Xinchi Huang | The University of Tokyo (original) (raw)

Papers by Xinchi Huang

Research paper thumbnail of Determination of source or initial values for acoustic equations with a time-fractional attenuation

Analysis and Applications

Research paper thumbnail of Initial-boundary value problems for coupled systems of time-fractional diffusion equations

This article deals with the initial-boundary value problem for a moderately coupled system of tim... more This article deals with the initial-boundary value problem for a moderately coupled system of time-fractional equations. Defining the mild solution, we establish fundamental unique existence, limited smoothing property and long-time asymptotic behavior of the solution, which mostly inherit those of a single equation. As an application of the coupling effect, we also obtain the uniqueness for an inverse problem on determining all the fractional orders by the single point observation of a single component of the solution.

Research paper thumbnail of Long-time asymptotic estimate and a related inverse source problem for time-fractional wave equations

Lying between traditional parabolic and hyperbolic equations, time-fractional wave equations of o... more Lying between traditional parabolic and hyperbolic equations, time-fractional wave equations of order α ∈ (1, 2) in time inherit both decaying and oscillating properties. In this article, we establish a long-time asymptotic estimate for homogeneous time-fractional wave equations, which readily implies the strict positivity/negativity of the solution for t ≫ 1 under some sign conditions on initial values. As a direct application, we prove the uniqueness for a related inverse source problem on determining the temporal component.

Research paper thumbnail of Carleman estimates for a magnetohydrodynamics system and application to inverse source problems

Mathematical Control & Related Fields

In this article, we consider a linearized magnetohydrodynamics system for incompressible flow in ... more In this article, we consider a linearized magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. We first prove two kinds of Carleman estimates. This is done by combining the Carleman estimates for the parabolic and the elliptic equations. Then we apply the Carleman estimates to prove Hölder type stability results for some inverse source problems.

Research paper thumbnail of Initial-boundary value problems for multi-term time-fractional diffusion equations with x-dependent coefficients

In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fracti... more In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fractional diffusion equation with x-dependent coefficients. By means of the Mittag-Leffler functions and the eigenfunction expansion, we reduce the IBVP to an equivalent integral equation to show the unique existence and the analyticity of the solution for the equation. Especially, in the case where all the coefficients of the time-fractional derivatives are non-negative, by the Laplace and inversion Laplace transforms, it turns out that the decay rate of the solution for long time is dominated by the lowest order of the time-fractional derivatives. Finally, as an application of the analyticity of the solution, the uniqueness of an inverse problem in determining the fractional orders in the multi-term time-fractional diffusion equations from one interior point observation is established.

Research paper thumbnail of Stability for inverse source problems by Carleman estimates

In this article, we provide a modified argument for proving conditional stability for inverse pro... more In this article, we provide a modified argument for proving conditional stability for inverse problems of determining spatially varying functions in evolution equations by Carleman estimates. Our method needs not any cut-off procedures and can simplify the existing proofs. We establish the conditional stability for inverse source problems for a hyperbolic equation and a parabolic equation, and our method is widely applicable to various evolution equations.

Research paper thumbnail of Carleman estimate for the Schrödinger equation and application to magnetic inverse problems

We prove that the stationary magnetic potential vector and the electrostatic potential entering t... more We prove that the stationary magnetic potential vector and the electrostatic potential entering the dynamic magnetic Schr\"odinger equation can be Lipschitz stably retrieved through finitely many local boundary measurements of the solution. The proof is by means of a specific global Carleman estimate for the Schr\"odinger equation, established in the first part of the paper.

Research paper thumbnail of Well-posedness and asymptotic estimate for a diffusion equation with time-fractional derivative

In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional dif... more In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional diffusion equation in a bounded domain subject to the homogeneous Dirichlet boundary condition. Firstly, the unique existence and regularity estimates of solution to the initial-boundary value problem are considered. Then combined with some important properties, including a maximum principle for a time-fractional ordinary equation and a coercivity inequality for fractional derivatives, the energy method shows that the decay in time of the solution is dominated by the term t^-α as t→∞.

Research paper thumbnail of Carleman estimates for the time-fractional advection-diffusion equations and applications

In this article, we prove Carleman estimates for the generalized time-fractional advection-diffus... more In this article, we prove Carleman estimates for the generalized time-fractional advection-diffusion equations by considering the fractional derivative as perturbation for the first order time-derivative. As a direct application of the Carleman estimates, we show a conditional stability of a lateral Cauchy problem for the time-fractional advection-diffusion equation, and we also investigate the stability of an inverse source problem.

Research paper thumbnail of A modified model on a traffic network and signal optimization

Research paper thumbnail of Inverse coefficients problem for a magnetohydrodynamics system

arXiv: Analysis of PDEs, 2018

In this article, we consider a magnetohydrodynamics system for incompressible flow in a three-dim... more In this article, we consider a magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. Firstly, we give the stability results for our inverse coefficients problem. Secondly, we establish and prove two Carleman estimates for both direct problem and inverse problem. Finally, we complete the proof of stability result in terms of the above Carleman estimates.

Research paper thumbnail of Determination of source and initial values for acoustic equations with a time-fractional attenuation

We consider the inverse problem of determining the initial states or the source term of a hyperbo... more We consider the inverse problem of determining the initial states or the source term of a hyperbolic equation damped by some non-local time-fractional derivative. This framework is relevant to medical imaging such as thermoacoustic or photoacoustic tomography. We prove a stability estimate for each of these two problems, with the aid of a Carleman estimate specifically designed for the governing equation.

Research paper thumbnail of Initial-boundary value problems for multi-term time-fractional diffusion equations with x-dependent coefficients

arXiv: Analysis of PDEs, 2018

In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fracti... more In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fractional diffusion equation with x-dependent coefficients. By means of the Mittag-Leffler functions and the eigenfunction expansion, we reduce the IBVP to an equivalent integral equation to show the unique existence and the analyticity of the solution for the equation. Especially, in the case where all the coefficients of the time-fractional derivatives are non-negative, by the Laplace and inversion Laplace transforms, it turns out that the decay rate of the solution for long time is dominated by the lowest order of the time-fractional derivatives. Finally, as an application of the analyticity of the solution, the uniqueness of an inverse problem in determining the fractional orders in the multi-term time-fractional diffusion equations from one interior point observation is established.

Research paper thumbnail of Carleman Estimate for a General Second-Order Hyperbolic Equation

In this article, we consider a general second-order hyperbolic equation. We first establish a mod... more In this article, we consider a general second-order hyperbolic equation. We first establish a modified Carleman estimate for this equation by adding some functions of adjustment. Then general conditions imposed on the principal parts, mixed with the weight function and the functions of adjustment are derived. Finally, we give the realizations of the weight functions by choosing suitable adjustments such that the above general conditions are satisfied in some specific cases.

Research paper thumbnail of Inverse coefficient problem for a magnetohydrodynamics system by Carleman estimates

Applicable Analysis, 2019

In this article, we consider a magnetohydrodynamics system for incompressible flow in a three-dim... more In this article, we consider a magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. Firstly, we state the stability results for our inverse coefficient problem. Secondly, we establish and prove two Carleman type inequalities both for the solutions and for the unknown coefficients. Finally, we complete the proofs of the stability results in terms of the above Carleman estimates.

Research paper thumbnail of Well-posedness and asymptotic estimate for a diffusion equation with time-fractional derivative

In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional dif... more In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional diffusion equation in a bounded domain subject to the homogeneous Dirichlet boundary condition. Firstly, the unique existence and regularity estimates of solution to the initial-boundary value problem are considered. Then combined with some important properties, including a maximum principle for a time-fractional ordinary equation and a coercivity inequality for fractional derivatives, the energy method shows that the decay in time of the solution is dominated by the term t−α as t → ∞.

Research paper thumbnail of Stability for inverse source problems by Carleman estimates

In this article, we provide a modified argument for proving conditional stability for inverse pro... more In this article, we provide a modified argument for proving conditional stability for inverse problems of determining spatially varying functions in evolution equations by Carleman estimates. Our method needs not any cut-off procedures and can simplify the existing proofs. We establish the conditional stability for inverse source problems for a hyperbolic equation and a parabolic equation, and our method is widely applicable to various evolution equations.

Research paper thumbnail of A stability result for the determination of order in time-fractional diffusion equations

This paper deals with an inverse problem of the determination of the fractional order in time-fra... more This paper deals with an inverse problem of the determination of the fractional order in time-fractional diffusion equations from one interior point observation. We give a representation of the solution via the Mittag-Leffler function and eigenfunction expansion, from which the Lipschitz stability of the fractional order with respect to the measured data at the interior point is established.

Research paper thumbnail of Inverse problems for a half-order time-fractional diffusion equation in arbitrary dimension by Carleman estimates

We consider a half-order time-fractional diffusion equation in an arbitrary dimension and investi... more We consider a half-order time-fractional diffusion equation in an arbitrary dimension and investigate inverse problems of determining the source term or the diffusion coefficient from spatial data at an arbitrarily fixed time under some additional assumptions. We establish the stability estimate of Lipschitz type in the inverse problems and the proofs are based on the Bukhgeim-Klibanov method by using Carleman estimates.

Research paper thumbnail of Conditional stability for an inverse coefficient problem of a weakly coupled time-fractional diffusion system with half order by Carleman estimate

Under a priori boundedness conditions of solutions and coefficients, we prove a Hölder stability ... more Under a priori boundedness conditions of solutions and coefficients, we prove a Hölder stability estimate for an inverse problem of determining two spatially varying zeroth order non-diagonal elements of a coefficient matrix in a one-dimensional fractional diffusion system of half order in time. The proof relies on the conversion of the fractional diffusion system to a system of order 4 in the space variable and the Carleman estimate.

Research paper thumbnail of Determination of source or initial values for acoustic equations with a time-fractional attenuation

Analysis and Applications

Research paper thumbnail of Initial-boundary value problems for coupled systems of time-fractional diffusion equations

This article deals with the initial-boundary value problem for a moderately coupled system of tim... more This article deals with the initial-boundary value problem for a moderately coupled system of time-fractional equations. Defining the mild solution, we establish fundamental unique existence, limited smoothing property and long-time asymptotic behavior of the solution, which mostly inherit those of a single equation. As an application of the coupling effect, we also obtain the uniqueness for an inverse problem on determining all the fractional orders by the single point observation of a single component of the solution.

Research paper thumbnail of Long-time asymptotic estimate and a related inverse source problem for time-fractional wave equations

Lying between traditional parabolic and hyperbolic equations, time-fractional wave equations of o... more Lying between traditional parabolic and hyperbolic equations, time-fractional wave equations of order α ∈ (1, 2) in time inherit both decaying and oscillating properties. In this article, we establish a long-time asymptotic estimate for homogeneous time-fractional wave equations, which readily implies the strict positivity/negativity of the solution for t ≫ 1 under some sign conditions on initial values. As a direct application, we prove the uniqueness for a related inverse source problem on determining the temporal component.

Research paper thumbnail of Carleman estimates for a magnetohydrodynamics system and application to inverse source problems

Mathematical Control & Related Fields

In this article, we consider a linearized magnetohydrodynamics system for incompressible flow in ... more In this article, we consider a linearized magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. We first prove two kinds of Carleman estimates. This is done by combining the Carleman estimates for the parabolic and the elliptic equations. Then we apply the Carleman estimates to prove Hölder type stability results for some inverse source problems.

Research paper thumbnail of Initial-boundary value problems for multi-term time-fractional diffusion equations with x-dependent coefficients

In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fracti... more In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fractional diffusion equation with x-dependent coefficients. By means of the Mittag-Leffler functions and the eigenfunction expansion, we reduce the IBVP to an equivalent integral equation to show the unique existence and the analyticity of the solution for the equation. Especially, in the case where all the coefficients of the time-fractional derivatives are non-negative, by the Laplace and inversion Laplace transforms, it turns out that the decay rate of the solution for long time is dominated by the lowest order of the time-fractional derivatives. Finally, as an application of the analyticity of the solution, the uniqueness of an inverse problem in determining the fractional orders in the multi-term time-fractional diffusion equations from one interior point observation is established.

Research paper thumbnail of Stability for inverse source problems by Carleman estimates

In this article, we provide a modified argument for proving conditional stability for inverse pro... more In this article, we provide a modified argument for proving conditional stability for inverse problems of determining spatially varying functions in evolution equations by Carleman estimates. Our method needs not any cut-off procedures and can simplify the existing proofs. We establish the conditional stability for inverse source problems for a hyperbolic equation and a parabolic equation, and our method is widely applicable to various evolution equations.

Research paper thumbnail of Carleman estimate for the Schrödinger equation and application to magnetic inverse problems

We prove that the stationary magnetic potential vector and the electrostatic potential entering t... more We prove that the stationary magnetic potential vector and the electrostatic potential entering the dynamic magnetic Schr\"odinger equation can be Lipschitz stably retrieved through finitely many local boundary measurements of the solution. The proof is by means of a specific global Carleman estimate for the Schr\"odinger equation, established in the first part of the paper.

Research paper thumbnail of Well-posedness and asymptotic estimate for a diffusion equation with time-fractional derivative

In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional dif... more In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional diffusion equation in a bounded domain subject to the homogeneous Dirichlet boundary condition. Firstly, the unique existence and regularity estimates of solution to the initial-boundary value problem are considered. Then combined with some important properties, including a maximum principle for a time-fractional ordinary equation and a coercivity inequality for fractional derivatives, the energy method shows that the decay in time of the solution is dominated by the term t^-α as t→∞.

Research paper thumbnail of Carleman estimates for the time-fractional advection-diffusion equations and applications

In this article, we prove Carleman estimates for the generalized time-fractional advection-diffus... more In this article, we prove Carleman estimates for the generalized time-fractional advection-diffusion equations by considering the fractional derivative as perturbation for the first order time-derivative. As a direct application of the Carleman estimates, we show a conditional stability of a lateral Cauchy problem for the time-fractional advection-diffusion equation, and we also investigate the stability of an inverse source problem.

Research paper thumbnail of A modified model on a traffic network and signal optimization

Research paper thumbnail of Inverse coefficients problem for a magnetohydrodynamics system

arXiv: Analysis of PDEs, 2018

In this article, we consider a magnetohydrodynamics system for incompressible flow in a three-dim... more In this article, we consider a magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. Firstly, we give the stability results for our inverse coefficients problem. Secondly, we establish and prove two Carleman estimates for both direct problem and inverse problem. Finally, we complete the proof of stability result in terms of the above Carleman estimates.

Research paper thumbnail of Determination of source and initial values for acoustic equations with a time-fractional attenuation

We consider the inverse problem of determining the initial states or the source term of a hyperbo... more We consider the inverse problem of determining the initial states or the source term of a hyperbolic equation damped by some non-local time-fractional derivative. This framework is relevant to medical imaging such as thermoacoustic or photoacoustic tomography. We prove a stability estimate for each of these two problems, with the aid of a Carleman estimate specifically designed for the governing equation.

Research paper thumbnail of Initial-boundary value problems for multi-term time-fractional diffusion equations with x-dependent coefficients

arXiv: Analysis of PDEs, 2018

In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fracti... more In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fractional diffusion equation with x-dependent coefficients. By means of the Mittag-Leffler functions and the eigenfunction expansion, we reduce the IBVP to an equivalent integral equation to show the unique existence and the analyticity of the solution for the equation. Especially, in the case where all the coefficients of the time-fractional derivatives are non-negative, by the Laplace and inversion Laplace transforms, it turns out that the decay rate of the solution for long time is dominated by the lowest order of the time-fractional derivatives. Finally, as an application of the analyticity of the solution, the uniqueness of an inverse problem in determining the fractional orders in the multi-term time-fractional diffusion equations from one interior point observation is established.

Research paper thumbnail of Carleman Estimate for a General Second-Order Hyperbolic Equation

In this article, we consider a general second-order hyperbolic equation. We first establish a mod... more In this article, we consider a general second-order hyperbolic equation. We first establish a modified Carleman estimate for this equation by adding some functions of adjustment. Then general conditions imposed on the principal parts, mixed with the weight function and the functions of adjustment are derived. Finally, we give the realizations of the weight functions by choosing suitable adjustments such that the above general conditions are satisfied in some specific cases.

Research paper thumbnail of Inverse coefficient problem for a magnetohydrodynamics system by Carleman estimates

Applicable Analysis, 2019

In this article, we consider a magnetohydrodynamics system for incompressible flow in a three-dim... more In this article, we consider a magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. Firstly, we state the stability results for our inverse coefficient problem. Secondly, we establish and prove two Carleman type inequalities both for the solutions and for the unknown coefficients. Finally, we complete the proofs of the stability results in terms of the above Carleman estimates.

Research paper thumbnail of Well-posedness and asymptotic estimate for a diffusion equation with time-fractional derivative

In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional dif... more In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional diffusion equation in a bounded domain subject to the homogeneous Dirichlet boundary condition. Firstly, the unique existence and regularity estimates of solution to the initial-boundary value problem are considered. Then combined with some important properties, including a maximum principle for a time-fractional ordinary equation and a coercivity inequality for fractional derivatives, the energy method shows that the decay in time of the solution is dominated by the term t−α as t → ∞.

Research paper thumbnail of Stability for inverse source problems by Carleman estimates

In this article, we provide a modified argument for proving conditional stability for inverse pro... more In this article, we provide a modified argument for proving conditional stability for inverse problems of determining spatially varying functions in evolution equations by Carleman estimates. Our method needs not any cut-off procedures and can simplify the existing proofs. We establish the conditional stability for inverse source problems for a hyperbolic equation and a parabolic equation, and our method is widely applicable to various evolution equations.

Research paper thumbnail of A stability result for the determination of order in time-fractional diffusion equations

This paper deals with an inverse problem of the determination of the fractional order in time-fra... more This paper deals with an inverse problem of the determination of the fractional order in time-fractional diffusion equations from one interior point observation. We give a representation of the solution via the Mittag-Leffler function and eigenfunction expansion, from which the Lipschitz stability of the fractional order with respect to the measured data at the interior point is established.

Research paper thumbnail of Inverse problems for a half-order time-fractional diffusion equation in arbitrary dimension by Carleman estimates

We consider a half-order time-fractional diffusion equation in an arbitrary dimension and investi... more We consider a half-order time-fractional diffusion equation in an arbitrary dimension and investigate inverse problems of determining the source term or the diffusion coefficient from spatial data at an arbitrarily fixed time under some additional assumptions. We establish the stability estimate of Lipschitz type in the inverse problems and the proofs are based on the Bukhgeim-Klibanov method by using Carleman estimates.

Research paper thumbnail of Conditional stability for an inverse coefficient problem of a weakly coupled time-fractional diffusion system with half order by Carleman estimate

Under a priori boundedness conditions of solutions and coefficients, we prove a Hölder stability ... more Under a priori boundedness conditions of solutions and coefficients, we prove a Hölder stability estimate for an inverse problem of determining two spatially varying zeroth order non-diagonal elements of a coefficient matrix in a one-dimensional fractional diffusion system of half order in time. The proof relies on the conversion of the fractional diffusion system to a system of order 4 in the space variable and the Carleman estimate.