On the Uniqueness of the Solution to an Inverse Problem for a Stationary Boltzmann-Type Equation

Year 2025, Volume: 8 Issue: 4, 235 - 243, 30.12.2025

Muhammed Hasdemir , Özlem Kaytmaz

https://doi.org/10.33401/fujma.1833055

Abstract

Stationary Boltzmann-type equations appear in many physical modeling contexts, such as time-independent neutron transport, rarefied gas flows, radiative transfer problems, and plasma equilibrium configurations. In this work, we investigate the uniqueness of the solution to an inverse source problem for a stationary Boltzmann-type equation posed in a phase-space domain. The inverse problem is supplemented with boundary conditions and additional interior point data. The proof of the uniqueness of the solution to the inverse problem is based on a divergence-type identity that couples the transport, collision, and source terms and makes essential use of the additional interior information together with the boundary conditions.

Keywords

Boltzmann-type equations , Neumann boundary conditions , Inverse problem

References

• [1] C. Cercignani, The Boltzmann equation and its applications, Springer, New York, (1988). $ \href{https://link.springer.com/chapter/10.1007/978-1-4612-1039-9_2}{\mbox{[Web]}} $
• [2] C. Villani, A review of mathematical topics in collisional kinetic theory, Elsevier, Amsterdam, (2002). $ \href{https://doi.org/10.1016/S1874-5792(02)80004-0}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/67649344144?origin=resultslist}{\mbox{[Scopus]}} $
• [3] C.A. Agbata, A Mathematical Model for the Control of Ebola Virus Disease with Vaccination Effect, J. Math. Epidemiol., 1(1) (2025), 26–48. $ \href{https://doi.org/10.64891/jome.6}{\mbox{[CrossRef]}} $
• [4] A.Ç. Yar, E. Yılmaz and T. Gülşen, On Some Spectral Properties of Discrete Sturm-Liouville Problem, Fundam. J. Math. Appl., 6(1) (2023), 61–69. $ \href{https://doi.org/10.33401/fujma.1242330}{\mbox{[CrossRef]}} $
• [5] C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations, Rev. Mod. Phys., 63(2) (1991), 221–287. $ \href{https://doi.org/10.1007/BF01026608}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/0001076950?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1991FM52300020}{\mbox{[Web of Science]}} $
• [6] M. Kawamoto and M. Machida, Global stability for multidimensional fractional transport equations, J. Differ. Equ., 286 (2021), 377–412.
• [7] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199(2) (2011), 493–525. $ \href{https://doi.org/10.1007/s00205-010-0354-2}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/78751650015?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000286463600006}{\mbox{[Web of Science]}} $
• [8] P.J. Morrison, Hamiltonian and action principle formulations of plasma physics, Phys. Plasmas, 12(5) (2005), 058102. $\href{https://doi.org/10.1063/1.1882353}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/20844461902?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000229276200173}{\mbox{[Web of Science]}} $
• [9] J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, Anal. Appl., 13(4) (2015), 557–598. $ \href{https://homepage.univie.ac.at/christian.schmeiser/DMS-0-13.pdf}{\mbox{[Web]}} $
• [10] Q. Li and W. Sun, Applications of kinetic tools to inverse transport problems, Inverse Probl., 36(3) (2020), 035011. $\href{https://doi.org/10.1088/1361-6420/ab59b8}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85080050639?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000537450500001}{\mbox{[Web of Science]}} $
• [11] T. Balehowsky, A. Kujanpaa, M. Lassas and T. Liimatainen, An inverse problem for the relativistic Boltzmann equation, Commun. Math. Phys., 396(3) (2022), 983–1049. $ \href{https://doi.org/10.1007/s00220-022-04486-8}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85137893185?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000852353800001}{\mbox{[Web of Science]}} $
• [12] Y. Wang, Inverse source problem for the Boltzmann equation in cosmology, Ann. Henri Poincare (2025). $ \href{https://doi.org/10.1007/s00023-025-01543-5}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85217882822?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001415631100001}{\mbox{[Web of Science]}} $
• [13] R. Galeano Andrades and J. Torres del Valle, Stationary Boltzmann equation: An approach via Morse theory, Proyecciones, 40(6) (2021), 1473–1487. $ \href{https://doi.org/10.22199/ISSN.0717-6279-4034}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85121805667?origin=resultslist}{\mbox{[Scopus]}} $
• [14] L. Arkeryd and A. Nouri, L1 solutions to the stationary Boltzmann equation in a slab, Ann. Fac. Sci. Toulouse (6), 9(3) (2000), 375–413. $\href{https://www.numdam.org/item/AFST_2000_6_9_3_375_0.pdf}{\mbox{[Web]}} $
• [15] G. Busoni, F. Glose and B. Perthame, Stationary Boltzmann’s equation with a source term, Commun. Partial Differ. Equ., 15(12) (1990), 471–481. $ \href{https://doi.org/10.1080/03605309908820747}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84954826738?origin=resultslist}{\mbox{[Scopus]}} $
• [16] L. Arkeryd, and A. Nouri, Stationary solutions to the Boltzmann equation in the plane, Unpublished manuscript.
• [17] R. Esposito, Y. Guo, C. Kim and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Commun. Math. Phys., 323(1) (2013), 177–239. $ \href{https://doi.org/10.1007/s00220-013-1766-2}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84881250133?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000322873800005}{\mbox{[Web of Science]}} $
• [18] R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, 4(1) (2018). $ \href{https://doi.org/10.1007/s40818-017-0037-5}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85086345208?origin=resultslist}{\mbox{[Scopus]}} $
• [19] S. Ukai and K. Asano, Steady solutions of the Boltzmann equation for a gas flow past an obstacle. I. Existence, Arch. Ration. Mech. Anal., 84(3) (1983), 249–291. $ \href{https://doi.org/10.1007/BF00281521}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/0020906174?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1983RV44700004}{\mbox{[Web of Science]}} $
• [20] S. Ukai and K. Asano, Steady solutions of the Boltzmann equation for a gas flow past an obstacle. II. Stability, Publ. Res. Inst. Math. Sci., 22(6) (1986), 1035–1062. $ \href{https://doi.org/10.2977/prims/1195177061}{\mbox{[Web]}} $
• [21] A. Kh. Amirov, Integral geometry and inverse problems for kinetic equations, VSP, Utrecht, (2001). $\href{https://books.google.com.tr/books?hl=tr&lr=&id=PPtvE4jfBEsC&oi=fnd&pg=PA1&dq=Integral+geometry+and+inverse+problems+for+kinetic+equations&ots=q6VDaDiBVU&sig=JPr2ykbtATPvngdEy9Y3EQ7BKOE&redir_esc=y#v=onepage&q=Integral%20geometry%20and%20inverse%20problems%20for%20kinetic%20equations&f=false}{\mbox{[Web]}} $
• [22] A. Kh. Amirov and F. Gölgeleyen, Solvability of an inverse problem for the kinetic equation and a symbolic algorithm, Comput. Model. Eng. Sci., 65(2) (2010), 179–190. $ \href{https://www.scopus.com/pages/publications/78249287575?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000284298900003}{\mbox{[Web of Science]}} $
• [23] M. Yıldız, B. Heydarov and I. Gölgeleyen, Approximate solution of an inverse problem for a non-stationary general kinetic equation, Comput. Model. Eng. Sci., 62(3) (2010), 255–264. $ \href{https://www.scopus.com/pages/publications/77957586268?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000281939900002}{\mbox{[Web of Science]}} $
• [24] A. Kh. Amirov, F. Gölgeleyen and A. Rahmanova, An inverse problem for the general kinetic equation and a numerical method, Comput. Model. Eng. Sci., 43(2) (2009), 131–147. $ \href{https://www.scopus.com/pages/publications/76249103949?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000267359300002}{\mbox{[Web of Science]}} $
• [25] İ. Gölgeleyen, An inverse problem for a generalized kinetic equation in semi-geodesic coordinates, J. Geom. Phys., 168 (2021), 104318. $ \href{https://doi.org/10.1016/j.geomphys.2021.104318}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85109523854?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000687953900007}{\mbox{[Web of Science]}} $
• [26] İ. Gölgeleyen, and M. Hasdemir, A solution algorithm for an inverse source problem for the kinetic equation, Int. J. Mod. Phys. C, 33(11) (2022), 2250151. $ \href{https://doi.org/10.1142/S0129183122501510}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85131039163?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000849413800001}{\mbox{[Web of Science]}} $
• [27] Ö. Kaytmaz, The problem of determining source term in a kinetic equation in an unbounded domain, AIMS Math., 9(4) (2024), 9184–9194. $ \href{https://doi.org/10.3934/math.2024447}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85186856017?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001179743900003}{\mbox{[Web of Science]}} $
• [28] F. Gölgeleyen, and M. Yamamoto, Stability for some inverse problems for transport equations, SIAM J. Math. Anal., 48(4) (2016), 2319– 2344. $\href{https://doi.org/10.1137/15M1038128}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84984941722?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000385023400002}{\mbox{[Web of Science]}} $
• [29] P. Cannarsa, G. Floridia, F. Gölgeleyen, and M. Yamamoto, Inverse coefficient problems for a transport equation by local Carleman estimate, Inverse Probl., 35(10) (2019), 105013. $ \href{https://doi.org/10.1088/1361-6420/ab1c69}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85068777068?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000504443800001}{\mbox{[Web of Science]}} $
• [30] M.V. Klibanov and S.E. Pamyatnykh, Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate, J. Math. Anal. Appl., 343(1) (2008), 352–365. $ \href{https://doi.org/10.1016/j.jmaa.2008.01.071}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/41449102778?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000255653900030}{\mbox{[Web of Science]}} $
• [31] I. Gölgeleyen, An inverse problem for a generalized transport equation in polar coordinates and numerical applications, Inverse Probl., 29(9) (2013), 095006. $ \href{https://doi.org/10.1088/0266-5611/29/9/095006}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84884150170?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000324495100006}{\mbox{[Web of Science]}} $
• [32] F. Gölgeleyen, and M. Hasdemir, Numerical solution of an inverse problem for the Liouville equation, TWMS J. Appl. Eng. Math., 9(4) (2019), 909–920. $ \href{https://www.scopus.com/pages/publications/85098540753?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000488642900023}{\mbox{[Web of Science]}} $
• [33] F. Gölgeleyen, İ. Gölgeleyen and M. Hasdemir, A hybrid solution method for an inverse problem for the general transport equation, Comput. Math. Appl., 192 (2025), 172–188. $ \href{https://doi.org/10.1016/j.camwa.2025.05.014}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/105005838355?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001503193000002}{\mbox{[Web of Science]}} $
• [34] L. Li and Z. Ouyang, Determining the collision kernel in the Boltzmann equation near the equilibrium, Proc. Amer. Math. Soc., 151(11) (2023), 4855–4865. $ \href{https://doi.org/10.1090/proc/16522}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85171439687?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001052204400001}{\mbox{[Web of Science]}} $
• [35] R.Y. Lai and L. Yan, Stable determination of time-dependent collision kernel in the nonlinear Boltzmann equation, SIAM J. Appl. Math., 84(5) (2024), 1937–1956. $\href{https://doi.org/10.1137/23M1604060}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85204681422?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001380638100003}{\mbox{[Web of Science]}} $