y at the time t?We will introduce an evaluation operator ε_t[Y] to define the value of Y given by this agent at time t.Thisoperator ε_t[·] assigns an (X_s)0(?)s(?)T-dependent random variable Y to an (X_s)0(?)s(?)t-dependent random variableε_t[Y].We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) withthe drift coefficient b and diffusion coefficient σcontaining an unknown parameter θ=θ_t.We then consider theso called super evaluation when the agent is a seller of the asset Y.We will prove that such super evaluation is afiltration consistent nonlinear expectation.In some typical situations,we will prove that a filtration consistentnonlinear evaluation dominated by this super evaluation is a g-evaluation.We also consider the correspondingnonlinear Markovian situation.     

On the validity of the Ginzburg-Landau equation

Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(ε ²) away from the critical valueR _c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-L's equation on a finite interval of theO(1/ε²)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(ε²) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact solution with a relatively small error.     

Homogenization of Nonlinear Degenerate Non-monotone Elliptic Operators in Domains Perforated with Tiny Holes

The paper deals with the homogenization problem beyond the periodic setting, for a degenerated nonlinear non-monotone elliptic type operator on a perforated domain Ω ^ε in ℝ ^N with isolated holes. While the space variable in the coefficients a ₀ and a is scaled with size ε (ε>0 a small parameter), the system of holes is scaled with ε ² size, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The homogenization problem is formulated in terms of the general, so-called deterministic homogenization theory combining real homogenization algebras with the Σ-convergence method. We present a new approach based on the Besicovitch type spaces to solve deterministic homogenization problems, and we obtain a very general abstract homogenization results. We then illustrate this abstract setting by providing some concrete applications of these results to, e.g., the periodic homogenization, the almost periodic homogenization, and others.     

An ε-relaxation method for separable convex cost generalized network flow problems

We generalize the ε-relaxation method of [14] for the single commodity, linear or separable convex cost network flow problem to network flow problems with positive gains. The method maintains ε-complementary slackness at all iterations and adjusts the arc flows and the node prices so as to satisfy flow conservation upon termination. Each iteration of the method involves either a price change on a node or a flow change along an arc or a flow change along a simple cycle. Complexity bounds for the method are derived. For one implementation employing ε-scaling, the bound is polynomial in the number of nodes N, the number of arcs A, a certain constant Γ depending on the arc gains, and ln(ε⁰/), where ε⁰ and denote, respectively, the initial and the final tolerance ε. Received: November 10, 1996 / Accepted: October 1999?Published online April 20, 2000     

Representing a Functional Curve by Curves with Fewer Peaks

In this paper, we study the problems of (approximately) representing a functional curve in 2-D by a set of curves with fewer peaks. Representing a function (or its curve) by certain classes of structurally simpler functions (or their curves) is a basic mathematical problem. Problems of this kind also find applications in applied areas such as intensity-modulated radiation therapy (IMRT). Let f\bf f be an input piecewise linear functional curve of size n. We consider several variations of the problems. (1) Uphill–downhill pair representation (UDPR): Find two nonnegative piecewise linear curves, one nondecreasing (uphill) and one nonincreasing (downhill), such that their sum exactly or approximately represents f\bf f. (2) Unimodal representation (UR): Find a set of unimodal (single-peak) curves such that their sum exactly or approximately represents f\bf f. (3) Fewer-peak representation (FPR): Find a piecewise linear curve with at most k peaks that exactly or approximately represents f\bf f. Furthermore, for each problem, we consider two versions. For the UDPR problem, we study its feasibility version: Given ε>0, determine whether there is a feasible UDPR solution for f\bf f with an approximation error ε; its min-ε version: Compute the minimum approximation error ε ^∗ such that there is a feasible UDPR solution for f\bf f with error ε ^∗. For the UR problem, we study its min-k version: Given ε>0, find a feasible solution with the minimum number k ^∗ of unimodal curves for f\bf f with an error ε; its min-ε version: given k>0, compute the minimum error ε ^∗ such that there is a feasible solution with at most k unimodal curves for f\bf f with error ε ^∗. For the FPR problem, we study its min-k version: Given ε>0, find one feasible curve with the minimum number k ^∗ of peaks for f\bf f with an error ε; its min-ε version: given k≥0, compute the minimum error ε ^∗ such that there is a feasible curve with at most k peaks for f\bf f with error ε ^∗. Little work has been done previously on solving these functional curve representation problems. We solve all the problems (except the UR min-ε version) in optimal O(n) time, and the UR min-ε version in O(n+mlog m) time, where m<n is the number of peaks of f\bf f. Our algorithms are based on new geometric observations and interesting techniques.     

Existence and multiplicity of solutions for a nonlinear Neumann problem

We consider a Neumann problem of the type -εΔu+F ^′(u(x))=0 in an open bounded subset Ω of R ⁿ , where F is a real function which has exactly k maximum points. Using Morse theory we find that, for ε suitably small, there are at least 2k nontrivial solutions of the problem and we give some qualitative information about them. Received: October 30, 1999 Published online: December 19, 2001     

Convergence theorems for solutions and energy functionals of boundary value problems in thick multilevel junctions of a new type with perturbed neumann conditions on the boundary of thin rectangles

Boundary value problems for the Poisson equation are considered in a multilevel thick junction consisting of a junction body and a lot of alternating thin rectangles of two levels depending on their lengths. Rectangles of the first level have a finite length, whereas rectangles of the second level have a length ε ^α , 0 < α < 1, where ε is the alternation period. On the boundary of thin rectangles, an inhomogeneous Neumann boundary condition involving additional perturbation parameters is imposed. We prove convergence theorems for solutions and energy integrals. Regarding the convergence of solutions of the original problem to solutions of the homogenized problem, we establish some (auxiliary) estimates necessary for obtaining the convergence rate. Bibliography: 48 titles. Illustrations: 3 figures. Dedicated to Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 113–132.     

A theorem of the Phragmén-Lindelöf type for solutions of an evolution equation of the second order with respect to time variable

We consider a solution u(x, t) of the general linear evolution equation of the second order with respect to time variable given on the ball Π(T) = {(x,t): xε R ⁿ, t ε [0, T]} and study the dependence of the behavior of this solution on the behavior of the functions at infinity. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 724–731, May, 1998.     

Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws

We consider a new way of establishing Navier wall laws. Considering a bounded domain Ω of R ^N , N=2,3, surrounded by a thin layer Σ _ε , along a part Γ₂ of its boundary ∂Ω, we consider a Navier-Stokes flow in Ω∪∂Ω∪Σ _ε with Reynolds’ number of order 1/ε in Σ _ε . Using Γ-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier law involving a matrix of Borel measures having the same support contained in the interface Γ₂. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider an optimal control problem within this context.     

Random perturbations of transformations of an interval

Let μ^ε be invariant measures of the Markov chainsx _n ^F which are small random perturbations of an endomorphismf of the interval [0,1] satisfying the conditions of Misiurewicz [6]. It is shown here that in the ergodic case μ^ε converges as ε→0 to the smoothf-invariant measure which exists according to [6]. This result exhibits the first example of stability with respect to random perturbations while stability with respect to deterministic perturbations does not take place.     

Oscillation of forced neutral differential equations with positive and negative coefficients

In this paper the forced neutral difterential equation with positive and negative coefficients d/dt [x(t)-R(t)x(t-r)] P(t)x(t-x)-Q(t)x(t-σ)=f(t),t≥t0,is considered,where f∈L^1(t0,∞)交集C([t0,∞],R^ )and r,x,σ∈(0,∞),The sufficient conditions to oscillate for all solutions of this equation are studied.     

The central connection problem at turning points of linear differential equations

A system of linear differential equations of the vectorial form εdy/dx=A (x, ε) y is considered, where ε is a positive parameter, and the matrixA (x, ε) is holomorphic in |x|⩽x ₀, 0 < ε ⩽ ε₀ , with an asymptotic expansionsA (x, ε) ∼ ∑ _r=0 ^∞ A _r (x) ε ^r , as ε→0. The eigenvalues ofA ₀(x) are supposed to coalesce atx=0 so as to make this point a simple turning point. With the help of refinements of the representations for the inner and outer asymptotic solutions, as ε→0, that were introduced in the articles [9] and [10] by the author (see the references at the end of the paper), explicit connection formulas between these solutions are calculated. As part of this derivation it is shown that only the diagonal entries of the connection matrix are asymptotically relevant.     

Rank-two update algorithms for the minimum volume enclosing ellipsoid problem

We consider the problem of computing a (1+ε)-approximation to the minimum volume enclosing ellipsoid (MVEE) of a given set of m points in R ⁿ . Based on the idea of sequential minimal optimization (SMO) method, we develop a rank-two update algorithm. This algorithm computes an approximate solution to the dual optimization formulation of the MVEE problem, which updates only two weights of the dual variable at each iteration. We establish that this algorithm computes a (1+ε)-approximation to MVEE in O(mn ³/ε) operations and returns a core set of size O(n ²/ε) for ε∈(0,1). In addition, we give an extension of this rank-two update algorithm. Computational experiments show the proposed algorithms are very efficient for solving large-scale problem with a high accuracy.     

Strongly nonlinear degenerate elliptic equations with discontinuous coefficients. II

We use energy methods to prove the existence and uniqueness of solutions of the Dirichlet problem for an elliptic nonlinear second-order equation of divergence form with a superlinear tem [i.e., g(x, u)=v(x) a(x)⋎u⋎ ^p−1u,p>1] in unbounded domains. Degeneracy in the ellipticity condition is allowed. Coefficients a _i,j(x,r) may be discontinuous with respect to the variable r. University of Catania, Italy. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1601–1609, December, 1997.     

Asymptotic analysis of a parabolic semilinear problem with nonlinear boundary multiphase interactions in a perforated domain

We consider a parabolic semilinear problem with rapidly oscillating coefficients in a domain Ω_ε that is ε-periodically perforated by small holes of size O\mathcal {O}(ε). The holes are divided into two ε-periodical sets depending on the boundary interaction at their surfaces, and two different nonlinear Robin boundary conditions σ_ε(u _ε) + εκ _m (u _ε) = εg ^(m) _ε, m = 1, 2, are imposed on the boundaries of holes. We study the asymptotics as ε → 0 and establish a convergence theorem without using extension operators. An asymptotic approximation of the solution and the corresponding error estimate are also obtained. Bibliography: 60 titles. Illustrations: 1 figure.     

Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data

We are interested in proving Monte-Carlo approximations for 2d Navier-Stokes equations with initial data u ₀ belonging to the Lorentz space L ^2,∞ and such that curl u ₀ is a finite measure. Giga, Miyakawaand Osada [7] proved that a solution u exists and that u=K* curl u, where K is the Biot-Savartkernel and v = curl u is solution of a nonlinear equation in dimension one, called the vortex equation. In this paper, we approximate a solution v of this vortex equationby a stochastic interacting particlesystem and deduce a Monte-Carlo approximation for a solution of the Navier-Stokesequation. That gives in this case a pathwise proofof the vortex algorithm introducedby Chorin and consequently generalizes the works ofMarchioro-Pulvirenti [12] and Méléardv [15] obtained in the case of a vortex equation with bounded density initial data. Received: 6 October 1999 / Revised version: 15 September 2000 / Published online: 9 October 2001     

Low regularity solutions of two fifth-order KDV type equations

The Kawahara and modified Kawahara equations are fifth-order KdV type equations that have been derived to model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness of the initial-value problem for the Kawahara equation in H ^s (R) with s ≥ − 7/4 and the local well-posedness for the modified Kawahara equation in H ^s (R) with s ≥ − 1/4. To prove these results, we derive a fundamental estimate on dyadic blocks for the Kawahara equation through the [k; Z]_multiplier norm method of Tao [14] and use this to obtain new bilinear and trilinear estimates in suitable Bourgain spaces.     

Implicit Gamma Theorems (I): Pseudoroots and Pseudospectra

   Abstract. Let g : E → F be an analytic function between two Hilbert spaces E and F. We study the set g(B(x, ε)) ⊂ E, the image under g of the closed ball about x∈ E with radius ε . When g(x) expresses the solution of an equation depending on x , then the elements of g(B(x,ε )) are ε -pseudosolutions. Our aim is to investigate the size of the set g(B(x,ε )) . We derive upper and lower bounds of the following form: g(x) + Dg (x) ( B(0, c ₁ ε N))

Su una condizione di fattorialità debole e l’annullamento del gruppo di Picard

Summary We give some characterizations of noetherian domains A such that ? every irreducible element generates a primary ideal ?. This condition, called (α)-property, is equivalent to the unique factorization if A is normal or a polynomial ring A=B[T]. If A is a1-dimensional k-algebra, the property (α) is equivalent to the vanishing of some Picard groups asPicA,Pic (A[T, T⁻¹]),Pic (A|T|_s), where S={Tⁿ, n εZ}. We give not trivial examples of (α)-rings which aren’t factorial.

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Entrata in Redazione il 6 febbraio 1976.

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Lavoro eseguito nell’ambito della sezione n. 3 del G.N.S.A.G.A. del C.N.R.