Stability estimate for a partial data inverse problem for the convection-diffusion equation

Soumen Senapati; Manmohan Vashisth

doi:10.3934/eect.2021060

Article Contents

2022, Volume 11, Issue 5: 1681-1699. Doi: 10.3934/eect.2021060

This issue Previous Article Controlled singular evolution equations and Pontryagin type maximum principle with applications Next Article Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

Stability estimate for a partial data inverse problem for the convection-diffusion equation

Soumen Senapati^1, and
Manmohan Vashisth^2, ,

1.
TIFR Centre for Applicable Mathematics, Bangalore 560065, India
2.
Department of Mathematics, Indian Institute of Technology, Jammu 181221, India

^*Corresponding author: Manmohan Vashisth
^*Corresponding author: Manmohan Vashisth

Received: April 2021

Revised: September 2021

Early access: December 2021

Published: October 2022

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Abstract

In this article, we study the stability in the inverse problem of determining the time-dependent convection term and density coefficient appearing in the convection-diffusion equation, from partial boundary measurements. For dimension $ n\ge 2 $, we show the convection term (modulo the gauge term) admits log-log stability, whereas log-log-log stability estimate is obtained for the density coefficient.

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Mathematics Subject Classification: 35R30, 35K20.

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[1]	S. A. Avdonin and M. I. Belishev, Dynamical inverse problem for the Schrödinger equation (BC-method), Proceedings of the St. Petersburg Mathematical Society, Amer. Math. Soc. Transl. Ser. Vol. X, 214 (2005), 1-14. doi: 10.1090/trans2/214/01.
[2]	S. A. Avdonin and T. I. Seidman, Identication of $q(x)$ in $u_t = \Delta u - qu$, from boundary observations, SIAM J. Control Optim., 33 (1995), 1247-1255. doi: 10.1137/S0363012993249729.
[3]	L. Baudouin and J.-P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 23 (2007), 1327-1328. doi: 10.1088/0266-5611/23/3/C01.
[4]	M. I. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1–R67. doi: 10.1088/0266-5611/23/5/R01.
[5]	M. Bellassoued and I. Ben Aïcha, Stable determination outside a cloaking region of two time-dependent coefficients in an hyperbolic equation from Dirichlet to Neumann map, J. Math. Anal. Appl., 449 (2017), 46-76. doi: 10.1016/j.jmaa.2016.11.082.
[6]	M. Bellassoued and O. Ben Fraj, Stably determining time-dependent convection-diffusion coefficients from a partial Dirichlet-to-Neumann map, Inverse Problems, 37 (2021), 045011, 35pp. doi: 10.1088/1361-6420/abe10d.
[7]	M. Bellassoued and D. Dos Santos Ferreira, Stable determination of coefficients in the dynamical anisotropic Schrödinger equation from the Dirichlet-to-Neumann map, Inverse Problems, 26 (2010), 125010, 30pp. doi: 10.1088/0266-5611/26/12/125010.
[8]	M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal., 85 (2006), 1219-1243. doi: 10.1080/00036810600787873.
[9]	M. Bellassoued, D. Jellali and M. Yamamoto, Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, J. Math. Anal. Appl., 343 (2008), 1036-1046. doi: 10.1016/j.jmaa.2008.01.098.
[10]	M. Bellassoued, Y. Kian and E. Soccorsi, An inverse stability result for non-compactly supported potentials by one arbitrary lateral Neumann observation, J. Differential Equations, 260 (2016), 7535-7562. doi: 10.1016/j.jde.2016.01.033.
[11]	M. Bellassoued, Y. Kian and E. Soccorsi, An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publ. Res. Inst. Math. Sci., 54 (2018), 679-728. doi: 10.4171/PRIMS/54-4-1.
[12]	M. Bellassoued and I. Rassas, Stability estimate for an inverse problem of the convection-diffusion equation, J. Inverse Ill-Posed Probl., 28 (2020), 71-92. doi: 10.1515/jiip-2018-0072.
[13]	I. Ben Aïcha, Stability estimate for a hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010, 21pp. doi: 10.1088/0266-5611/31/12/125010.
[14]	I. Ben Aïcha, Stability estimate for an inverse problem for the Schrödinger equation in a magnetic field with time-dependent coefficient, J. Math. Phys., 58 (2017), 071508, 21pp. doi: 10.1063/1.4995606.
[15]	A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large class of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269–272.
[16]	A. L. Bukhge${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$m and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations, 27 (2002), 653-668. doi: 10.1081/PDE-120002868.
[17]	P. Caro and Y. Kian, Determination of convection terms and quasi-linearities appearing in diffusion equations, preprint, arXiv: 1812.08495.
[18]	J. Cheng and M. Yamamoto, The global uniqueness for determining two convection coefficients from Dirichlet to Neumann map in two dimensions, Inverse Problems, 16 (2000), L25–L30. doi: 10.1088/0266-5611/16/3/101.
[19]	J. Cheng and M. Yamamoto, Identification of convection term in a parabolic equation with a single measurement, Nonlinear Anal., 50 (2002), 163-171. doi: 10.1016/S0362-546X(01)00742-8.
[20]	J. Cheng and M. Yamamoto, Determination of Two Convection Coefficients from Dirichlet to Neumann Map in the Two-Dimensional Case, SIAM J. Math. Anal., 35 (2004), 1371-1393. doi: 10.1137/S0036141003422497.
[21]	M. Choulli, An abstract inverse problem, J. Appl. Math. Stoc. Ana., 4 (1991), 117-128. doi: 10.1155/S1048953391000084.
[22]	M. Choulli, An abstract inverse problem and application, J. Math. Anal. Appl., 160 (1991), 190-202. doi: 10.1016/0022-247X(91)90299-F.
[23]	M. Choulli, Une Introduction Aux problèmes Inverses Elliptiques et Paraboliques, , Mathématiques & Applications, 65. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3.
[24]	M. Choulli and Y. Kian, Stability of the determination of a time-dependent coefficient in parabolic equations, Math. Control Relat. Fields, 3 (2013), 143-160. doi: 10.3934/mcrf.2013.3.143.
[25]	M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl., 114 (2018), 235-261. doi: 10.1016/j.matpur.2017.12.003.
[26]	M. Choulli, Y. Kian and E. Soccorsi, Stable determination of time-dependent scalar potential from boundary measurements in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558. doi: 10.1137/140986268.
[27]	Z.-C. Deng, J.-N. Yu and Y. Liu, Identifying the coefficient of first-order in parabolic equation from final measurement data, Math. Comput. Simulation, 77 (2008), 421-435. doi: 10.1016/j.matcom.2008.01.002.
[28]	D. Dos Santos Ferreira, C. E. Kenig, J. Sjöstrand and G. Uhlmann, Determining a magnetic Schrödinger operator from partial {C}auchy data, Comm. Math. Phys., 271 (2007), 467-488. doi: 10.1007/s00220-006-0151-9.
[29]	G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 022105, 18pp. doi: 10.1063/1.2841329.
[30]	P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 065006, 18pp. doi: 10.1088/0266-5611/29/6/065006.
[31]	G. Hu and Y. Kian, Determination of singular time-dependent coefficients for wave equations from full and partial data, Inverse Probl. Imaging, 12 (2018), 745-772. doi: 10.3934/ipi.2018032.
[32]	V. Isakov, Completeness of products of solutions and some inverse problems for PDE, J. Differential Equations, 92 (1991), 305-316. doi: 10.1016/0022-0396(91)90051-A.
[33]	V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12. doi: 10.1007/BF00392201.
[34]	V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127. Springer-Verlag, New York, 1998.
[35]	A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, FL, 2001.
[36]	Y. Kian, Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging, 8 (2014), 713-732. doi: 10.3934/ipi.2014.8.713.
[37]	Y. Kian, Stability in the determination of a time-dependent coefficient for wave equations from partial data, J. Math. Anal. Appl., 436 (2016), 408–428. doi: 10.1016/j.jmaa.2015.12.018.
[38]	Y. Kian, Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021–4046. doi: 10.1137/16M1076708.
[39]	Y. Kian, Unique determination of a time-dependent potential for wave equations from partial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 973–990. doi: 10.1016/j.anihpc.2016.07.003.
[40]	Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations, Int. Math. Res. Not. IMRN, 2019 (2019), 5087-5126. doi: 10.1093/imrn/rnx263.
[41]	Y. Kian, Q. Sang Phan and E. Soccorsi, A Carleman estimate for infinite cyclindrical quantum domains and the application to inverse problems, Inverse Problems, 30 (2014), 055016, 16 pp. doi: 10.1088/0266-5611/30/5/055016.
[42]	Y. Kian, Q. Sang Phan and E. Soccorsi, Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains, J. Math. Anal. Appl., 426 (2015), 194–210. doi: 10.1016/j.jmaa.2015.01.028.
[43]	Y. Kian and E. Soccorsi, Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, SIAM J. Math. Anal., 51 (2019), 627-647. doi: 10.1137/18M1197308.
[44]	Y. Kian and A. Tetlow, Hölder stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic schrödinger equation, Inverse Probl. Imaging, 14 (2020), 819-839. doi: 10.3934/ipi.2020038.
[45]	Y. Kian and M. Yamamoto, Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations, Inverse Problems, 35 (2019), 115006, 24pp doi: 10.1088/1361-6420/ab2d42.
[46]	V. P. Krishnan and M. Vashisth, An inverse problem for the relativistic Schrödinger equation with partial boundary data, Appl. Anal., 99 (2020), 1889-1909. doi: 10.1080/00036811.2018.1549321.
[47]	J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. 3. (French) Travaux et Recherches Mathématiques, Dunod, Paris, 1970.
[48]	R. K. Mishra and M. Vashisth, Determining the Time Dependent Matrix Potential in A Wave Equation From Partial Boundary Data, Appl. Anal., 100 (2021), 3492-3508. doi: 10.1080/00036811.2020.1721476.
[49]	N. S. Nadirašvili, A generalization of Hadamard's three circles theorem, (Russian) Vestnik Moskov. Univ. Ser. I Mat. Meh., 31 (1976), 39-42.
[50]	G. Nakamura and S. Sasayama, Inverse boundary value problem for the heat equation with discontinuous coefficients, J. Inverse Ill-Posed Probl., 21 (2013), 217-232. doi: 10.1515/jip-2012-0073.
[51]	V. Pohjola, A uniqueness result for an inverse problem of the steady state convection-diffusion equation, SIAM J. Math. Anal., 47 (2015), 2084-2103. doi: 10.1137/140970926.
[52]	Ra kesh and W. W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 87-96. doi: 10.1080/03605308808820539.
[53]	S. K. Sahoo and M. Vashisth, A partial data inverse problem for Convection-Diffusion equation, Inverse Probl. Imaging, 14 (2020), 53-75. doi: 10.3934/ipi.2019063.
[54]	R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17pp. doi: 10.1088/0266-5611/29/9/095015.
[55]	S. Senapati, Stability estimates for the relativistic Schrödinger equation from partial boundary data, Inverse Problems, 37 (2021), 015001, 25pp. doi: 10.1088/1361-6420/abca7f.
[56]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.
[57]	S. Vessella, A continuous dependence result in the analytic continuation problem, Forum Math., 11 (1999), 695-703. doi: 10.1515/form.1999.020.