New asymptotic expansion algorithm for singularly perturbed evolution equations

A new asymptotic expansion algorithm related to the Chapman-Enskog expansion in kinetic theory is applied to systems of linear evolution equations. The uniform convergence of the asymptotic solution to the exact one is shown. The algorithm is applied to the linearized Carleman model of the Boltzmann equation, to the neutron transport equation, and to the Fokker-Planck equation.     

On the solution of a certain class of nonlinear singular integro-differential equations with Carleman shift

The solution of a class of nonlinear singular integro-differential equations with Carleman shift is obtained in the space H_μ⁽ⁿ⁾(Γ). The approach followed depends essentially on solution of linear singular integro-differential equation with shift. A criterion for the Noetherity of a correspondence singular integral functional operator of second order with Carleman shift preserving orientation is obtained and the index formula is given.     

An asymptotic derivation of weakly nonlinear ray theory

Using a method of expansion similar to Chapman-Enskog expansion, a new formal perturbation scheme based on high frequency approximation has been constructed. The scheme leads to an eikonal equation in which the leading order amplitude appears. The transport equation for the amplitude has been deduced with an errorO(ε²) where ε is the small parameter appearing in the high frequency approximation. On a length scale over which Choquet-Bruhat’s theory is valid, this theory reduces to the former. The theory is valid on a much larger length scale and the leading order terms give the weakly nonlinear ray theory (WNLRT) of Prasad, which has been very successful in giving physically realistic results and also in showing that the caustic of a linear theory is resolved when nonlinear effects are included. The weak shock ray theory with infinite system of compatibility conditions also follows from this theory.     

On systems of linear functional equations of the second kind in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

We consider a general system of functional equations of the second kind in L ₂ with a continuous linear operator T satisfying the condition that zero lies in the limit spectrum of the adjoint operator T*. We show that this condition holds for the operators of a wide class containing, in particular, all integral operators. The system under study is reduced by means of a unitary transformation to an equivalent system of linear integral equations of the second kind in L ₂ with Carleman matrix kernel of a special kind. By a linear continuous invertible change, this system is reduced to an equivalent integral equation of the second kind in L ₂ with quasidegenerate Carleman kernel. It is possible to apply various approximate methods of solution for such an equation.     

Existence of Insensitizing Controls for a Semilinear Heat Equation with a Superlinear Nonlinearity

Abstract

In this paper we consider a semilinear heat equation (in a bounded domain Ω of ? ^N ) with a nonlinearity that has a superlinear growth at infinity. We prove the existence of a control, with support in an open set ω ? Ω, that insensitizes the L ² ? norm of the observation of the solution in another open subset 𝒪 ? Ω when ω ∩ 𝒪 ≠ ?, under suitable assumptions on the nonlinear term f(y) and the right hand side term ξ of the equation. The proof, involving global Carleman estimates and regularizing properties of the heat equation, relies on the sharp study of a similar linearized problem and an appropriate fixed-point argument. For certain superlinear nonlinearities, we also prove an insensitivity result of a negative nature. The crucial point in this paper is the technique of construction of L ^r -controls (r large enough) starting from insensitizing controls in L ².     

Steady state bifurcations and exact multiplicity conditions via Carleman linearization

The problem of steady-state bifurcations of vector fields under parameter perturbation is resolved by a linear algebraic method. Exact multiplicity conditions for any steady state are obtained in terms of the system parameters. No reduction of the steady-state system to one equation is required. Instead the one-dimensional case is included as a subspace in this generalized framework. The key point that this paper highlights is that the order of the steady multiplicity at bifurcation can be determined by examining the dimension of the kernel of the successive Carleman linear operators for all cases of practical interest. In particular, the dimension of the kernel of any Carleman linear operator of order l, equals l if l is less than the multiplicity, μ. However, the μth order Carleman operator retains a (μ − 1)-dimensional kernel.     

Asymptotic Analysis of Abstract Linear Kinetic Equations

In this paper we provide a theory of the asymptotic expansion for an abstract kinetic equation. We show that the modified Chapman–Enskog procedure, introduced in [19], gives the error of order of ε², uniformly in time, on the second level of approximation and in particular that the zeroth moment of the solution (the spatial density of particles) can be approximated by the solution of the diffusion-type equation with the same accuracy. The theory is applied to several types of kinetic equations with the Fokker–Planck collision operator.     

Stackelberg–Nash exact controllability for the Kuramoto–Sivashinsky equation

This paper deals with a hierarchical control problem for the Kuramoto–Sivashinsky equation following a Stackelberg–Nash strategy. We assume that there is a main control, called the leader, and two secondary controls, called the followers. The leader tries to drive the solution to a prescribed target and the followers intend to be a Nash equilibrium for given functionals. It is known that this problem is equivalent to a null controllability result for an optimality system consisting of three non-linear equations. One of the novelties is a new Carleman estimate for a fourth-order equation with right-hand sides in Sobolev spaces of negative order, which allows to relax some geometric conditions for the observation sets for the followers.     

Carleman estimate for a strongly damped wave equation and applications to an inverse problem

In this paper, we establish a Carleman estimate for a strongly damped wave equation in order to solve a coefficient inverse problems of retrieving a stationary potential from a single time‐dependent Neumann boundary measurement on a suitable part of the boundary. This coefficient inverse problem is for a strongly damped wave equation. We prove the uniqueness and the local stability results for this inverse problem. The proof of the results relies on Carleman estimate and a certain energy estimates for hyperbolic equation with strongly damped term. Moreover, this method could be used for a similar inverse problem for an integro‐differential equation with hyperbolic memory kernel. Copyright © 2012 John Wiley & Sons, Ltd.     

A Relaxation Scheme for Solving the Boltzmann Equation Based on the Chapman-Enskog Expansion

Abstract In [16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jinand Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weaklyparabolic, has a linearly hyperbolic convection part, and is endowed with a generalized eotropy inequality. Itagrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion. We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system forthe Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme,and those obtained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and bythe extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments showthat the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equationobtained by the DSMC, for a range of Mach numbers for hypersonic flows, th     

Some controllability results for the 2D Kolmogorov equation

In this article, we prove the null controllability of the 2D Kolmogorov equation both in the whole space and in the square. The control is a source term in the right-hand side of the equation, located on a subdomain, that acts linearly on the state. In the first case, it is the complementary of a strip with axis x and in the second one, it is a strip with axis x.The proof relies on two ingredients. The first one is an explicit decay rate for the Fourier components of the solution in the free system. The second one is an explicit bound for the cost of the null controllability of the heat equation with potential that the Fourier components solve. This bound is derived by means of a new Carleman inequality.     

Uniqueness and stability in inverse parabolic equations with memory

First we establish a Carleman estimate for parabolic equations with second order spatial memory. Then we prove the stability results for the coefficient q from a measurement of the solution with respect to the normal derivative on an arbitrary part of the boundary and certain spatial derivatives at t=θ. Further we deduce the uniqueness result under some equivalence conditions on the solutions about the potential q. The proof of the results rely on Carleman estimates and certain energy estimates for parabolic equations with memory.     

On splitting up singularities of fundamental solutions to elliptic equations in ℂ2

It is known that the fundamental solution to an elliptic differential equation with analytic coefficients exists, is determined up to the kernel of the differential operator, and has singularities on characteristics of the equation in ℂ². In this paper we construct a representation of fundamental solution as a sum of functions, each of those has singularity on a single characteristic.      

Carleman Estimates for Second-Order Hyperbolic Equations

In the space of variables (x, t) ∈ ? ⁿ⁺¹, we consider a linear second-order hyperbolic equation with coefficients depending only on x. Given a domain D ? ? ⁿ⁺¹ whose projection to the x-space is a compact domain Ω, we consider the question of construction of a stability estimate for a solution to the Cauchy problem with data on the lateral boundary S of D. The well-known method for obtaining such estimates bases on the Carleman estimates with an exponential-type weight function exp(2τ?(x, t)) whose construction faces certain difficulties in case of hyperbolic equations with variable coefficients. We demonstrate that if D is symmetric with respect to the plane t = 0 then we can take ?(x, t) to be the function ?(x, t) = s ²(x, x ⁰) ? pt ², where s(x, x ⁰) is the distance between points x and x ⁰ in the Riemannian metric induced by the differential equation, p is some positive number less than 1, and the fixed point x ⁰ can either belong to the domain Ω or lie beyond it. As for the metric, we suppose that the sectional curvature of the corresponding Riemannian space is bounded above by some number k ₀ ≥ 0. In case of space of nonpositive curvature the parameter p can be taken arbitrarily close to 1; in this case as p → 1 the stability estimates lead to a uniqueness theorem which describes exactly the domain of the solution continuation through S. It turns out that, in case of space of bounded positive curvature, construction of a Carleman estimate is possible only if the product of k ₀ and sup _x∈Ω s ²(x, x ⁰) satisfies some smallness condition.     

Regularization of solutions of the Cauchy problem for systems of elasticity theory in infinite domains

We construct the Carleman matrix for the Cauchy problem for the Helmholtz equation in an unbounded domain ℝ³ with piecewise smooth boundaries. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 548–553, October, 2000.     

Hardy Approximation to L p Functions on Subsets of the Circle with 1< =p< = infinity

We consider approximation of L ^p functions by Hardy functions on subsets of the circle for . After some preliminaries on the possibility of such an approximation which are connected to recovery problems of the Carleman type, we prove existence and uniqueness of the solution to a generalized extremal problem involving norm constraints on the complementary subset. December 6, 1995. Date revised: August 26, 1996.     

Controllability results for the two-dimensional heat equation with mixed boundary conditions using Carleman inequalities: a linear and a semilinear case

In this paper, we prove controllability results for a two-dimensional semilinear heat equation with mixed boundary conditions. It is well-known that mixed boundary conditions can present a singular behaviour of the solution. First, we will prove global Carleman estimates then we will use these inequalities to obtain controllability results.     

Numerical solution of a minimax ergodic optimal control problem

In this work we consider an L^∞ minimax ergodic optimal control problem with cumulative cost. We approximate the cost function as a limit of evolutions problems. We present the associated Hamilton-Jacobi-Bellman equation and we prove that it has a unique solution in the viscosity sense. As this HJB equation is consistent with a numerical procedure, we use this discretization to obtain a procedure for the primitive problem. For the numerical solution of the ergodic version we need a perturbation of the instantaneous cost function. We give an appropriate selection of the discretization and penalization parameters to obtain discrete solutions that converge to the optimal cost. We present numerical results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Smoothness of solutions of nonlinear ODE

Smoothness of a -function f is measured by (Carleman) sequence we say if with As an extension of our results in [2] we prove the following type statements: Let u be -function on [0,1] such that where is an entire function of order 1, finite type, on z and of a nonanalytic Carleman class on t. Then u is in the same Carleman class. Received: 16 January 2001 / Revised version: 10 August 2001 / Published online: 6 August 2002