# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2022024
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## Well-posedness of an inverse problem for two- and three-dimensional convective Brinkman-Forchheimer equations with the final overdetermination

 Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, India

*Corresponding author: Manil T. Mohan

Received  August 2021 Revised  April 2022 Early access May 2022

Fund Project: The authors are supported by INSPIRE Faculty Award Grant IFA17-MA110

In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations:
 \begin{align*} \boldsymbol{u}_t-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p = \boldsymbol{F}: = \boldsymbol{f} g, \ \ \ \nabla\cdot\boldsymbol{u} = 0, \end{align*}
in bounded domains
 $\Omega\subset\mathbb{R}^d$
(
 $d = 2, 3$
) with smooth boundary, where
 $\alpha, \beta, \mu>0$
and
 $r\in[1, \infty)$
. The CBF equations describe the motion of incompressible fluid flows in a saturated porous medium. The inverse problem under our consideration consists of reconstructing the vector-valued velocity function
 $\boldsymbol{u}$
 $\nabla p$
and the vector-valued function
 $\boldsymbol{f}$
. We prove the well-posedness result (existence, uniqueness and stability) of an inverse problem for 2D and 3D CBF equations with the final overdetermination condition using Schauder's fixed point theorem for arbitrary smooth initial data. The well-posedness results hold for
 $r\geq 1$
in two dimensions and for
 $r \geq 3$
in three dimensions. The global solvability results available in the literature helped us to obtain the uniqueness and stability results for the model with fast growing nonlinearities.
Citation: Pardeep Kumar, Manil T. Mohan. Well-posedness of an inverse problem for two- and three-dimensional convective Brinkman-Forchheimer equations with the final overdetermination. Inverse Problems and Imaging, doi: 10.3934/ipi.2022024
##### References:
 [1] S. N. Antontsev and H. B. de Oliveira, The Navier-Stokes problem modified by an absorption term, Appl. Anal., 89 (2010), 1805-1825.  doi: 10.1080/00036811.2010.495341. [2] M. Badra, F. Caubet and J. Dardé, Stability estimates for Navier-Stokes equations and application to inverse problems, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2379-2407.  doi: 10.3934/dcdsb.2016052. [3] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993. [4] V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, J. Math. Anal. Appl., 255 (2001), 281-307.  doi: 10.1006/jmaa.2000.7256. [5] I. Bushuyev, Global uniqueness for inverse parabolic problems with final observation, Inverse Problems, 11 (1995), 11-16.  doi: 10.1088/0266-5611/11/4/001. [6] X. Cai and Q. Jiu, Weak and Strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041. [7] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Mat. Sem. Univ. Padova, 31 (1961), 308-340. [8] A. Y. Chebotarev, Inverse problem for Navier-Stokes systems with finite-dimensional overdetermination, Differ. Uravn., 48 (2012), 1153-1160.  doi: 10.1134/S0012266112080101. [9] A. Y. Chebotarev, Inverse problems for stationary Navier-Stokes systems, Comput. Math. Math. Phys., 54 (2014), 537-545.  doi: 10.1134/S0965542514030038. [10] M. Chouli, O. Y. Imanuvilov, J.-P. Puel and M. Yamamoto, Inverse source problem for linearized Navier-Stokes equations with data in arbitrary sub-domain, Appl. Anal., 92 (2013), 2127-2143.  doi: 10.1080/00036811.2012.718334. [11] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM Philadelphia, 2013. [12] J. Fan, M. Di Cristo, Y. Jiang and G. Nakamura, Inverse viscosity problem for the Navier-Stokes equation, J. Math. Anal. Appl., 365 (2010), 750-757.  doi: 10.1016/j.jmaa.2009.12.012. [13] J. Fan and G. Nakamura, Local solvability of an inverse problem to the density-dependent Navier-Stokes equations, Appl. Anal., 87 (2008), 1255-1265.  doi: 10.1080/00036810802428920. [14] 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), 565-584.  doi: 10.1515/JIIP.2009.035. [15] C. L. Fefferman, K. W. Hajduk and J. C. Robinson, Simultaneous Approximation in Lebesgue and Sobolev Norms Via Eigenspaces, https://arXiv.org/abs/1904.03337. [16] D. Fujiwara and H. Morimoto, An $L^r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700. [17] N. L. Gol'dman, Determination of the right-hand side in a quasilinear parobolic equation with a terminal observation, Diff. Eqns., 41 (2005), 384-392. [18] K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer equations, J. Differential Equations, 263 (2017), 7141-7161.  doi: 10.1016/j.jde.2017.08.001. [19] K. W. Hajduk, J. C. Robinson and W. Sadowski, Robustness of regularity for the 3D convective Brinkman-Forchheimer equations, J. Math. Anal. Appl., 500 (2021), 125058, 23 pp. doi: 10.1016/j.jmaa.2021.125058. [20] O. Y. Imanuvilov and M. Yamamoto, Global uniqueness in inverse boundary value problems for the Navier-Stokes equations and Lamé system in two dimensions, Inverse Problems, 31 (2015), 035004, 46 pp. doi: 10.1088/0266-5611/31/3/035004. [21] V. Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math., 44 (1991), 185-209.  doi: 10.1002/cpa.3160440203. [22] V. Isakov, Inverse Problems for Partial Differential Equation, 2$^{ed}$ edition, Now York, Springer, 2006. [23] Y. Jiang, J. Fan, S. Nagayasu and G. Nakamura, Local solvability of an inverse problem to the Navier-Stokes equation with memory term, Inverse Problems, 36 (2020), 065007, 14 pp. doi: 10.1088/1361-6420/ab7e05. [24] V. K. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054.  doi: 10.3934/cpaa.2012.11.2037. [25] A. I. Korotkii, Inverse problems of reconstructing parameters of the Navier-Stokes system, J. Math. Sci., 140 (2007), 808-831.  doi: 10.1007/s10958-007-0019-3. [26] P. Kumar, K. Kinra and M. T. Mohan, A local in time existence and uniqueness result of an inverse problem for the Kelvin-Voigt fluids, Inverse Problems, 37 (2021), 085005, 34 pp. doi: 10.1088/1361-6420/ac1050. [27] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. [28] R. Y. Lai, G. Uhlmann and J. N. Wang, Inverse boundary value problem for the Stokes and the Navier-Stokes equations in the plane, Arch. Ration. Mech. Anal., 215 (2015), 811-829.  doi: 10.1007/s00205-014-0794-1. [29] P. A. Markowich, E. S. Titi and S. Trabelsi, Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 29 (2016), 1292-1328.  doi: 10.1088/0951-7715/29/4/1292. [30] M. T. Mohan, On the convective Brinkman-Forchheimer equations, (submitted). [31] M. T. Mohan, Stochastic convective Brinkman-Forchheimer equations (submitted), https://arXiv.org/abs/2007.09376. [32] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 2000. [33] A. I. Prilepko and D. S. Tkachenko, Well-posedness of the inverse source problem for parabolic systems, Diff. Eqns., {40} (2004), 1619–1626. doi: 10.1007/s10625-005-0080-y. [34] J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations, Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 2016.  doi: 10.1017/CBO9781139095143. [35] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984. [36] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^{nd}$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050. [37] I. A. Vasin and A. I. Prilepko, The solvability of three dimensional inverse problem for the nonlinear Navier-Stokes equations, U. S. S. R. Comput. Math. and Math. Phys., 30 (1990), 189-199.  doi: 10.1016/0041-5553(90)90177-T. [38] Z. Zhang, X. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414-419.  doi: 10.1016/j.jmaa.2010.11.019. [39] Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (2012), 1822-1825.  doi: 10.1016/j.aml.2012.02.029. [40] H. Zou, et al, Handbook of Differential Equations: Stationary Partial Differential Equations, Volume VI, Elsevier.

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##### References:
 [1] S. N. Antontsev and H. B. de Oliveira, The Navier-Stokes problem modified by an absorption term, Appl. Anal., 89 (2010), 1805-1825.  doi: 10.1080/00036811.2010.495341. [2] M. Badra, F. Caubet and J. Dardé, Stability estimates for Navier-Stokes equations and application to inverse problems, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2379-2407.  doi: 10.3934/dcdsb.2016052. [3] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993. [4] V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, J. Math. Anal. Appl., 255 (2001), 281-307.  doi: 10.1006/jmaa.2000.7256. [5] I. Bushuyev, Global uniqueness for inverse parabolic problems with final observation, Inverse Problems, 11 (1995), 11-16.  doi: 10.1088/0266-5611/11/4/001. [6] X. Cai and Q. Jiu, Weak and Strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041. [7] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Mat. Sem. Univ. Padova, 31 (1961), 308-340. [8] A. Y. Chebotarev, Inverse problem for Navier-Stokes systems with finite-dimensional overdetermination, Differ. Uravn., 48 (2012), 1153-1160.  doi: 10.1134/S0012266112080101. [9] A. Y. Chebotarev, Inverse problems for stationary Navier-Stokes systems, Comput. Math. Math. Phys., 54 (2014), 537-545.  doi: 10.1134/S0965542514030038. [10] M. Chouli, O. Y. Imanuvilov, J.-P. Puel and M. Yamamoto, Inverse source problem for linearized Navier-Stokes equations with data in arbitrary sub-domain, Appl. Anal., 92 (2013), 2127-2143.  doi: 10.1080/00036811.2012.718334. [11] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM Philadelphia, 2013. [12] J. Fan, M. Di Cristo, Y. Jiang and G. Nakamura, Inverse viscosity problem for the Navier-Stokes equation, J. Math. Anal. Appl., 365 (2010), 750-757.  doi: 10.1016/j.jmaa.2009.12.012. [13] J. Fan and G. Nakamura, Local solvability of an inverse problem to the density-dependent Navier-Stokes equations, Appl. Anal., 87 (2008), 1255-1265.  doi: 10.1080/00036810802428920. [14] 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), 565-584.  doi: 10.1515/JIIP.2009.035. [15] C. L. Fefferman, K. W. Hajduk and J. C. Robinson, Simultaneous Approximation in Lebesgue and Sobolev Norms Via Eigenspaces, https://arXiv.org/abs/1904.03337. [16] D. Fujiwara and H. Morimoto, An $L^r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700. [17] N. L. Gol'dman, Determination of the right-hand side in a quasilinear parobolic equation with a terminal observation, Diff. Eqns., 41 (2005), 384-392. [18] K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer equations, J. Differential Equations, 263 (2017), 7141-7161.  doi: 10.1016/j.jde.2017.08.001. [19] K. W. Hajduk, J. C. Robinson and W. Sadowski, Robustness of regularity for the 3D convective Brinkman-Forchheimer equations, J. Math. Anal. Appl., 500 (2021), 125058, 23 pp. doi: 10.1016/j.jmaa.2021.125058. [20] O. Y. Imanuvilov and M. Yamamoto, Global uniqueness in inverse boundary value problems for the Navier-Stokes equations and Lamé system in two dimensions, Inverse Problems, 31 (2015), 035004, 46 pp. doi: 10.1088/0266-5611/31/3/035004. [21] V. Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math., 44 (1991), 185-209.  doi: 10.1002/cpa.3160440203. [22] V. Isakov, Inverse Problems for Partial Differential Equation, 2$^{ed}$ edition, Now York, Springer, 2006. [23] Y. Jiang, J. Fan, S. Nagayasu and G. Nakamura, Local solvability of an inverse problem to the Navier-Stokes equation with memory term, Inverse Problems, 36 (2020), 065007, 14 pp. doi: 10.1088/1361-6420/ab7e05. [24] V. K. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054.  doi: 10.3934/cpaa.2012.11.2037. [25] A. I. Korotkii, Inverse problems of reconstructing parameters of the Navier-Stokes system, J. Math. Sci., 140 (2007), 808-831.  doi: 10.1007/s10958-007-0019-3. [26] P. Kumar, K. Kinra and M. T. Mohan, A local in time existence and uniqueness result of an inverse problem for the Kelvin-Voigt fluids, Inverse Problems, 37 (2021), 085005, 34 pp. doi: 10.1088/1361-6420/ac1050. [27] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. [28] R. Y. Lai, G. Uhlmann and J. N. Wang, Inverse boundary value problem for the Stokes and the Navier-Stokes equations in the plane, Arch. Ration. Mech. Anal., 215 (2015), 811-829.  doi: 10.1007/s00205-014-0794-1. [29] P. A. Markowich, E. S. Titi and S. Trabelsi, Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 29 (2016), 1292-1328.  doi: 10.1088/0951-7715/29/4/1292. [30] M. T. Mohan, On the convective Brinkman-Forchheimer equations, (submitted). [31] M. T. Mohan, Stochastic convective Brinkman-Forchheimer equations (submitted), https://arXiv.org/abs/2007.09376. [32] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 2000. [33] A. I. Prilepko and D. S. Tkachenko, Well-posedness of the inverse source problem for parabolic systems, Diff. Eqns., {40} (2004), 1619–1626. doi: 10.1007/s10625-005-0080-y. [34] J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations, Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 2016.  doi: 10.1017/CBO9781139095143. [35] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984. [36] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^{nd}$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050. [37] I. A. Vasin and A. I. Prilepko, The solvability of three dimensional inverse problem for the nonlinear Navier-Stokes equations, U. S. S. R. Comput. Math. and Math. Phys., 30 (1990), 189-199.  doi: 10.1016/0041-5553(90)90177-T. [38] Z. Zhang, X. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414-419.  doi: 10.1016/j.jmaa.2010.11.019. [39] Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (2012), 1822-1825.  doi: 10.1016/j.aml.2012.02.029. [40] H. Zou, et al, Handbook of Differential Equations: Stationary Partial Differential Equations, Volume VI, Elsevier.
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