Cauchy Data Research Articles

Inverse problems for fractional equations with a minimal number of measurements

In this paper, we study several inverse problems associated with a fractional differential equation of the following form: \begin{document}$ (-\Delta)^s u(x)+\sum\limits_{k = 0}^N a^{(k)}(x) [u(x)]^k = 0, \ \ 0<s<1, \ N\in\mathbb{N}\cup\{0\}\cup\{\infty\}, $\end{document} which is given in a bounded domain $ \Omega\subset\mathbb{R}^n $, $ n\geq 1 $. For any finite $ N $, we show that $ a^{(k)}(x) $, $ k = 0, 1, \ldots, N $, can be uniquely determined by $ N+1 $ different pairs of Cauchy data in $ \Omega_e: = \mathbb{R}^n\backslash\overline{\Omega} $. If $ N = \infty $, the uniqueness result is established by using infinitely many pairs of Cauchy data. The results are highly intriguing in that it generally does not hold true in the local case, namely $ s = 1 $, even for the simplest case when $ N = 0 $, a fortiori $ N\geq 1 $. The nonlocality plays a key role in establishing the uniqueness result, and we do not utilize any linearization techniques. We also establish several other unique determination results by making use of a minimal number of measurements. Moreover, in the process we derive a novel comparison principle for nonlinear fractional differential equations as a significant byproduct.

Stability for an inverse source problem of the diffusion equation

We prove an increasing stability estimate for an inverse source problem of the diffusion equation with a nontrivial variable absorption coefficient in . The stability estimate consists of two parts: the Lipschitz type data discrepancy and the high frequency tail of the source function. The latter decreases when the upper bound of the frequency increases. The proof is based on an exact observability bound for the heat equation and the resolvent estimates for the elliptic operator. In particular, to justify the Fourier transform and obtain Plancherel’s theorem for the time-domain Cauchy data on the boundary, we derive certain appropriate time decay estimates for the heat equation using semigroup theory.

Fractional evolution equation with Cauchy data in L^{p} spaces

In this paper, we consider the Cauchy problem for fractional evolution equations with the Caputo derivative. This problem is not well posed in the sense of Hadamard. There have been many results on this problem when data is noisy in L^{2} and H^{s}. However, there have not been any papers dealing with this problem with observed data in L^{p} with p neq 2. We study three cases of source functions: homogeneous case, inhomogeneous case, and nonlinear case. For all of them, we use a truncation method to give an approximate solution to the problem. Under different assumptions on the smoothness of the exact solution, we get error estimates between the regularized solution and the exact solution in L^{p}. To our knowledge, L^{p} evaluations for the inverse problem are very limited. This work generalizes some recent results on this problem.

Painlevé-type asymptotics for the defocusing Hirota equation in transition region

We consider the long-time asymptotics for the defocusing Hirota equation with Schwartz Cauchy data in the transition region. On the basis of direct and inverse scattering transform of the Lax pair of Hirota equations, we first express the solution of the Cauchy problem in terms of the solution of a Riemann–Hilbert problem. Further, we apply nonlinear steepest descent analysis to obtain the long-time asymptotics of the solution in the critical transition region | x / t − ( α 2 / 3 β ) | t 2 / 3 ≤ M , where M is a positive constant. Our result shows that the long-time asymptotics of the Hirota equation can be expressed in terms of the solution of the Painlevé II equation.

The Carleman-based contraction principle to reconstruct the potential of nonlinear hyperbolic equations

We develop an efficient and convergent numerical method for solving the inverse problem of determining the potential of nonlinear hyperbolic equations from lateral Cauchy data. In our numerical method we construct a sequence of linear Cauchy problems whose corresponding solutions converge to a function that can be used to efficiently compute an approximate solution to the inverse problem of interest. The convergence analysis is established by combining the contraction principle and Carleman estimates. We numerically solve the linear Cauchy problems using a quasi-reversibility method. Numerical examples are presented to illustrate the efficiency of the method.

Determining the Potential Function of the Stationary Vector Burgers’ Equation

We consider the inverse problem of determining the potential function of the stationary vector Burgers equations Δu – (u · ∇)u – q(x)u = 0. We give the well-posedness of the solution in H2 a small boundary value. Then, by linearization, we prove that the potential function can be determined from the boundary Cauchy data.

Recovery of Piecewise Smooth Density and Lamé Parameters from High Frequency Exterior Cauchy Data

Recovery of Piecewise Smooth Density and Lamé Parameters from High Frequency Exterior Cauchy Data

Stability of inverse scattering problem for the damped biharmonic plate equation

This paper is concerned with the stability of the inverse scattering problem for the damped biharmonic plate equation in a bounded domain with the Cauchy data. A sharp estimate for the mass density is established using a priori information concerning Sobolev norms and a priori information about the support of the inhomogeneity. Our results improve previous estimates and explicitly depend on the damping coefficient. The proof mainly relies on the complex geometric optics solution method and the Fourier analysis.

Notes on Convergence Results for Parabolic Equations with Riemann–Liouville Derivatives

Fractional diffusion equations have applications in various fields and in this paper we consider a fractional diffusion equation with a Riemann–Liouville derivative. The main objective is to investigate the convergence of solutions of the problem when the fractional order tends to 1−. Under some suitable conditions on the Cauchy data, we prove the convergence results in a reasonable sense.

Propagation of smallness and size estimate in the second order elliptic equation with discontinuous complex Lipschitz conductivity

In this paper, we would like to derive three-ball inequalities and propagation of smallness for the complex second order elliptic equation with discontinuous Lipschitz coefficients. As an application of such estimates, we study the size estimate problem by one pair of Cauchy data on the boundary, that is, a pair of the Neumann and Dirichlet data of the solution on the boundary. The main ingredient in the derivation of three-ball inequalities and propagation of smallness is a local Carleman proved in our recent paper [13].

On the application of the Fourier method to solve the problem of correction of thermographic images

The work is devoted to the construction of computational algorithms implementing the method of correction of thermographic images. The correction is carried out on the basis of solving some ill-posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. This problem corresponds to the problem of the analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study towards the heat sources. The cylindrical region is bounded by an arbitrary surface and plane. On an arbitrary surface, a temperature distribution is measured (and thus is known). It is called a thermogram and reproduces an image of the internal heat-generating structure. On this surface, which is the boundary of the object under study, convective heat exchange with the external environment of a given temperature takes place, which is described by Newton’s law. This is the third boundary condition, which together with the first boundary condition corresponds to the Cauchy conditions - the boundary values of the desired function and its normal derivative. The problem is ill-posed. In this paper, using the Tikhonov regularization method, an approximate solution of the problem was obtained, stable with respect to the error in the Cauchy data, and which can be used to build effective computational algorithms. The paper considers algorithms that can significantly reduce the amount of calculations.

Global Conserved Quantities and Unfree Gauge Symmetry

We consider a class of theories with unfree gauge symmetry, whose gauge parameters are restricted by differential equations. We demonstrate that such theories admit global conserved quantities, whose on-shell values are defined by asymptotics of the fields rather than Cauchy data. The global conserved quantities can be deduced proceeding from the equations restricting gauge parameters, and they are treated differently by two BRST complexes corresponding to a system with unfree gauge symmetry.

On an inverse photoacoustic tomography problem of small absorbers with inhomogeneous sound speed

Abstract This work is devoted to the study of the inverse photoacoustic tomography (PAT) problem. It is an imaging technique similar to TAT studied in El Badia & Ha-Duong (2000); however, in this case, a high-frequency radiation is delivered into the biological tissue to be imaged, such as visible or near infra red light that are characterized by their high frequency compared with that of radio waves that are used in TAT. As in the case of TAT El Badia & Ha-Duong (2000), the inverse problem we are concerned in is the reconstruction of small absorbers in an open, bounded and connected domain $\Omega \subset{\mathbb{R}}^3$. Again, we follow the algebraic algorithm, initially proposed in El Badia & Jebawy (2020), that allows us to resolve the problem from a single Cauchy data and without the knowledge of the Grüneisen’s coefficient. However, the high-frequency radiation used in this case makes some changes in the context of the problem and allows us to give our results using partial boundary observations and in both cases of constant and variable acoustic speed. Finally, we establish the corresponding Hölder stability result.

Regularization of Inverse Initial Problem for Conformable Pseudo-Parabolic Equation with Inhomogeneous Term

The main goal of the paper is to approximate two types of inverse problems for conformable heat equation (or called parabolic equation with conformable operator); as follows, we considered two cases: the right hand side of equation such that F x , t and F x , t = φ t f x . Up to now, there are very few surveys working on the results of regularization in L p spaces. Our paper is the first work to investigate the inverse problem for conformable parabolic equations in such spaces. For the inverse source problem and the backward problem, use the Fourier truncation method to approximate the problem. The error between the regularized solution and the exact solution is obtained in L p under some suitable assumptions on the Cauchy data.

Fast spectral solver for the inversion of boundary data problem of Poisson equation in a doubly connected domain

In this paper, we study the inversion of the inner boundary data for Poisson equation in a doubly connected domain by fast Fourier ultraspherical spectral solver. The solver depends on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by an ultraspherical spectral method. Because this problem is seriously ill‐posed, the Cauchy data with noise will lead to ill‐conditioned linear system. Hence, we apply Tikhonov regularization to solve the obtained linear system and use generalized cross‐validation (GCV) criterion to select regularization parameters. The accuracy and efficiency of the proposed method are illustrated by several numerical results of different regions.

Asymptotic behavior of solutions to a chemotaxis-logistic model with transitional end-states

We study Cauchy problem of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. Our Cauchy data connect two different end-states for the chemical signal while the cell density takes its typical carrying capacity at the far fields. We are interested in the time-asymptotic behavior of the solution. We show that in the borderline, the component representing the chemical signal converges to a permanent, diffusive background wave, which connects the two end-states monotonically. On the other hand, the cell component converges to the spatial derivative of a heat kernel. The asymptotic solution has explicit formulation and is common to all solutions sharing the same end-states. Optimal L2 and L∞ convergence rates are obtained. We first convert the model into a 2×2 hyperbolic-parabolic system via inverse Hopf-Cole transformation. Then we apply Chapman-Enskog expansion to identify the asymptotic solution. After extracting the asymptotic solution, we use a variety of analytic tools to study the remainder and obtain optimal rates. These include time-weighted energy method, spectral analysis, Green's function estimate and iterations. Our results apply to a general class of Cauchy data for the model and for its transformed system. In particular, our results apply to large data solutions.

A large data theory for nonlinear wave on the Schwarzschild background

We study both of the scattering and Cauchy problems for the semilinear wave equation with a null quadratic form on the Schwarzschild background. Prescribing the scattering data that are given by the short pulse data on the future null infinity and are trivial on the future event horizon, we construct a class of global solutions backwards up to any finite time and show that the wave travels in such a way that almost all of the (large) energy is focusing in an outgoing null strip, while little radiates out of this strip. In reverse, considering a class of Cauchy data with large energy norms, there exists a unique and global solution in the future development, and the radiation field along the future null infinity exists. Furthermore, most of the wave packet is confined in an incoming null strip and reflected to the future event horizon, whereas little is transmitted to the future null infinity.

A Scattering Theory Approach to Cauchy Horizon Instability and Applications to Mass Inflation

Motivated by the strong cosmic censorship conjecture, we study the linear scalar wave equation in the interior of subextremal strictly charged Reissner–Nordström black holes by analyzing a suitably defined “scattering map” at 0 frequency. The method can already be demonstrated in the case of spherically symmetric scalar waves on Reissner–Nordström: we show that assuming suitable (\(L^2\)-averaged) upper and lower bounds on the event horizon, one can prove (\(L^2\)-averaged) polynomial lower bound for the solution (1) on any radial null hypersurface transversally intersecting the Cauchy horizon, and (2) along the Cauchy horizon toward timelike infinity. Taken together with known results regarding solutions to the wave equation in the exterior, (1) above in particular provides yet another proof of the linear instability of the Reissner–Nordström Cauchy horizon. As an application of (2) above, we prove a conditional mass inflation result for a nonlinear system, namely the Einstein–Maxwell-(real)-scalar field system in spherical symmetry. For this model, it is known that for a generic class of Cauchy data \(\mathcal G\), the maximal globally hyperbolic future developments are \(C^2\)-future-inextendible. We prove that if a (conjectural) improved decay result holds in the exterior region, then for the maximal globally hyperbolic developments arising from initial data in \(\mathcal G\), the Hawking mass blows up identically on the Cauchy horizon.

Uniqueness of a spherically symmetric refractive index from modified transmission eigenvalues

In the present work we study the modified transmission eigenvalue problem for the case of the spherically symmetric refractive index. This problem occurs as a modification of the classic transmission eigenvalue problem for a fixed wavenumber, by introducing a new spectral parameter. We prove the existence of infinite and discrete eigenvalues using asymptotic expressions and monotonicity properties of the characteristic functions. Next, we define the inverse spectral problem of determining the refractive index from the knowledge of modified transmission eigenvalues. We show that one eigenvalue corresponding to each characteristic function is sufficient to uniquely determine the fraction of the boundary Cauchy data. The knowledge of these data, in combination with a Dirichlet-to-Neumann mapping property allows us to establish uniqueness for the inverse problem.

Lifespan of classical solutions to one dimensional fully nonlinear wave equations with time-dependent damping

This paper deals with the lower bound estimate for the maximal time T˜(ε) of the classical solutions to one dimensional fully nonlinear wave equations □u+b(t)ut=F(u,Du,DxDu) with Cauchy data of small size ε, where the coefficient b(t)=μ(1+t)−α is “effective” with μ>0 and α∈(0,1). If F is sufficiently smooth and vanishes at order p+1 in a neighborhood of zero, it is proved thatT˜(ε){≥Cε−p1−K,K<1,≥eCε−p,K=1,=∞,K>1, where K=(p−1)(1+α)2 and C is a positive constant independent of ε.