×
Kuehn, Christian; Neamţu, Alexandra

Center manifolds for rough partial differential equations. (English) Zbl 1528.60112

Electron. J. Probab. 28, Paper No. 48, 31 p. (2023).
In this paper, the authors deal with the existence of center manifolds for stochastic partial differential equations with nonlinear drift driven by \(\gamma\)-Holder rough paths with \(\gamma \in ({\frac 13}, {\frac 12})\). The main result is proved by the Lyapunov-Perron method along with the rough paths theory and the semigroup theory. As an application, the authors show the existence of center manifolds for a class of reaction-diffusion equations and the Swift-Hohenberg equations.
Reviewer: Bixiang Wang (Socorro)

Cited in 2 Documents

MSC:

60L20 Rough paths
35K57 Reaction-diffusion equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G22 Fractional processes, including fractional Brownian motion
60L50 Rough partial differential equations
37L55 Infinite-dimensional random dynamical systems; stochastic equations

Keywords:

center manifold; rough path; evolution equation; interpolation spaces; Lyapunov-Perron method
Cite Review PDF
Full Text: DOI arXiv Link

References:

[1] H. Amann. Linear and quasilinear parabolic problems. Birkhäuser Verlag, 1995. · Zbl 0819.35001
[2] L. Arnold. Random Dynamical Systems. Springer, Berlin Heidelberg, Germany, 2003.
[3] I. Bailleul. Flows driven by Banach space-valued rough paths. Séminaire de Probabilités XLVI, pp. 195-205, 2014. · Zbl 1390.60200
[4] I. Bailleul, S. Riedel and M. Scheutzow. Random dynamical system, rough paths and rough flows. J. Differ. Equations, 262(12):5792-5823, 2017. · Zbl 1382.37050
[5] P.W. Bates and C.K.R.T. Jones. Invariant manifolds for semilinear partial differential equations. In U. Kirchgraber and H.O. Walther, editors, Dynamics Reported, volume 2, pages 1-37. Wiley, 1989. · Zbl 0674.58024
[6] D. Blömker and M. Hairer. Amplitude equations for SPDEs: Approximate centre manifolds and invariant measures. Probability and partial differential equations in modern applied mathematics. Springer, 2005. · Zbl 1098.60057
[7] D. Blömker and A. Neamţu. Amplitude equations for SPDEs driven by fractional additive noise for small Hurst parameter. 2109.09387. · Zbl 1494.60037
[8] D. Blömker and W. Wang. Qualitative properties of local random invariant manifolds for SPDEs with quadratic nonlinearity. J. Dyn. Differ. Equations, 22(4):677-695, 2010. · Zbl 1205.35347
[9] A. Blumenthal, M. Engel and A. Neamţu. On the pitchfork bifurcation for the Chafee-Infante equation with additive noise. 2108.11073.
[10] T. Bonnefoi, A. Chandra, A. Moinat and H. Weber. A priori bounds for rough differential equations with a non-linear damping term. 2011.06645. · Zbl 1498.60400
[11] P. Boxler. A stochastic version of the center manifold theory. Probab. Theory Related Fields. 83(4):509-545, 1989. · Zbl 0671.58045
[12] P. Boxler. How to construct stochastic center manifolds on the level of vector fields. Lecture Notes in Mathematics. 1486:141-158, 1991. · Zbl 0741.34023
[13] T. Caraballo, J. Duan, K. Lu, B. Schmalfuß. Invariant manifolds for random and stochastic partial differential equations. Adv. Nonlinear Studies, 10(1):23-52, 2010. · Zbl 1209.37094
[14] T. Caraballo, J.A. Langa and J.C. Robinson. Stability and random attractors for a reactiondiffusion equation with multiplicative noise. Discrete Contin. Dynam. Systems, 6(4):875-892, 2000. · Zbl 1011.37031
[15] J. Cardona, M.Hofmanová, T. Nilssen and N. Rana. Random dynamical system generated by the 3D Navier-Stokes equation with rough transport noise. 2104.14312, pages 1-27, 2021. · Zbl 1498.35382
[16] J. Carr. Applications of Centre Manifold Theory. Springer, 1981. · Zbl 0464.58001
[17] M.D. Chekroun, H. Liu and S. Wang. Approximation of stochastic invariant manifolds. Stochastic manifolds for nonlinear SPDEs I. Springer, 2015. · Zbl 1319.60002
[18] X. Chen, A.J. Roberts and J. Duan. Centre manifolds for stochastic evolution equations. J. Differ. Equ. Appl., 21(7):602-632. · Zbl 1325.60103
[19] P.-L. Chow. Stochastic Partial Differential Equations. Chapman & Hall / CRC, 2007. · Zbl 1134.60043
[20] S.-N. Chow, K. Lu and G.R. Sell. Smoothness of inertial manifolds. J. Math. Anal. Appl., 169(1):283-312, 1992. · Zbl 0767.58026
[21] L. Coutin and A. Lejay. Sensitivity of rough differential equations: an approach through the Omega lemma. 1712.04705v1, pages 1-, 2017. · Zbl 1423.60088
[22] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. CUP, 2014. · Zbl 1317.60077
[23] A. Deya, M. Gubinelli and S. Tindel. Non-linear rough heat equations. Probab. Theory Related Fields. 153(1-2):97-147, 2012. · Zbl 1255.60106
[24] A. Du and J. Duan. Invariant manifold reduction for stochastic dynamical systems. Dynamic Systems and Applications 16:681-696, 2007. · Zbl 1142.37038
[25] J. Duan, K. Lu, B. Schmalfuß. Smooth Stable and Unstable Manifolds for Stochastic Evolutionary Equations. J. Dynam. Diff. Eq., 16(4) 949-972, 2004. · Zbl 1065.60077
[26] J. Duan and W. Wang. Effective Dynamics of Stochastic Partial Differential Equations. Elsevier, 2014. · Zbl 1298.60006
[27] B. Fehrman and B. Gess. Well-posedness of stochastic porous media equations with nonlinear, conservative noise. To appear in Arch. Ration. Mech. Anal. · Zbl 1448.35586
[28] N. Fenichel. Persistence and smoothness of invariant manifolds for flows. J. Indiana Math., 21(3):193-226, 1972. · Zbl 0246.58015
[29] P.K. Friz and M. Hairer. A Course on Rough Paths. Second ed., Springer, 2020. · Zbl 1437.60002
[30] P.K. Friz and N.B. Victoir. Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Studies in Advanced Mathematics, 2010. · Zbl 1193.60053
[31] H. Fu and D. Blömker. The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations. Nonlinearity, 33(8):3905, 2020. · Zbl 1464.60058
[32] T. Gallay. A center-stable manifold theory for differential equations in Banach spaces. Comm. Math. Phys., 152:249-2689, 1993. · Zbl 0776.34051
[33] M.J. Garrido-Atienza, K. Lu and B. Schmalfuß. Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion. J. Differential Equat., 248(7):1637-1667, 2010. · Zbl 1186.37094
[34] M.J. Garrido-Atienza, K. Lu and B. Schmalfuß. Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parametes \[H\in (1/ 3,1/ 2]\]. SIAM J. Appl. Dyn. Syst. 15(1), 625-654, 2016. · Zbl 1336.60103
[35] A. Gerasimovics and M. Hairer. Hörmander’s theorem for semilinear SPDEs. Electron. J. Probab., 24:1-56, 2019. · Zbl 1462.60084
[36] A. Gerasimovics, A. Hocquet and T. Nilssen. Non-autonomous rough semilinear PDEs and the multiplicative Sewing Lemma. J. Func. Anal., 281(10):109200, 2021. · Zbl 1507.60148
[37] M. Ghani Varzaneh, S. Riedel and M. Scheutzow. A dynamical theory for singular stochastic delay differential equations I: Linear equations and a Multiplicative Ergodic Theorem on fields of Banach spaces. SIAM J. Appl.Dyn. Syst., 21(1):10.1137/21M1433435, 2022. · Zbl 1490.34103
[38] M. Ghani Varzaneh, S. Riedel and M. Scheutzow. A dynamical theory for singular stochastic delay differential equations II: Nonlinear equations and invariant manifolds. Discrete Contin. Dyn. Syst. B, 26(8):4587-4612, 2021. · Zbl 1469.60213
[39] M. Gubinelli. Controlling rough paths. J. Func. Anal. 216(1):86-140, 2004. · Zbl 1058.60037
[40] M. Gubinelli, A. Lejay and S. Tindel. Young integrals and SPDEs. Potential Anal. 25(4):307-326, 2006. · Zbl 1103.60062
[41] M. Gubinelli and S. Tindel. Rough evolution equations. Ann. Probab. 38(1):1-75, 2010. · Zbl 1193.60070
[42] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, 1983. · Zbl 0515.34001
[43] M. Hairer. An introduction to stochastic PDEs. 0907.4178, pages 1-78, 2009.
[44] M. Hairer. A theory of regularity structures. Invent. Math., 198(2):269-504, 2014. · Zbl 1332.60093
[45] M. Hairer. Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Probab., 33(2):703-758, 2005. · Zbl 1071.60045
[46] D. Henry. Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin Heidelberg, Germany, 1981. · Zbl 0456.35001
[47] R. Hesse and A. Neamţu. Local mild solutions for rough stochastic partial differential equations. J. Differential Equat., 267(11):6480-6538, 2019. · Zbl 1448.60139
[48] R. Hesse and A. Neamţu. Global solutions and random dynamical systems for rough evolution equations. Discrete Contin. Dyn. Syst., 25(7):2723-2748, 2020. · Zbl 1434.60156
[49] R. Hesse and A. Neamţu. Global solutions for semilinear rough partial differential equations. Stoch. Dyn., 22(2):22400011, 2022. · Zbl 1491.35274
[50] Y. Hu, D. Nualart, Rough path analysis via fractional calculus. Trans. Am. Math. Soc., 361(5):2689-2718, 2009. · Zbl 1175.60061
[51] A. Hocquet and A. Neamţu. Quasilinear rough evolution equations. 2207.04787, pages 1-49, 2022.
[52] M. Hofmanová, J.-M. Leahy and T. Nilssen. On a rough perturbation of the Navier-Stokes system and its vorticity formulation. Annals Appl. Probab., 31(2), 2021. · Zbl 1477.35122
[53] C. Kuehn. Multiple Time Scale Dynamics. Springer, Berlin Heidelberg, Germany, 2015. · Zbl 1335.34001
[54] C. Kuehn and A. Neamţu. Rough center manifolds. SIAM J. Math. Anal., 53(4):3912-3957, 2021. · Zbl 1480.37066
[55] Y.A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer, Berlin Heidelberg, Germany, 2013.
[56] H. Kunita. Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, 1990. · Zbl 0743.60052
[57] Z. Lian and K. Lu. Lyapunov exponents and invariant manifolds for random dynamical systems. Mem. Amer. Math. Soc., 206(2010), vi+106 pp. · Zbl 1200.37047
[58] K. Lu and B. Schmalfuß. Invariant manifolds for stochastic wave equations. J. Differential Equat., 236(2):460-492, 2007. · Zbl 1113.37056
[59] T. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana, 14(2):215-310, 1998. · Zbl 0923.34056
[60] A. Lunardi. Analytic semigroups and optimal regularity in parabolic problems. Birkhäuser, 1995. · Zbl 0816.35001
[61] B. Maslowski and D. Nualart. Evolution equations driven by a fractional Brownian motion. J. Funct. Anal., 202(1):277-305, 2003. · Zbl 1027.60060
[62] S. Mohammed, T. Zhang and H. Zhao. The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations. Memoirs of the AIMS, vol. 196, nr. 197, 2008. · Zbl 1169.60014
[63] A. Neamţu. Random invariant manifolds for ill-posed stochastic evolution equations. Stoch. Dynam., 20(2):2050013, 2020. · Zbl 1441.37088
[64] A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer Applied Mathematical Series. Springer-Verlag, Berlin, 1983. · Zbl 0516.47023
[65] A.J. Roberts. Normal form transforms separate slow and fast modes in stochastic dynamical systems. Physica A, 387(1):12-38, 2008.
[66] M. Scheutzow. On the perfection of crude cocycles. Random Comput. Dynam., 4(4):235-255, 1996. · Zbl 0881.60062
[67] G. Sell and Y. You. Dynamics of evolutionary equations. Springer-Verlag New York, 2002. · Zbl 1254.37002
[68] G. Simonett. Center manifolds for quasilinear reaction-diffusion systems. Differ. Integral Equ., 8(4):753-796, 1995. · Zbl 0815.35054
[69] A. Vanderbauwhede. Center manifold, normal forms and elementary bifurcations. Dynamics Reported. pp. 89-169, 1989. · Zbl 0677.58001
[70] A. Vanderbauwhede and G. Iooss. Center manifold theory in infinite dimensions. Dynamics Reported. Springer-Verlag Berlin, pp. 125-163, 1992. · Zbl 0751.58025
[71] W. Wang and J. Duan. A dynamical approximation of stochastic partial differential equations. J. Math. Phys., 48(10):102701-102701-14, 2007 · Zbl 1152.81629
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.