Abstract
In this paper, the existence and uniqueness of a strong solution of the inverse problem of determining a coefficient of right-hand side of the integro-differential Kelvin–Voigt equation are investigated. The unknown coefficient that we search defends on space variables. Additional information on a solution of the inverse problem is given here as an integral overdetermination condition. The original inverse problem is reduced to study an equivalent inverse problem with homogeneous initial condition. Then the equivalences of the last inverse problem to an operator equation of second kind is proved. We establish the sufficient conditions for the unique solvability of the operator equation of second kind.
Funding source: Ministry of Education and Science of the Republic of Kazakhstan
Award Identifier / Grant number: AP09057950
Award Identifier / Grant number: AP08052425
Funding statement: Both the first and second authors were supported by Grant no. AP09057950 of the Ministry of Education and Science of the Republic of Kazakhstan (MES RK), Kazakhstan. The first author also was partially supported by Grant no. AP08052425 of MES RK, Kazakhstan.
References
[1] U. U. Abylkairov and K. Khompysh, An inverse problem of identifying the coefficient in Kelvin–Voight equations, Appl. Math. Sci. 9 (2015), 5079–5089. 10.12988/ams.2015.57464Suche in Google Scholar
[2] A. B. Al’shin, M. O. Korpusov and A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, De Gruyter Ser. Nonlinear Anal. Appl. 15, Walter de Gruyter, Berlin, 2011. 10.1515/9783110255294Suche in Google Scholar
[3] S. N. Antontsev, H. B. de Oliveira and K. Khompysh, Generalized Kelvin–Voigt equations for nonhomogeneous and incompressible fluids, Commun. Math. Sci. 17 (2019), no. 7, 1915–1948. 10.4310/CMS.2019.v17.n7.a7Suche in Google Scholar
[4] S. N. Antontsev, H. B. de Oliveira and K. Khompysh, Kelvin–Voigt equations perturbed by anisotropic relaxation, diffusion and damping, J. Math. Anal. Appl. 473 (2019), no. 2, 1122–1154. 10.1016/j.jmaa.2019.01.011Suche in Google Scholar
[5] S. N. Antontsev, H. B. de Oliveira and K. Khompysh, Kelvin–Voigt equations with anisotropic diffusion, relaxation and damping: Blow-up and large time behavior, Asymptot. Anal. 121 (2021), no. 2, 125–157. 10.3233/ASY-201597Suche in Google Scholar
[6] S. N. Antontsev and K. Khompysh, Generalized Kelvin–Voigt equations with p-Laplacian and source/absorption terms, J. Math. Anal. Appl. 456 (2017), no. 1, 99–116. 10.1016/j.jmaa.2017.06.056Suche in Google Scholar
[7] S. N. Antontsev and K. Khompysh, Kelvin–Voight equation with p-Laplacian and damping term: existence, uniqueness and blow-up, J. Math. Anal. Appl. 446 (2017), no. 2, 1255–1273. 10.1016/j.jmaa.2016.09.023Suche in Google Scholar
[8] A. Asanov and E. R. Atamanov, Nonclassical and Inverse Problems for Pseudoparabolic Equations, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 1997. 10.1515/9783110900149Suche in Google Scholar
[9] H. A. Barnes, A Handbook of Elementary Rheology, University of Wales, Cardiff, 2000. Suche in Google Scholar
[10] A. Y. Chebotarëv, Determination of the right-hand side of the Navier–Stokes system and inverse problems for thermal convection equations, Comp. Math. Math. Phys. 51 (2011), 2146–2154. 10.1134/S0965542511120098Suche in Google Scholar
[11] J. Cheng and J. Liu, An inverse source problem for parabolic equations with local measurements, Appl. Math. Lett. 103 (2020), Article ID 106213. 10.1016/j.aml.2020.106213Suche in Google Scholar
[12] J. Fan and G. Nakamura, Well-posedness of an inverse problem of Navier–Stokes equations with the final overdetermination, J. Inverse Ill-Posed Probl. 17 (2009), no. 6, 565–584. 10.1515/JIIP.2009.035Suche in Google Scholar
[13] V. E. Fedorov and N. D. Ivanova, Inverse problem for Oskolkov’s system of equations, Math. Methods Appl. Sci. 40 (2017), no. 17, 6123–6126. 10.1002/mma.3807Suche in Google Scholar
[14] V. E. Fedorov and A. V. Urazaeva, An inverse problem for linear Sobolev type equations, J. Inverse Ill-Posed Probl. 12 (2004), no. 4, 387–395. 10.1515/1569394042248210Suche in Google Scholar
[15] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, Springer, New York, 2011. 10.1007/978-0-387-09620-9Suche in Google Scholar
[16] O. Y. Imanuvilov and M. Yamamoto, Remark on boundary data for inverse boundary value problems for the Navier–Stokes equations [Addendum to MR3319370], Inverse Problems 31 (2015), no. 10, Article Id 109401. 10.1088/0266-5611/31/10/109401Suche in Google Scholar
[17] V. Isakov, Inverse Problems for Partial Differential Equations, Appl. Math. Sci. 127, Springer, New York, 2006. Suche in Google Scholar
[18] S. I. Kabanikhin, Inverse and Ill-Posed Problems. Theory and Applications, Inverse Ill-posed Probl. Ser. 55, De Gruyter, Berlin, 2011. 10.1515/9783110224016Suche in Google Scholar
[19] N. A. Karazeeva, Solvability of initial boundary value problems for equations describing motions of linear viscoelastic fluids, J. Appl. Math. (2005), no. 1, 59–80. 10.1155/JAM.2005.59Suche in Google Scholar
[20] K. Khompysh, Inverse Problem with integral overdetermination for system of equations of Kelvin–Voight fluids, Adv. Mat. Res. 705 (2013), 15–20. 10.4028/www.scientific.net/AMR.705.15Suche in Google Scholar
[21] M. O. Korpusov and A. G. Sveshnikov, Blow-up of Oskolkov’s system of equations, Sb. Math. 200 (2009), no. 4, 549–572. 10.1070/SM2009v200n04ABEH004008Suche in Google Scholar
[22] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Nauka, Moscow, 1970. Suche in Google Scholar
[23] S. Lu, N. Chen, B. Hu and J. Cheng, On the inverse problems for the coupled continuum pipe flow model for flows in karst aquifers, Inverse Problems 28 (2012), no. 6, Article ID 065003. 10.1088/0266-5611/28/6/065003Suche in Google Scholar
[24] A. S. Lyubanova and A. Tani, An inverse problem for pseudoparabolic equation of filtration: The existence, uniqueness and regularity, Appl. Anal. 90 (2011), no. 10, 1557–1571. 10.1080/00036811.2010.530258Suche in Google Scholar
[25] A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 98–136. Suche in Google Scholar
[26] A. P. Oskolkov, Some nonstationary linear and quasilinear systems that arise in the study of the motion of viscous fluids, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 59 (1976), 133–177. Suche in Google Scholar
[27] A. P. Oskolkov, Theory of nonstationary flows of Kelvin–Voigt fluids, J. Math. Sci. 28 (1985), 751–758. 10.1007/BF02112340Suche in Google Scholar
[28] A. P. Oskolkov, Initial-boundary value problems for equations of motion of Kelvin–Voight fluids and Oldroyd fluids, Proc. Steklov Inst. Math. 179 (1989), 137–182. Suche in Google Scholar
[29] A. P. Oskolkov, Nonlocal problems for the equations of motion of the Kelvin–Voight fluids, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 197 (1992), 120–158. Suche in Google Scholar
[30] A. P. Oskolkov, Nonlocal problems for the equations of motion of the Kelvin–Voight fluids, J. Math. Sci. 75 (1995), no. 6, 2058–2077. 10.1007/BF02362946Suche in Google Scholar
[31] A. P. Oskolkov and R. D. Shadiev, Nonlocal problems in the theory of equations of motion for Kelvin–Voigt fluids, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 181 (1990), 146–185. Suche in Google Scholar
[32] V. A. Pavlovsky, On the theoretical description of weak water solutions of polymers, Dokl. Akad. Nauk SSSR. 200 (1971), no. 4, 809–812. Suche in Google Scholar
[33] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 2000. Suche in Google Scholar
[34] L. N. Pyatnitsky, Turbulence Nature and the Inverse Problem, Fluid Mech. Appl. 89, Springer, New York, 2009. 10.1007/978-90-481-2251-6Suche in Google Scholar
[35] G. A. Sviridyuk and A. S. Shipilov, On the stability of solutions of the Oskolkov equations on a graph, Differ. Equ. 46 (2010), 742–747. 10.1134/S0012266110050137Suche in Google Scholar
[36] Q. Wu, A new type of the Gronwall–Bellman inequality and its application to fractional stochastic differential equations, Cogent Math. 4 (2017), Article ID 1279781. 10.1080/23311835.2017.1279781Suche in Google Scholar
[37] V. G. Zvyagin and V. P. Orlov, On weak solutions of the equations of motion of a viscoelastic medium with variable boundary, Bound. Value Probl. 3 (2005), 215–245. 10.1155/BVP.2005.215Suche in Google Scholar
[38] V. G. Zvyagin and M. V. Turbin, Investigation of initial-boundary value problems for mathematical models of the motion of Kelvin–Voigt fluids, J. Math. Sci. 168 (2010), 157–308. 10.1007/s10958-010-9981-2Suche in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Variation method solving of the inverse problems for Schrödinger-type equation
- Stability estimate for scalar image velocimetry
- Inverse problems for equations of a mixed parabolic-hyperbolic type with power degeneration in finding the right-hand parts that depend on time
- Inverse problem for integro-differential Kelvin–Voigt equations
- A projected homotopy perturbation method for nonlinear inverse problems in Banach spaces
- Tensor tomography of the residual stress field in graded-index YAG’s single crystals
- Uniqueness and stability for inverse source problem for fractional diffusion-wave equations
- Reconstruction of modified transmission eigenvalues using Cauchy data
- Alternative fan-beam backprojection and adjoint operators
- The problem of determining multiple coefficients in an ultrahyperbolic equation
- A note on the degree of ill-posedness for mixed differentiation on the d-dimensional unit cube
- A uniqueness result for the inverse problem of identifying boundaries from weighted Radon transform
Artikel in diesem Heft
- Frontmatter
- Variation method solving of the inverse problems for Schrödinger-type equation
- Stability estimate for scalar image velocimetry
- Inverse problems for equations of a mixed parabolic-hyperbolic type with power degeneration in finding the right-hand parts that depend on time
- Inverse problem for integro-differential Kelvin–Voigt equations
- A projected homotopy perturbation method for nonlinear inverse problems in Banach spaces
- Tensor tomography of the residual stress field in graded-index YAG’s single crystals
- Uniqueness and stability for inverse source problem for fractional diffusion-wave equations
- Reconstruction of modified transmission eigenvalues using Cauchy data
- Alternative fan-beam backprojection and adjoint operators
- The problem of determining multiple coefficients in an ultrahyperbolic equation
- A note on the degree of ill-posedness for mixed differentiation on the d-dimensional unit cube
- A uniqueness result for the inverse problem of identifying boundaries from weighted Radon transform