Well-posedness of the water-waves equations

We prove that the water-waves equations (i.e., the inviscid Euler equations with free surface) are well-posed locally in time in Sobolev spaces for a fluid layer of finite depth, either in dimension or under a stability condition on the linearized equations. This condition appears naturally as the Lévy condition one has to impose on these nonstricly hyperbolic equations to insure well-posedness; it coincides with the generalized Taylor criterion exhibited in earlier works. Similarly to what happens in infinite depth, we show that this condition always holds for flat bottoms. For uneven bottoms, we prove that it is satisfied provided that a smallness condition on the second fundamental form of the bottom surface evaluated on the initial velocity field is satisfied.

We work here with a formulation of the water-waves equations in terms of the velocity potential at the free surface and of the elevation of the free surface, and in Eulerian variables. This formulation involves a Dirichlet-Neumann operator which we study in detail: sharp tame estimates, symbol, commutators and shape derivatives. This allows us to give a tame estimate on the linearized water-waves equations and to conclude with a Nash-Moser iterative scheme.

     

Belousov equations on ternary quasigroups

Summary The study of Belousov equations in binary quasigroups was initiated by V. D. Belousov. Krape and Taylor showed that every finite set of Belousov equations was equivalent to a single Belousov equation which was in some sense no longer than any single member of the set. This led to the concept of an irreducible Belousov equation, that is one which is not equivalent to an equation with fewer variables. Krape and Taylor determined the structure of the irreducible equations by establishing a correspondence between them and specific polynomials overZ ₂.In this paper it is shown that the structure of the ternary equations is richer than the binary counterpart, although the main result is similar to the binary case in as far as a system of ternary Belousov equations is equivalent to a single Belousov equation which is no longer than any member of the system or the system is equivalent to a pair of equations each with three variables.     

Mean-value type functional equations

Summary In this work the following two conjectures concerning mean-value type functional equations are proved: then-dimensional octahedron and cube equations are equivalent (conjectured by D. Z. Djokovi and H. Haruki), and the continuous solutions of these equations on ⁿ are linear combinations of a given harmonic polynomial (conjectured by H. Haruki).     

Nonlinear potentials for Hamilton-Jacobi-Bellman equations

A formalism is suggested which makes it possible to investigate Hamilton-Jacobi-Bellman-type equations of general form. For such equations, we construct certain families of nonlinear operators, which we call as nonlinear potentials. The suggested method of investigation forfully nonlinear equations is based on only information aboutlinear equations and their solutions. This is a generalization of N. V. Krylov's approach.     

On the geometry of soliton equations

The paper aims to suggest a geometric point of view in the theory of soliton equations. The belief is that a deeper understanding of the origin of these equations may provide a better understanding of their remarkable properties. According to the geometric point of view, soliton equations are the outcome of a specific reduction process of a bi-Hamiltonian manifold. The suggestion of the paper is to pay attention also to the unreduced form of soliton equations.This work has been supported by the Italian MURST and by the GNFM of the Italian CNR.     

An approximation theorem for second-order evolution equations

Summary A class of approximation schemes of arbitrary accuracy, generated by a two-step recurrence relation, is devised for evolution equations of the second order. The schemes are effected via a specially constructed family of rational approximations to cos for 0 and yield computationally efficient methods for systems of second-order ordinary differential equations and semidiscrete approximations for initial-boundary value problems for second-order hyperbolic equations.Research supported by ONR grant N00014-57-A-0298-0015Research supported by USARO grant DAAG 29-278-C-0024     

Solutions of Navier equations and their representation structure

Navier equations are used to describe the deformation of a homogeneous, isotropic and linear elastic medium in the absence of body forces. Mathematically, the system is a natural vector -invariant generalization of the classical Laplace equation. In this paper, we decompose the space of polynomial solutions of Navier equations into a direct sum of irreducible -submodules and construct an explicit basis for each irreducible summand. Moreover, we explicitly solve the initial value problems for Navier equations and their wave-type extension—Lamé equations by Fourier expansion and Xu's method of solving flag partial differential equations. Our work might be counted as a continuation of Olver's important work on the algebraic study of elasticity in a certain sense.     

On Fredholm integral equations,Toeplitz equations and Kalman-Bucy filtering

In this paper we consider some new algorithms for computing the Kalman-Bucy gain in stationary systems requiring a number of equations of ordern (rather thann ² ) whenever the ordern of the system is much larger than the dimension of the output. These equations were independently obtained by Kailath and Lindquist in continuous and discrete time respectively. We briefly discuss the relations with some recent related results due to Casti, Kalaba & Murthy and Rissanen. Some of the reasons for these reductions are inherent in the properties of general stationary processes, and therefore a considerable portion of the paper is devoted to exploring the connections with some previous work by Levinson, Whittle and Wiggins & Robinson, and also with the Szegö theory of polynomials orthogonal on the unit circle and some continuous analogs due to Krein. We demonstrate that the Bellman-Krein formula is the fundamental relation in continuous time, the trick being to introduce a reversed time counterpart of the weighting function (Fredholm resolvent). This is suggested by the forward and backward innovation approach in a previous paper by the author, the essential relations of which we reformulate in terms of Fredholm integral equations (in continuous time) and Toeplitz equations (in discrete time). Therefore we also derive the discrete-time Bellman-Krein formulas of which there are actually two—one corresponding to the one-step predictor and one to the pure filter. In this way we shall be able to pin down the reasons for the striking discrepancies between the continuous-time and the discrete-time cases. Finally we clarify the relations between Levinson's equations and Chandrasekhar'sX- andY-functions.This research was carried out while the author held a visiting position at the Division of Applied Mathematics, Brown University, Providence, Rhode Island, and was supported by the National Science Foundation under grant NSF-46614.     

Fluid-dynamic equations for reacting gas mixtures

Starting from the Grad 13-moment equations for a bimolecular chemical reaction, Navier-Stokes-type equations are derived by asymptotic procedure in the limit of small mean paths. Two physical situations of slow and fast reactions, with their different hydrodynamic variables and conservation equations, are considered separately, yielding different limiting results.This work was performed in the frame of the activities sponsored by MIUR (Project Mathematical Problems of Kinetic Theories), by INdAM, by GNFM, by the University of Parma (Italy), and by the European TMR Network Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Analysis.This revised version was published online in April 2005 with a corrected issue number.     

Theory of equations of neutral type

In the paper we present a survey of the investigations on the theory of equations of neutral type, i.e., equations for which the value of the derivative at the present moment depends explicitly on the prehistory of the behavior of the derivative. The paper consists of eleven sections.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 19, pp. 55–126, 1982.     

Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices

The Cauchy problems for Navier-Stokes equations and nonlinear heat equations are studied in modulation spaces . Though the case of the derivative index s=0 has been treated in our previous work, the case s≠0 is also treated in this paper. Our aim is to reveal the conditions of s, q and σ of for the existence of local and global solutions for initial data .     

On the approximation of singular integral equations by equations with smooth kernels

Singular integral equations with Cauchy kernel and piecewise-continuous matrix coefficients on open and closed smooth curves are replaced by integral equations with smooth kernels of the form(t–)[(t–) ²–n ² (t) ²]^–1,0, wheren(t), t , is a continuous field of unit vectors non-tangential to . we give necessary and sufficient conditions under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the spaceL ₂() these conditions coincide with the strong ellipticity of the given equation.This work was fulfilled during the first author's visit to the Weierstrass Institute for Applied Analysis and Stochastics, Berlin in October 1993.     

Lamé equations with algebraic solutions

In this paper we study Lamé equations L_n,By=0 in so-called algebraic form, having only algebraic functions as solution. In particular we provide a complete list of all finite groups that occur as the monodromy groups, together with a list of examples of such equations. We show that the set of such Lamé equations with is countable, up to scaling of the equation. This result follows from the general statement that the set of equivalent second-order equations, having algebraic solutions and all of whose integer local exponent differences are 1, is countable.     

Midpoint collocation for Cauchy singular integral equations

Summary A Cauchy singular integral equation on a smooth closed curve may be solved numerically using continuous piecewise linear functions and collocation at the midpoints of the underlying grid. Even if the grid is non-uniform, suboptimal rates of convergence are proved using a discrete maximum principle for a modified form of the collocation equations. The same techniques prove negative norm estimates when midpoint collocation is used to determine piecewise constant approximations to the solution of first kind equations with the logarithmic potential.This work was supported by the Australian Research Council through the program grant Numerical analysis for integrals, integral equations and boundary value problems     

Mutational equations in metric spaces

This paper summarizes an extension of differential calculus to a mutational calculus for maps from one metric space to another. The simple idea is to replace half-lines allowing to define difference quotients of maps and their various limits in the case of vector space by transitions with which we can also define differential quotients of a map. Their various limits are called mutations of a map. Many results of differential calculus and set-valued analysis, including the Inverse Function Theorem, do not really rely on the linear structure and can be adapted to the nonlinear case of metric spaces and exploited. Furthermore, the concept of differential equation can be extended tomutational equation governing the evolution in metric spaces. Basic Theorems as the Nagumo Theorem, the Cauchy-Lipschitz Theorem, the Center Manifold Theorem and the second Lyapunov Method hold true for mutational equations.This work was motivated by evolution equations of tubes in visual servoing on one hand, mathematical morphology on the other, when the metric spaces are power spaces. This paper begins by listing some consequences of general theorems concerning mutational equations for tubes.     

Arithmetic differential equations on $$GL_n$$, III Galois groups

Differential equations have arithmetic analogues (Buium in Arithmetic differential equations, Mathematical Surveys and Monographs, vol 118. American Mathematical Society, Providence 2005) in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations, and the present paper is concerned with the “linear” ones. The equations themselves were introduced in a previous paper (Buium and Dupuy, in Arithmetic differential equations on \(GL_{n}\), II: arithmetic Lie–Cartan theory, arXiv:1308.0744). In the present paper we deal with the solutions of these equations as well as with the Galois groups attached to the solutions.