Stability estimate and the Tikhonov regularization method for the Kuramoto–Sivashinsky equation backward in time

References

[1] R. Alonso-Sanz and J. P. Cárdenas, The effect of memory in cellular automata on scale-free networks: The K ¯ = 4 case, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 (2008), no. 8, 2477–2486. 10.1142/S0218127408021841Suche in Google Scholar

[2] A. R. Barron, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans. Inform. Theory 39 (1993), no. 3, 930–945. 10.1109/18.256500Suche in Google Scholar

[3] D. J. Benney, Long waves on liquid films, J. Math. Phys. 45 (1966), 150–155. 10.1002/sapm1966451150Suche in Google Scholar

[4] K. Cao and D. Lesnic, Determination of the time-dependent effective ion collision frequency from an integral observation, J. Inverse Ill-Posed Probl. 32 (2024), no. 4, 761–793. 10.1515/jiip-2023-0024Suche in Google Scholar

[5] B. I. Cohen, J. A. Krommes, W. M. Tang and M. N. Rosenbluth, Non-linear saturation of the dissipative trapped-ion mode by mode coupling, Nuclear Fusion 16 (1976), no. 6, 971–992. 10.1088/0029-5515/16/6/009Suche in Google Scholar

[6] P. Colet, K. Wiesenfeld and S. Strogatz, Frequency locking in josephson arrays: Connection with the Kuramoto model, Phys. Rev. E 57 (1998), Article ID 1563. 10.1103/PhysRevE.57.1563Suche in Google Scholar

[7] A. V. Coward, D. T. Papageorgiou and Y. S. Smyrlis, Nonlinear stability of oscillatory core-annular flow: A generalized Kuramoto–Sivashinsky equation with time periodic coefficients, Z. Angew. Math. Phys. 46 (1995), no. 1, 1–39. 10.1007/BF00952254Suche in Google Scholar

[8] S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002), no. 2, 430–455. 10.1006/jcph.2002.6995Suche in Google Scholar

[9] G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems 2 (1989), no. 4, 303–314. 10.1007/BF02551274Suche in Google Scholar

[10] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press,New York, 1982. Suche in Google Scholar

[11] N. V. Duc and N. V. Thang, Stability results for semi-linear parabolic equations backward in time, Acta Math. Vietnam. 42 (2017), no. 1, 99–111. 10.1007/s40306-015-0163-7Suche in Google Scholar

[12] A. Gonzalez and A. Castellanos, Nonlinear electrohydrodynamic waves on films falling down an inclined plane, Phys. Rev. E 53 (1996), no. 4, Article ID 3573. 10.1103/PhysRevE.53.3573Suche in Google Scholar PubMed

[13] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conf. Ser. in Appl. Math. 26, Society for Industrial and Applied Mathematics, Philadelphia, 1977. Suche in Google Scholar

[14] J. Gustafsson and B. Protas, Regularization of the backward-in-time Kuramoto–Sivashinsky equation, J. Comput. Appl. Math. 234 (2010), no. 2, 398–406. 10.1016/j.cam.2009.12.032Suche in Google Scholar

[15] P. Guzmán Meléndez, Lipschitz stability in an inverse problem for the main coefficient of a Kuramoto–Sivashinsky type equation, J. Math. Anal. Appl. 408 (2013), no. 1, 275–290. 10.1016/j.jmaa.2013.05.050Suche in Google Scholar

[16] D. N. Hào, N. V. Duc and N. V. Thang, Stability estimates for Burgers-type equations backward in time, J. Inverse Ill-Posed Probl. 23 (2015), no. 1, 41–49. 10.1515/jiip-2013-0050Suche in Google Scholar

[17] D. N. Hào, N. V. Duc and N. V. Thang, Backward semi-linear parabolic equations with time-dependent coefficients and local Lipschitz source, Inverse Problems 34 (2018), no. 5, Article ID 055010. 10.1088/1361-6420/aab8cbSuche in Google Scholar

[18] A. P. Hooper and R. Grimshaw, Nonlinear instability at the interface between two viscous fluids, Phys. Fluids 28 (1985), no. 1, 37–45. 10.1063/1.865160Suche in Google Scholar

[19] K. Hornik, M. Stinchcombe and H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2 (1989), no. 5, 359–366. 10.1016/0893-6080(89)90020-8Suche in Google Scholar

[20] J. M. Hyman, B. Nicolaenko and S. Zaleski, Order and complexity in the Kuramoto–Sivashinsky model of weakly turbulent interfaces, Phys. D 23 (1986), 265–292. 10.1016/0167-2789(86)90136-3Suche in Google Scholar

[21] E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University, Cambridge, 2023. Suche in Google Scholar

[22] G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics, Numer. Math. Sci. Comput., Oxford University, New York, 2005. 10.1093/acprof:oso/9780198528692.001.0001Suche in Google Scholar

[23] A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput. 26 (2005), no. 4, 1214–1233. 10.1137/S1064827502410633Suche in Google Scholar

[24] I. Kukavica, On the behavior of the solutions of the Kuramoto–Sivashinsky equation for negative time, J. Math. Anal. Appl. 166 (1992), no. 2, 601–606. 10.1016/0022-247X(92)90319-9Suche in Google Scholar

[25] I. Kukavica and M. Malcok, Backward behavior of solutions of the Kuramoto–Sivashinsky equation, J. Math. Anal. Appl. 307 (2005), no. 2, 455–464. 10.1016/j.jmaa.2005.01.057Suche in Google Scholar

[26] Y. Kuramoto, Diffusion-induced chaos in reaction systems, Progr. Theoret. Phys. Suppl. 64 (1978), 346–367. 10.1143/PTPS.64.346Suche in Google Scholar

[27] Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems: Reductive perturbation approach, Progr. Theoret. Phys. 54 (1975), no. 3, 687–699. 10.1143/PTP.54.687Suche in Google Scholar

[28] D. M. Michelson and G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames. II. Numerical experiments, Acta Astronaut. 4 (1977), no. 11–12, 1207–1221. 10.1016/0094-5765(77)90097-2Suche in Google Scholar

[29] D. T. Papageorgiou, C. Maldarelli and D. S. Rumschitzki, Nonlinear interfacial stability of core-annular film flows, Phys. Fluids A 2 (1990), no. 3, 340–352. 10.1063/1.857784Suche in Google Scholar

[30] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math. 23, Springer, Berlin, 2008. Suche in Google Scholar

[31] M. Raissi, P. Perdikaris and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378 (2019), 686–707. 10.1016/j.jcp.2018.10.045Suche in Google Scholar

[32] T. Shlang and G. I. Sivashinsky, Irregular flow of a liquid film down a vertical column, J. Phys. 43 (1982), no. 3, 459–466. 10.1051/jphys:01982004303045900Suche in Google Scholar

[33] G. I. Sivashinsky, Instabilities, pattern formation, and turbulence in flames, Annu. Rev. Fluid Mech. 15 (1983), no. 1, 179–199. 10.1146/annurev.fl.15.010183.001143Suche in Google Scholar

[34] G. I. Sivashinsky and D. M. Michelson, On irregular wavy flow of a liquid film down a vertical plane, Progr. Theoret. Phys. 63 (1980), no. 6, 2112–2114. 10.1143/PTP.63.2112Suche in Google Scholar

[35] M. Suzuki, Statistical mechanics of non-equilibrium systems. III. Fluctuation and relaxation of quantal macrovariables, Progr. Theoret. Phys. 55 (1976), no. 4, 1064–1081. 10.1143/PTP.55.1064Suche in Google Scholar

[36] W. C. Troy, Phaselocked solutions of the finite size Kuramoto coupled oscillator model, SIAM J. Math. Anal. 49 (2017), no. 3, 1912–1931. 10.1137/16M1055542Suche in Google Scholar