一类耦合Korteweg-de Vries方程组输运系数反演问题的Lipschitz稳定性

数学物理学报 ›› 2021, Vol. 41 ›› Issue (3): 740-761.

一类耦合Korteweg-de Vries方程组输运系数反演问题的Lipschitz稳定性

吴斌*(),陈群

^{南京信息工程大学数学与统计学院南京 210044}

收稿日期:2020-03-23 出版日期:2021-06-26 发布日期:2021-06-09
通讯作者: 吴斌 E-mail:binwu@nuist.edu.cn
基金资助:
国家自然科学基金(11601240)

Lipschitz Stability for a Transport Coefficient Inverse Problem of a Linearly Coupled Korteweg-de Vries System

Bin Wu*(),Qun Chen

^{School of Mathematics and Statistics, College of Science, Nanjing University of Information Science and Technology, Nanjing 210044}

Received:2020-03-23 Online:2021-06-26 Published:2021-06-09
Contact: Bin Wu E-mail:binwu@nuist.edu.cn
Supported by:
the NSFC(11601240)

摘要/Abstract

摘要：

该文研究了一类耦合Korteweg-de Vries（KdV）方程组中两个仅依赖空间变量的输运系数的反演问题.为证明在单个内部测量数据下反问题的稳定性，该文先证明了该耦合KdV方程组的一个仅含单个局部积分项的卡勒曼估计，然后进一步得到了在先验信息下的反问题的Lipschitz稳定性.

关键词: 卡勒曼估计, 耦合KdV方程组, Lipschitz稳定性, 系数反问题

Abstract:

This paper concerns an inverse problem of determining two spatially varying transport coefficients simultaneously in a linearly coupled Korteweg-de Vries (KdV) system with the first order terms. To obtain the stability result for the inverse problem with only one internal measurement data, we first prove a Carleman estimate including only one local integral for this coupled KdV system. Based on this Carleman estimate, we then obtain Lipschitz stability for the inverse problem under some priori information.

Key words: Carleman estimate, Coupled KdV system, Lipschitz stability, Coefficient inverse problem

中图分类号:

O175.28

吴斌,陈群. 一类耦合Korteweg-de Vries方程组输运系数反演问题的Lipschitz稳定性[J]. 数学物理学报, 2021, 41(3): 740-761.

Bin Wu,Qun Chen. Lipschitz Stability for a Transport Coefficient Inverse Problem of a Linearly Coupled Korteweg-de Vries System[J]. Acta mathematica scientia,Series A, 2021, 41(3): 740-761.

参考文献 35

1	Baudouin L , Cerpa E , Crépeau E , Mercado A . On the determination of the principal coefficient from boundary measurements in a KdV equation. J Inverse Ill-Posed Prob, 2014, 22, 819- 845
2	Bellassoued M , Yamamoto M . Lipschitz stability in determining density and two Lamé coefficients. J Math Anal Appl, 2017, 329, 1240- 1259
3	Bellassoued M , Yamamoto M . Carleman estimates and an inverse heat source problem for the thermoelasticity system. Inverse Probl, 2011, 27, 015006 doi: 10.1088/0266-5611/27/1/015006
4	Bellassoued M , Yamamoto M . Carleman estimate and inverse source problem for Biot's equations describing wave propagation in porous media. Inverse Probl, 2013, 29, 115002 doi: 10.1088/0266-5611/29/11/115002
5	Bukhgeim A L , Klibanov M V . Uniqueness in the large of a class of multidimensional inverse problems. Soviet Math Dokl, 1981, 17, 244- 247
6	Caicedo M A , Capistrano-Filho R , Zhang B Y . Neumann boundary controllability of the Korteweg-de Vries equation on a bounded domain. SIAM J Control Optim, 2017, 55, 3503- 3532 doi: 10.1137/15M103755X
7	Capistrano-Filho R , Pazoto A F , Rosier L . Internal controllability of the Korteweg-de Vries equation on a bounded domain. ESAIM Control Optim Calc Var, 2015, 21, 1076- 1107 doi: 10.1051/cocv/2014059
8	Carréno N , Guerrero S . Uniform null controllability of a linear KdV equation using two controls. J Math Anal Appl, 2018, 457, 922- 943 doi: 10.1016/j.jmaa.2017.08.039
9	Cerpa E , Rivas I , Zhang B Y . Boundary controllability of the Korteweg-de Vries equation on a bounded domain. SIAM J Control Optim, 2013, 51, 2976- 3010 doi: 10.1137/120891721
10	Chen M . Lipschitz stability in an inverse problem for the Korteweg-de Vries equation on a finite domain. Boundary Value Problems, 2007, 2017, 48 doi: 10.1186/s13661-017-0779-8
11	Coron J M , Crépeau E . Exact boundary controllability of a nonlinear KdV equation with a critical length. J Eur Math Soc, 2004, 31, 367- 398
12	Dias F , Il'ichev A . Interfacial waves with free-surface boundary conditions: an approach via a model equation. Physica D, 2001, 150, 278- 300 doi: 10.1016/S0167-2789(01)00149-X
13	Fochesato C , Dias F , Grimshaw R . Generalized solitary waves and fronts in coupled Korteweg-de Vries systems. Physica D, 2005, 210, 96- 117 doi: 10.1016/j.physd.2005.07.010
14	Gao P . Carleman estimate and unique continuation property for the linear stochastic Korteweg-de Vries equation. Bull Aust Math Soc, 2014, 90, 283- 294 doi: 10.1017/S0004972714000276
15	Glass O , Guerrero S . Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit. Asymptot Anal, 2008, 60, 61- 100 doi: 10.3233/ASY-2008-0900
16	Grimshaw R , Pelinovsky E , Talipova T . The modified Korteweg-de Vries equation in the theory of the large-amplitude internal waves. Nonlinear Process Geophys, 1997, 4, 237- 250 doi: 10.5194/npg-4-237-1997
17	Imanuvilov O Y . Controllability of parabolic equations. Sbornik Math, 1995, 186, 879- 900 doi: 10.1070/SM1995v186n06ABEH000047
18	Imanuvilov O Y , Yamamoto M . Carleman estimates for the non-stationary Lemé system and application to an inverse problem. ESAIM Control Optim Calc Var, 2005, 11, 1- 56 doi: 10.1051/cocv:2004030
19	Imanuvilov O Y , Yamamoto M . Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Probl, 2001, 17, 717- 728 doi: 10.1088/0266-5611/17/4/310
20	Isakov V . Inverse Problems for Partial Differential Equations. Berlin: Springer-Verlag, 1998
21	Isakov V , Kim N . Carleman estimates with second large parameter and applications to elasticity with residual stress. Applicationes Mathematicae, 2008, 35, 447- 465 doi: 10.4064/am35-4-4
22	Klibanov M V , Timonov A . Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. Utrecht: VSP, 2004
23	Klibanov M V . Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J Inverse Ill-Posed Prob, 2013, 21, 477- 560
24	Klibanov M V , Kolesov A E . Convexification of a 3-D coefficient inverse scattering problem. Computers and Mathematics with Applications, 2019, 77, 1681- 1702 doi: 10.1016/j.camwa.2018.03.016
25	Klibanov M V , Kolesov A E , Nguyen L , Sullivan A . Globally strictly convex cost functional for a 1D inverse medium scattering problem with experimental data. SIAM J Applied Mathematics, 2017, 77, 1733- 1755 doi: 10.1137/17M1122487
26	Kumarasamy S , Hasanov A . An inverse problem for the KdV equation with Neumann boundary measured data. J Inverse Ill-Posed Prob, 2018, 26, 133- 151 doi: 10.1515/jiip-2017-0038
27	Romanov V G , Yamamoto M . Recovering a Lamé kernel in viscoelastic equation by a single boundary measurement. Appl Anal, 2010, 89, 377- 390 doi: 10.1080/00036810903518975
28	Rosier L . Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line. SIAM J Control Optim, 2000, 39, 331- 351 doi: 10.1137/S0363012999353229
29	Rousseau J , Lebeau G . On Carleman estimates for elliptic and parabolic operators Applications to unique continuation and control of parabolic equations. ESAIM Control Optim Calc Var, 2012, 18, 712- 747 doi: 10.1051/cocv/2011168
30	Saut J , Scheurer B . Unique continuation for some evolution equations. J Differential Equations, 1987, 66, 118- 139 doi: 10.1016/0022-0396(87)90043-X
31	Tang S , Zhang X . Null controllability for forward and backward stochastic parabolic equations. SIAM J Control Optim, 2009, 48, 2191- 2216 doi: 10.1137/050641508
32	Wu B , Gao Y , Wang Z , Chen Q . Unique continuation for a reaction-diffusion system with cross diffusion. J Inverse Ill-Posed Prob, 2019, 27, 511- 525 doi: 10.1515/jiip-2017-0094
33	Wu B , Liu J . Conditional stability and uniqueness for determining two coefficients in a hyperbolic-parabolic system. Inverse Probl, 2011, 27, 075013 doi: 10.1088/0266-5611/27/7/075013
34	Wu B , Yu J , Wang Z . A coefficient identification problem for a mathematical model related to ductal carcinoma in situ. Stud Appl Math, 2019, 143, 356- 372 doi: 10.1111/sapm.12281
35	Yamamoto M . Carleman estimates for parabolic equations and applications. Inverse Probl, 2009, 25, 123013 doi: 10.1088/0266-5611/25/12/123013