Mukhtar Karazym, Tohru Ozawa, Durvudkhan Suragan
We discuss inverse problems to finding the time-dependent coefficient for the multidimensional Cauchy problems for both strictly hyperbolic equations and polyharmonic heat equations. We also extend our techniques to the general inverse Cauchy problems for evolution equations.
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An analogue of the Cauchy problem for the iterated multidimensional Klein- Gordon-Fock equation with a time-dependent Bessel operator is investigated. Applying the generalized Erdelyi-Kober operator of fractional order, the problem posed is reduced to the Cauchy problem for the polywave equation. An explicit formula for solving this problem is constructed by the spherical mean method. On the basis of the solution obtained, an integral representation of the solution of the problem is found.
We study the Cauchy problem for a multidimensional scalar conservation law on the Bohr compactification of $\R^n$. The existence and uniqueness of entropy solutions are established in the general case of merely continuous flux vector. We propose also the necessary and sufficient condition for the decay of entropy solutions as time $t\to+\infty$.
We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the $C_0$-semigroup (this is a standard assumption in the evolution equations theory, which guarantees the existence of the solution) and then find the representation (based on the family of translation operators) of the solution in terms of coefficients of the equation and initial condition. It is proved that if the coefficients of the equation are bounded, infinite…
This text deals with multidimensional Borg-Levinson inverse theory. Its main purpose is to establish that the Dirichlet eigenvalues and Neumann boundary data of the Dirichlet Laplacian acting in a bounded domain of dimension 2 or greater, uniquely determine the real-valued bounded potential. We first address the case of incomplete spectral data, where finitely many boundary spectral eigen-pairs remain unknown. Under suitable summability condition on the Neumann data, we also consider the case where only the asymptotic behavior of the eigenvalues is known. Finally, we use the multidimensional B…
We prove locally in time the existence of a smooth solution for multidimensional two-phase Stefan problem for degenerate parabolic equations of the porous medium type. We establish also natural Hölder class for the boundary conditions in the Cauchy-Dirichlet problem for a degenerate parabolic equation.
We study the Cauchy problem for a multidimensional scalar conservation law with merely continuous flux vector in the class of Besicovitch almost periodic functions. The existence and uniqueness of entropy solutions are established. We propose also the necessary and sufficient condition for the decay of almost periodic entropy solutions as time $t\to+\infty$.
Direct and inverse abstract Cauchy problems with fractional powers of almost sectorial operators
本文研究了具有分数幂的近似扇形算子的直接和反向抽象柯西问题，推导了显式解算子并探讨了时间变量系数的求解方法。
In this paper we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered elliptic Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with deviating argument.
Lipschitz stability of an inverse conductivity problem with two Cauchy data pairs
本文证明了具有两对 Cauchy 数据的反导电率问题中线性化前向映射的单射性，推导出非线性反问题解的 Lipschitz 稳定性，并指出绝缘包含物仅需单对数据即可。
An inverse Cauchy problem of a stochastic hyperbolic equation
本文研究随机双曲型方程的反柯西问题，利用 Carleman 恒等式建立观测性估计并构造 Tikhonov 型泛函求正则化近似，结合核学习理论提出数值重构算法，实验验证了其可行性与准确性。
In this paper, we study the performance of Full Waveform Inversion (FWI) from time-harmonic Cauchy data via conditional well-posedness driven iterative regularization. The Cauchy data can be obtained with dual sensors measuring the pressure and the normal velocity. We define a novel misfit functional which, adapted to the Cauchy data, allows the independent location of experimental and computational sources. The conditional well-posedness is obtained for a hierarchy of subspaces in which the inverse problem with partial data is Lipschitz stable. Here, these subspaces yield piecewise linear rep…
An inverse obstacle problem with a single pair of Cauchy data: Laplace's equation case
本文研究Laplace方程的逆障碍问题，提出基于factorization method的domain-defined sampling method与迭代混合策略，从单对柯西数据中恢复常数电导率系数和Dirichlet多边形障碍物，并证明了唯一性与高效性。
The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998, based on Carleman estimates, seems hard to apply to the case of Grushin-type operators of interest to this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse source problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a mixed strategy which combines the appraoch du…
In $L_2(\mathbb{R}^d)$, we consider an elliptic differential operator $\mathcal{A}_\varepsilon = - \operatorname{div} g(\mathbf{x}/\varepsilon) \nabla + \varepsilon^{-2} V(\mathbf{x}/\varepsilon)$, $ \varepsilon > 0$, with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian $\mathcal{A}_\varepsilon$, analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator $\mathcal{A}_1$ are studied (the so called high-energy homogenization). For the solutions of the Cauchy problems for these equations with special initial d…
Inverse Random Source and Cauchy Problems for Semi-Discrete Stochastic Parabolic Equations in Arbitrary Dimensions
本文研究了半离散随机抛物方程在任意维度中的逆源和Cauchy逆问题，利用新的Carleman估计获得稳定性结果。
In this paper we study the Cauchy problem for one multidimensional compressible nonlocal model of the dissipative quasi-geostrophic equations. First, we obtain the local existence and uniqueness of the smooth non-negative solution or the strong solution in time. Secondly, for the sub-critical and critical case $1\leα\leq 2$, we obtain the global existence and uniqueness results of the nonnegative smooth solution. Then, we prove the global existence of the weak solution for $0\le α\le 2$ and $ν\ge 0$. Finally, for the sub-critical case, we establish the global $H^1$ and $L^p, p>2,$ decay rate o…
Basing on our results [1] on a representation of solutions to the Cauchy problem for multidimensional non-viscous Burgers equation obtained by a method of stochastic perturbation of the associated Langevin system, we deduce an explicit asymptotic formula for smooth solutions to the Cauchy problem for any genuinely nonlinear hyperbolic system of equations written in the Riemann invariants.
We consider an inverse source problem for the Helmholtz equation in a bounded domain. The problem is to reconstruct the shape of the support of a source term from the Cauchy data on the boundary of the solution of the governing equation. We prove that if the shape is a polygon, one can calculate its support function from such data. An application to the inverse boundary value problem is also included.
In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation with inverse-square potential \[iu_{t} +Δu-c|x|^{-2}u=λ|x|^{-b} |u|^{σ} u,\; u(0)=u_{0} \in H^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $λ=\pm1$, $0<b<2$, $σ=\frac{4-2b}{d-2}$ and $c>-c(d):=-\left(\frac{d-2}{2}\right)^{2}$. We first prove the local well-posedness as well as small data global well-posedness and scattering in $H^{1}$ for $c>-\frac{(d+2-2b)^{2}-4}{(d+2-2b)^{2}}c(d)$ and $0<b<\frac{4}{d}$, by using the contraction mapping principle based on the Strich…
Local Hölder stabilities for inverse problems of first-order hyperbolic equations
本文针对时间依赖系数的一阶双曲方程Cauchy问题，利用Carleman估计，在侧向子边界给定数据下，建立了反源与反系数问题的局部Hölder稳定性。
On the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation with inverse-power potential
本文研究带逆幂势的非齐次非线性薛定谔方程的柯西问题，借助索伯列夫-洛伦兹空间的施特里哈茨估计与压缩映射原理建立局部适定性，还探究了H¹解的全局存在性与爆破，推广了已有相关结果。
In this paper we investigate the problem of identifying the source term in an elliptic system from a single noisy measurement couple of the Neumann and Dirichlet data. A variational method of Tikhonov-type regularization with specific misfit term of Kohn-Vogelius-type and quadratic stabilizing penalty term is suggested to tackle this linear inverse problem. The method also appears as a variant of the Lavrentiev regularization. For the occurring linear inverse problem in infinite dimensional Hilbert spaces, convergence and rate results can be found from the general theory of classical Tikhonov …
We study the Cauchy problem for $p$-adic nonlinear evolutionary pseudo-differential equations for complex-valued functions of a real positive time variable and p-adic spatial variables. Among the equations under consideration there is the p-adic analog of the porous medium equation (or more generally, the nonlinear filtration equation) which arise in numerous application in mathematical physics and mathematical biology. Our approach is based on the construction of a linear Markov semigroup on a p-adic ball and the proof of m-accretivity of the appropriate nonlinear operator. The latter result …
This paper is concerned with reconstruction issue of some typical inverse problems and consists of three parts. First a framework of the enclosure method for an inverse source problem governed by the Helmholtz equation at a fixed wave number in three dimensions is introduced. It is based on the nonvanishing of the coefficient of the leading profile of an oscillatory integral over a domain having a conical singularity. Second an explicit formula of the coefficient for a domain having a circular cone singularity and its implication under the framework are given. Third, an application under the f…
The paper is devoted to the development of the theory of inverse problems for evolution equations with terms rapidly oscillating in time. A new approach to setting such problems is developed for the case in which additional constraints are imposed only on several first terms of the asymptotics of the solution rather that on the whole solution. This approach is realized in the case of a multidimensional hyperbolic equation with unknown absolute term.
We prove that the multidimensional Schrödinger equation is exactly controllable in infinite time near any point which is a finite linear combination of eigenfunctions of the Schrödinger operator. We prove that, generically with respect to the potential, the linearized system is controllable in infinite time. Applying the inverse mapping theorem, we prove the controllability of the nonlinear system.
In this short paper we prove a global logarithmic stability of the Cauchy problem for H 2-solutions of an anisotropic elliptic equation in a Lip-schitz domain. The result we obtained is based on tools borrowed from the existing technics to establish stability estimate for the Cauchy problem [4] (see also [1]) combined with tools we already used in [6] to study an inverse medium problem.
Feynman's inverse problem
本文研究了费曼提出的水波逆问题，通过分析重力-表面张力波的逆Cauchy问题，证明了逆问题的唯一性，并通过数值实验验证了结果的稳定性。
The paper is concerned with sticky weak solutions to the equations of pressureless gases in two or more space dimensions. Various initial data are constructed, showing that the Cauchy problem can have (i) two distinct sticky solutions, or (ii) no sticky solution, not even locally in time. In both cases the initial density is smooth with compact support, while the initial velocity field is continuous.
We consider an inverse problem of determining coefficient matrices in an $N$-system of second-order elliptic equations in a bounded two dimensional domain by a set of Cauchy data on arbitrary subboundary. The main result of the article is as follows: If two systems of elliptic operators generate the same set of partial Cauchy data on an arbitrary subboundary, then the coefficient matrices of the first-order and zero-order terms satisfy the prescribed system of first-order partial differential equations. The main result implies the uniqueness of any two coefficient matrices provided that the on…
This paper presents a full classification of the short-time behavior of the interfaces in the Cauchy problem for the nonlinear second order degenerate parabolic PDE \[ u_t-Δu^m +b u^β=0, \ x\in \mathbb{R}^N, 0<t<T \] with nonnegative initial function $u_0$ such that \[ supp~u_0 = \{|x|<R\}, \ u_0 \sim C(R-|x|)^α, \quad{as} \ |x|\to R-0, \] where $m>1, C,α, β>0, b \in \mathbb{R}$. Interface surface $t=η(x)$ may shrink, expand or remain stationary depending on the relative strength of the diffusion and reaction terms near the boundary of support, expressed in terms of the parameters $m,β, α, sig…
This paper is devoted to the inverse problem of recovering simultaneously a potential and a point source in a Shrödinger equation from the associated nonlinear Dirichlet to Neumann map. The uniqueness of the inversion is proved and logarithmic stability estimates are derived. It is well known that the inverse problem of determining only the potential while knowing the source, is ill-posed. In contrast the problem of identifying a point source when the potential is given is well posed. The obtained results show that the nonlinear Dirichlet to Neumann map contains enough information to determine…
We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.
Sampling methods for recovering buried corroded boundaries from partial electrostatic Cauchy data
本文将线性采样法和因子分解法推广到部分边界测量的静电问题中，通过推导边界积分方程恢复混合格林函数，解决埋藏腐蚀的反形状与参数问题，并证明了参数反演的唯一性和利普希茨稳定性。
We deal with an inverse problem arising in corrosion detection. The presence of corrosion damage is modeled by a nonlinear boundary condition on the inaccessible portion of the metal specimen. We propose a method for the approximate reconstruction of such a nonlinearity. A crucial step of this procedure, which encapsulates the major cause of the ill-posedness of the problem, consists of the solution of a Cauchy problem for an elliptic equation. For this purpose we propose an SVD approach.
Direct and inverse source problems for two-term time-fractional diffusion equation with Hilfer derivative
本文研究矩形域中具有两项广义分数阶导数（Hilfer导数）的扩散方程的正问题和反源问题，通过谱展开方法将问题转化为两项分数阶微分方程，并利用正弦函数和二元多项Mittag-Leffler函数表示解，在给定数据条件下证明无穷级数的一致收敛性。
Consider a real-analytic orientable connected complete Riemannian manifold $M$ with boundary of dimension $n\ge 2$ and let $k$ be an integer $1\le k\le n$. In the case when $M$ is compact of dimension $n\ge 3$, we show that the manifold and the metric on it can be reconstructed, up to an isometry, from the set of the Cauchy data for harmonic $k$-forms, given on an open subset of the boundary. This extends a result of [13] when $k=0$. In the two-dimensional case, the same conclusion is obtained when considering the set of the Cauchy data for harmonic $1$-forms. Under additional assumptions on t…
We consider an inverse problem for electrically conductive material occupying a domain $Ω$ in $\Bbb R^2$. Let $γ$ be the conductivity of $Ω$, and $D$ a subdomain of $Ω$. We assume that $γ$ is a positive constant $k$ on $D$, $k\not=1$ and is $1$ on $Ω\setminus D$; both $D$ and $k$ are unknown. The problem is to find a reconstruction formula of $D$ from the Cauchy data on $\partialΩ$ of a non-constant solution $u$ of the equation $\nabla\cdotγ\nabla u=0$ in $Ω$. We prove that if $D$ is known to be a convex polygon such that $\text{diam}\,D<\text{dist}\,(D,\partialΩ)$, there are two formulae for …
The aim of this article is to investigate the uniqueness of solution of an inverse problem for ultrahyperbolic equations. We first reduce the inverse problem to a Cauchy problem for an integro-differential equation and then by using a pointwise Carleman type inequality we prove the uniqueness.
We consider the inverse boundary value problem of determining the potential $q$ in the equation $Δu + qu = 0$ in $Ω\subset\mathbb{R}^n$, from local Cauchy data. A result of global Lipschitz stability is obtained in dimension $n\geq 3$ for potentials that are piecewise linear on a given partition of $Ω$. No sign, nor spectrum condition on $q$ is assumed, hence our treatment encompasses the reduced wave equation $Δu + k^2c^{-2}u=0$ at fixed frequency $k$.
We investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that we reduce nonlocal parabolic inverse problems to the corresponding local inverse problems with the lateral boundary Cauchy data. In addition, we derive a new equation and offer a novel proof of the unique continuation property for this new equation. We also build both uniqueness and non-uniqueness results for both nonlocal isotropic and anisotropic parabolic Calderón problems, resp…
In this article, we explore some of the main mathematical problems connected to multidimensional fractional conservation laws driven by Lévy processes. Making use of an adapted entropy formulation, a result of existence and uniqueness of a solution is established. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Lévy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing vis…
Inverse source problem for the Boltzmann equation in cosmology
该研究聚焦从宇宙微波背景各向异性恢复原初扰动的逆问题，将其转化为带柯西面测量的线性玻尔兹曼方程逆源问题，得到了针对一般吸收系数和散射核的两个稳定确定性结果。
We consider different types of processes obtained by composing Brownian motion $B(t)$, fractional Brownian motion $B_{H}(t)$ and Cauchy processes $% C(t)$ in different manners. We study also multidimensional iterated processes in $\mathbb{R}^{d},$ like, for example, $\left( B_{1}(|C(t)|),...,B_{d}(|C(t)|)\right) $ and $\left( C_{1}(|C(t)|),...,C_{d}(|C(t)|)\right) ,$ deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equa…
We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial stationary state. We prove the blowup results using the characteristics of the propagation of the solution in space and find upper and lower bounds for the density of a smooth solution in a given region of space in terms of the initial data. To solve the problems, we introduce a special family of integral functionals and study their temporal dynamics.
In this article, for a fourth-order parabolic equation which is closely related for example to the Cahn-Hilliard equation, we study an inverse source problem by interior data and the continuation of solution from lateral Cauchy data. Our method relies on a Carleman estimate and proves conditional stability for both problems.
This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The a…
Constructing steering-type solutions for higher order Cauchy-Riemann equations in $\mathbb{R}^{m+1}$
本文提出通过构造闭合于共轭和复Cauchy-Riemann算子的复值函数族，显式求解高阶Cauchy-Riemann方程在$\mathbb{R}^{m+1}$中的解，并证明部分解满足超复导数相关的齐次微分方程。
A Coefficient Inverse Problem for the Mean Field Games System
本文研究了二阶 Mean Field Games System 的系数反问题，利用单次测量生成的横向 Cauchy 数据和固定时刻解，通过 Carleman 估计方法证明了 Hölder 稳定性和解的唯一性。
We consider the spatially inhomogeneous quantum Boltzmann equation for bosons with a singular collision kernel, the weak-coupling limit of a large system of Bose-Einstein particles interacting through inverse power law. Global well-posedness of the corresponding Cauchy problem is proved in a periodic box near equilibrium for initial data satisfying high temperature condition.
We consider the two-dimensional Kirchhoff-Love plate equation in the context of elasticity modeling the stresses and deformations in thin plates subjected to forces and moments. We establish global recovery of the material parameters like bending stiffness, Poisson coefficient, Lamé parameters from the associated boundary Cauchy data of the equation.
In this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measu…
In this paper, we obtain the sharp uniqueness for an inverse $x$-source problem for a one-dimensional time-fractional diffusion equation with a zeroth-order term by the minimum possible lateral Cauchy data. The key ingredient is the unique continuation which holds for weak solutions.
Direct and inverse scattering for an isotropic medium with a second-order boundary condition
研究针对具有二级Robin边界条件的各向同性散射体，提出直接采样方法解决反向形状问题，并探讨传输特征值问题。
Uniqueness in determining multidimensional domains with unknown initial data
本文研究了初始数据未知的线性抛物型系统的几何反问题，通过边界或内部观测结合半群理论等工具，证明了多维子区域判定的唯一性并推导出初始数据信息。
The manuscript is concerned with uniqueness and stability for inverse source problem of determining spatially varying factor $f(x)$ of a source term given by $R(t)f(x)$ with suitable given $R(t)$ in the right hand side of the Schrödinger equation with time independent coefficients. In order to establish these results we provide a simple proof of a logarithmic conditional stability of the Cauchy problem for the Schrödinger equation with time-independent coefficients and the zero Dirichlet boundary conditions on the whole boundary and a proof of uniqueness of solution to the Cauchy problem for t…
Identification of an inclusion from local Cauchy data for time-harmonic elastic waves
针对时谐弹性波反问题，利用局部柯西数据，在无频谱假设下建立对数型稳定性估计，并给出小频率下局部Dirichlet-to-Neumann映射的稳定性结果。
We prove weighted $L^2$ estimates for the Klein-Gordon equation perturbed with singular potentials such as the inverse-square potential. We then deduce the well-posedness of the Cauchy problem for this equation with small perturbations, and go on to discuss local smoothing and Strichartz estimates which improve previously known ones.
We study in what sense one can determine the function $k=k(x)$ in the scalar hyperbolic conservation law $u_t+(k(x)f(u))_x=0$ by observing the solution $u(t,\dott)$ of the Cauchy problem with initial data $u|_{t=0}=u_o$.
Simultaneously recover two constant coefficients and a polygon with a single pair of Cauchy data for the Helmholtz equation
本文提出了一种基于一波因子分解方法，利用单对Cauchy数据同时恢复亥姆霍兹方程中两个常数系数和多边形障碍物的位置与形状。
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