Inverse viscosity problem for the Navier–Stokes equation

Fan, Jishan; Cristo, Michele Di; Jiang, Yu; Nakamura, Gen

doi:10.1016/j.jmaa.2009.12.012

Cited by 38 publications

(39 citation statements)

References 16 publications

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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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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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“…where in the last inequality we have used Lemma 2.3 again. Finally, substituting (16), (18) into (15) and taking N , C 0 D C e 2T e 3 T 2 and s N s, where N s can be taken of the form N s D s 0 C C.1 C T 2 /e 4T e 3 T 2 , we get the desired estimate (7). This ends the proof of Theorem 2.1.…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…Puel and Yamamoto [13] gave the first Lipschitz stability result for a multidimensional inverse problem for a hyperbolic equation. Later, this method was extended for many varieties of equations and system, such as [14,15], [16][17][18][19][20][21]. Especially, [14] concerned an inverse problem for a Schrödinger equation and where the method in the proof of the Lipschitz stability is the key ingredient to prove our stability result.…”

Section: Introductionmentioning

confidence: 99%

Carleman estimate for a strongly damped wave equation and applications to an inverse problem

2012

Math Methods in App Sciences

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In this paper, we establish a Carleman estimate for a strongly damped wave equation in order to solve a coefficient inverse problems of retrieving a stationary potential from a single time-dependent Neumann boundary measurement on a suitable part of the boundary. This coefficient inverse problem is for a strongly damped wave equation. We prove the uniqueness and the local stability results for this inverse problem. The proof of the results relies on Carleman estimate and a certain energy estimates for hyperbolic equation with strongly damped term. Moreover, this method could be used for a similar inverse problem for an integro-differential equation with hyperbolic memory kernel., provided that u 0 , u 1 are given.Many physical phenomena are properly described by the strongly damped wave equation as (1). Such as equations of type (1) can be considered as a class of linear evolution equations governing the motion of a viscoelastic solid (for example, a bar if the space dimension N D 1 and a plate if N D 2) composed of the material of the rate type; see [1][2][3][4]. They can also be seen as field equations governing the longitudinal motion of a viscoelastic bar obeying the simply Voigt model [5]. Mathematical studies on this direct problem have been extensively developed, for example, see [1,2,[6][7][8][9] and the references therein, where the global existence and other properties of solutions are obtained. However, very little work is concerned with the inverse problem for this strongly damped wave equation.The key ingredient we follow here to determine the coefficient relies on the global Carleman estimates. For the first time, the method of Carleman estimates was introduced in the field of inverse problems in the work of Bukhgeim and Klibanov [10], also for some followup publications [11,12] of these authors. A Carleman estimate is an efficient tool not only for the unique continuation, but also for

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“…Those global Carleman estimate is convenient for proving the Lipschitz stability for an inverse source problem (e.g., [3]) and the exact null controllability ( [6]), but is not suitable for proving the unique continuation, and such a weight function does not admit Carleman estimates for the Navier-Stokes equations coupled with first-order equation or hyperbolic equation such as a conservation law. As for Carleman estimates for the Navier-Stokes equations, see also Fan, Di Cristo, Jiang and Nakamura [4] and Fan, Jiang and Nakamura [5] with extra data in a neighborhood of the whole boundary, which is too much by considering the parabolicity of the equations. We prove Lemma 1.…”

Section: Introductionmentioning

confidence: 99%

Carleman estimate for the Navier–Stokes equations and an application to a lateral Cauchy problem

2016

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Inverse viscosity problem for the Navier–Stokes equation

Abstract: We consider an inverse problem of determining a viscosity coefficient in the Navier-Stokes equation by observation data in a neighborhood of the boundary. We prove the Lipschitz stability by the Carleman estimates in Sobolev spaces of negative order.

Cited by 38 publications

References 16 publications

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

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

Carleman estimate for a strongly damped wave equation and applications to an inverse problem

Carleman estimate for the Navier–Stokes equations and an application to a lateral Cauchy problem

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