Stability for Some Inverse Problems for Transport Equations

Gölgeleyen, Fikret; Yamamoto, Masahiro

doi:10.1137/15m1038128

Cited by 39 publications

(41 citation statements)

References 18 publications

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“…Carleman estimates for transport equations are proved in Gaitan and Ouzzane [4], Gölgeleyen and Yamamoto [5], Klibanov and Pamyatnykh [6], Machida and Yamamoto [7] to be applied to inverse problems of determining spatially varying coefficients, where coefficients of the first-order terms in x are assumed not to depend on t. In order to improve results for inverse problems by the application of Carleman estimates, we need a better choice of the weight function in the Carleman estimate. The works [4] and [6] use one weight function which is very conventional for a second-order hyperbolic equation but seems less useful to derive analogous results for a time-dependent function H(t).…”

Section: Main References and Outline Of The Papermentioning

confidence: 99%

Observability Inequalities for Transport Equations through Carleman Estimates

Cannarsa

Floridia

Yamamoto

2019

Springer INdAM Series

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We consider the transport equation ∂ t u(x,t) + H(t) · ∇u(x,t) = 0 in Ω × (0, T ), where T > 0 and Ω ⊂ R d is a bounded domain with smooth boundary ∂ Ω . First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. Then, under a further condition which guarantees that the orbits of H intersect ∂ Ω , we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.

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Section: Main References and Outline Of The Papermentioning

confidence: 99%

Observability Inequalities for Transport Equations through Carleman Estimates

Cannarsa

Floridia

Yamamoto

2019

Springer INdAM Series

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“…As for similar inverse problems for transport equations with k ≡ 0 in (1.1), we refer to Gaitan-Ouzzane [17]. Moreover one can consult Cannarsa, Floridia, Gölgeleyen and Yamamoto [12], Cannarsa, Floridia and Yamamoto [13] and Gölgeleyen and Yamamoto [18], where a linear weight function is used.…”

Section: Introductionmentioning

confidence: 99%

Global Lipschitz stability in determining coefficients of the radiative transport equation

Machida

Yamamoto

2014

Inverse Problems

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In this article, for the radiative transport equation, we study inverse problems of determining a time independent scattering coefficient or total attenuation by boundary data on the complementary sub-boundary after making one time input of a pair of a positive initial value and boundary data on a suitable sub-boundary. The main results are Lipschitz stability estimates. We can also prove the reverse inequality, which means that our estimates for the inverse problems are the best possible. The proof is based on a Carleman estimate with a linear weight function.

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“…As for similar inverse problems for the Navier-Stokes equations, see Bellassoued, Imanuvilov and Yamamoto [7], Choulli, Imanuvilov, Puel and Yamamoto [15], Fan, Di Cristo, Jiang and Nakamura [20], Fan, Jiang and Nakamura [21]. Gaitan and Ouzzane [23], and Gölgeleyen and Yamamoto [24] [4], Klibanov and Timonov [46], Lavrent'ev, Romanov and Shishat·skiȋ [49].…”

Section: Partial Observability Inequalitymentioning

confidence: 99%

Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case

2017

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We consider an anisotropic hyperbolic equation with memory term:for x ∈ Ω and t ∈ (0, T ) or ∈ (−T, T ), which is a model equation for viscoelasticity.First we establish a Carleman estimate for this equation with overdetermining boundary data on a suitable lateral subboundary Γ × (−T, T ). Second we apply the Carleman estimate to establish a both-sided estimate of u(·, 0) H 3 (Ω) by ∂ ν u| Γ×(0,T ) under the assumption that ∂ t u(·, 0) = 0 and T > 0 is sufficiently large, Γ ⊂ ∂Ω satisfies some geometric condition. Such an estimate is a kind of observability inequality and related to the exact controllability. Finally we apply the Carleman estimate to an inverse source problem of determining a spatial varying factor in F (x, t) and we establish a both-sided Lipschitz stability estimate.

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Stability for Some Inverse Problems for Transport Equations

Abstract: Abstract. In this article, we consider inverse problems of determining a source term and a coefficient of a first-order partial differential equation and prove conditional stability estimates with minimum boundary observation data and relaxed condition on the principal part.

Cited by 39 publications

References 18 publications

Observability Inequalities for Transport Equations through Carleman Estimates

Observability Inequalities for Transport Equations through Carleman Estimates

Global Lipschitz stability in determining coefficients of the radiative transport equation

Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case

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