Null controllability for the heat equation with singular inverse-square potentials

We prove the null controllability of the heat equation perturbed by a singular inverse-square potential arising in quantum mechanics and combustion theory. This is done within the range of subcritical coefficients of the singular potential, provided the control acts on an annular set around the singularity. Our proof uses a splitting argument on the domain, decomposition in spherical harmonics, new Carleman inequalities and refined Hardy inequalities.     

Determination of the insolation function in the nonlinear Sellers climate model

We are interested in the climate model introduced by Sellers in 1969 which takes the form of some nonlinear parabolic equation with a degenerate diffusion coefficient. We investigate here some inverse problem issue that consists in recovering the so-called insolation function. We not only solve the uniqueness question but also provide some strong stability result, more precisely unconditional Lipschitz stability in the spirit of the well-known result by Imanuvilov and Yamamoto (1998) [22]. The main novelties rely in the fact that the considered model is degenerate and above all nonlinear. Indeed we provide here one of the first result of Lipschitz stability in a nonlinear case.     

Inverse problems with partial data for a Dirac system: A Carleman estimate approach

We prove that the material parameters in a Dirac system with magnetic and electric potentials are uniquely determined by measurements made on a possibly small subset of the boundary. The proof is based on a combination of Carleman estimates for first and second order systems, and involves a reduction of the boundary measurements to the second order case. For this reduction a certain amount of decoupling is required. To effectively make use of the decoupling, the Carleman estimates are established for coefficients which may become singular in the asymptotic limit.     

Observability Inequalities by Internal Observation and Their Applications

By means of a Carleman estimate, we obtain several observability inequalities by internal observation for hyperbolic equations with lower-order terms. Also, we apply our observability results to an exact controllability problem and an inverse wave source problem.     

Global Uniqueness and Stability for an Inverse Plate Problem

This paper deals with the global uniqueness and stability of an inverse problem for the plate equation. By means of a suitable transformation, the problem is reduced to the observability estimate for the plate equations with memory. The main tool that we employ is a global Carleman-type estimate. This work was supported by FANEDD of China Project 200119 and by NSFC under Grant 10371084. The author thanks Professor X. Zhang for help.     

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

The focus of this article is on conditional stability estimates for ill-posed inverse problems in partial differential equations. Conditional stability estimates have been obtained in related literature by a couple different methods. In this article, we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We provide the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capabilities of our method are illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.     

Estimates of initial conditions of parabolic equations and inequalities in infinite domains via lateral Cauchy data

A parabolic equation/inequality in an infinite domain is considered. The lateral Cauchy data are given at an arbitrary C²-smooth lateral surface. The inverse problem of the interest of this paper consists in an estimate of the unknown initial condition via these Cauchy data.     

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.     

Monotonicity of input-output mappings in inverse coefficient and source problems for parabolic equations

This article presents a mathematical analysis of input-output mappings in inverse coefficient and source problems for the linear parabolic equation ut=(kx(x)ux)+F(x,t), (x,t)∈ΩT:=(0,1)×(0,T]. The most experimentally feasible boundary measured data, the Neumann output (flux) data f(t):=−k(0)ux(0,t), is used at the boundary x=0. For each inverse problems structure of the input-output mappings is analyzed based on maximum principle and corresponding adjoint problems. Derived integral identities between the solutions of forward problems and corresponding adjoint problems, permit one to prove the monotonicity and invertibility of the input-output mappings. Some numerical applications are presented.     

Attractors for non-autonomous parabolic problems with singular initial data

In this paper, we study the asymptotic behavior of solutions of non-autonomous parabolic problems with singular initial data. We first establish the well-posedness of the equation when the initial data belongs to Lr(Ω) (1<r<∞) and W1,r(Ω) (1<r<N), respectively. When the initial data belongs to Lr(Ω), we establish the existence of uniform attractors in Lr(Ω) for the family of processes with external forces being translation bounded but not translation compact in . When we consider the existence of uniform attractors in , the solution of equation lacks the higher regularity, so we introduce a new type of solution and prove the existence result. For the long time behavior of solutions of the equation in W1,r(Ω), we only obtain the uniform attracting property in the weak topology.