Abstract
By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa-Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bao, J., Ren, P., Wang, F.-Y.: Bismut formulas for Lions derivative of McKean-Vlasov SDEs with memory. J. Differential Equations 282, 285–329 (2021)
Huang, X., Röckner, M., Wang, F.-Y.: Nonlinear Fokker-Planck equations for probability measures on path space and path-distribution dependent SDEs. Discrete Contin. Dyn. Syst. 39, 3017–3035 (2019)
Ren, P., Wang, F.-Y.: Bismut formula for Lions derivative of distribution dependent SDEs and applications. J. Differential Euqations 267, 4745–4777 (2019)
Ren, P., Wang, F.-Y.: Donsker-Varadhan large deviations for path-distribution dependent SPDEs. J. Math. Anal. Appl. 499(1), 32, Paper No. 125000 (2021)
Wang, F.-Y.: Distribution dependent SDEs for Landau type equations. Stoch. Proc. Appl. 128, 595–621 (2018)
Wang, F.-Y.: A new type distribution-dependent SDE for singular nonlinear PDE. J. Evol. Equ. 23(2), 30, Paper No. 35 (2023)
Debussche A.: Ergodicity results for the stochastic Navier–Stokes equations: an introduction. In: Topics in Mathematical Fluid Mechanics, volume 2073 of Lecture Notes in Math, pp. 23–108, Springer, Heidelberg (2013)
Flandoli, F.: Random Perturbation of PDEs and Fluid Dynamic Models, Saint Flour Summer School Lectures 2010. Lecture Notes in Mathematics, vol. 2015. Springer, Berlin (2011)
Breit, D., Feireisl, E., Hofmanová, M.: Stochastically forced compressible fluid flows, De Gruyter Series in Applied and Numerical Mathematics 3, xii+330 (2018)
Kuksin, S., Shirikyan, A.: Mathematics of two-dimensional turbulence, Cambridge University Press, Cambridge, xvi+320 (2012)
Fedrizzi, E., Flandoli, F.: Noise prevents singularities in linear transport equations. J. Funct. Anal. 264(6), 1329–1354 (2013)
Fedrizzi, E., Neves, W., Olivera C.: On a class of stochastic transport equations for \(L^2_{loc}\) vector fields. Ann. Sc. Norm. Super. Pisa Cl. Sci. XVIII(5), 397–419 (2018)
Flandoli, F., Gubinelli, M., Priola, E.: Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180(1), 1–53 (2010)
Mollinedo, D., Olivera, C.: Stochastic continuity equation with nonsmooth velocity. Ann. Mat. Pura Appl. 196(4), 1669–1684 (2017)
Alonso-Orán, D., Bethencourt de León, A., Takao, S.: The Burgers’ equation with stochastic transport: Shock formation, local and global existence of smooth solutions. NoDEA Nonlinear Differential Equations Appl. 26(6), Paper No. 57, 33 (2019)
Miao, Y., Rohde, C., Tang, H.: Well-posedness for a stochastic Camassa-Holm type equation with higher order nonlinearities. Stoch. Partial Differ. Equ. Anal. Comput. (2023). https://doi.org/10.1007/s40072-023-00291-z
Tang, H.: On the stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noise. J. Funct. Anal. 285(9), 61, Paper No. 110075 (2023)
Neves, W., Olivera, C.: Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition. NoDEA Nonlinear Differential Equations Appl. 22, 1247–1258 (2015)
Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions, volume 152 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition (2014)
Krylov, N.V., Rozovskiĭ, B.L.: Stochastic evolution equations. In: Current Problems in Mathematics, Vol. 14 (Russian), pp. 71–147, 256. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow (1979)
Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905. Springer, Berlin (2007)
Pardoux, E.: Sur des equations aux dérivées partielles stochastiques monotones. C. R. Acad. Sci. 275, A101–A103 (1972)
Da Prato, G.: Kolmogorov Equations for Stochastic PDEs. Birkhäuser Verlag, Basel, Adv. Courses Math. CRM Barcelona (2004)
Glatt-Holtz, N., Vicol, V.: Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise. Ann. Probab. 42(1), 80–145 (2014)
Röckner, M., Zhu, R., Zhu, X.: Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise. Stoch. Proc. Appl. 124, 1974–2002 (2014)
Tang, H.: On the pathwise solutions to the Camassa-Holm equation with multiplicative noise. SIAM J. Math. Anal. 50(1), 1322–1366 (2018)
Li, J., Liu, H., Tang, H.: Stochastic MHD equations with fractional kinematic dissipation and partial magnetic diffusion in \({\mathbb{R} }^2\). Stochastic Process. Appl. 135, 139–182 (2021)
Tang, H., Wang, Z.: Strong solutions to nonlinear stochastic aggregation-diffusion equations. Commun. Contemp. Math. (2023). https://doi.org/10.1142/S0219199722500730
Flandoli, F., Gubinelli, M., Priola, E.: Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations. Stochastic Process. Appl. 121, 1445–1463 (2011)
Khas’minskii, R.Z.: Stability of systems of differential equations under random perturbations of their parameters. (Russian). Izdat. Nauka, Moscow (1969)
Brzeźniak, Z., Maslowski, B., Seidler, J.: Stochastic nonlinear beam equations. Probab. Theory Related Fields 132(1), 119–149 (2005)
Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 4(1), 47–66 (1981/82)
Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664 (1993)
Holm, D.D., Staley, M.F.: Nonlinear balance and exchange of stability of dynamics of solitons, peakons, ramps/cliffs and leftons in a \(1+1\) nonlinear evolutionary PDE. Phys. Lett. A 308(5–6), 437–444 (2003)
Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 192(1), 165–186 (2009)
Dullin, H.R., Gottwald, G.A., Holm, D.D.: An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87, 194501 (2001)
Gui, G., Liu, Y., Sun, J.: A nonlocal shallow-water model arising from the full water waves with the Coriolis effect. J. Math. Fluid Mech. 21(2), Paper No. 27, 29 (2019)
Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181(2), 229–243 (1998)
Constantin, A., Escher, J.: Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Comm. Pure Appl. Math. 51(5), 475–504 (1998)
Zhu, M., Liu, Y., Mi, Y.: Wave-breaking phenomena and persistence properties for the nonlocal rotation-Camassa-Holm equation. Ann. Mat. Pura Appl. 199, 355–377 (2020)
Alonso-Orán, D., Rohde, C., Tang, H.: A local-in-time theory for singular SDEs with applications to fluid models with transport noise. J. Nonlinear Sci. 31(6), Paper No. 98, 55 (2021)
Chen, Y., Duan, J., Gao, H.: Global well-posedness of the stochastic Camassa-Holm equation. Commun. Math. Sci. 19(3), 607–627 (2021)
Galimberti, L., Holden H., Karlsen, K.H., Pang, P.H.C.: Global existence of dissipative solutions to the Camassa–Holm equation with transport noise. arXiv:2211.07046 (2022)
Holden, H., Karlsen, K.H., Pang, P.H.C.: Global well-posedness of the viscous Camassa-Holm equation with gradient noise. Discrete Contin. Dyn. Syst. 43(2), 568–618 (2023)
Tang, H., Yang, A.: Noise effects in some stochastic evolution equations: global existence and dependence on initial data. Ann. Inst. Henri Poincaré Probab. Stat. 59(1), 378–410 (2023)
Ren, P.: Singular McKean-Vlasov SDEs: well-posedness, regularities and Wang’s Harnack inequality. Stoch. Proc. Appl. 156, 291–311 (2023)
Huang, X., Wang, F.-Y.: Distribution dependent SDEs with singular coefficients. Stoch. Proc. Appl. 129, 4747–4770 (2019)
Kurtz, T.: Weak and strong solutions of general stochastic models. Electron. Commun. Probab. 19(58), 16 (2014)
Himonas, A., Kenig, C.: Non-uniform dependence on initial data for the CH equation on the line. Diff. Integr. Eqns. 22, 201–224 (2009)
Kenig, C.E., Ponce, G., Vega, L.: Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Amer. Math. Soc. 4(2), 323–347 (1991)
Acknowledgements
We would like to thank the referee for helpful comments as well as Mr. Wei Hong for careful check and corrections.
Funding
Open access funding provided by University of Oslo (incl Oslo University Hospital). Feng-Yu Wang is supported in part by the National Key R &D Program of China (No. 2022YFA1006000, 2020YFA0712900) and NNSFC (11831014, 11921001). Panpan Ren is supported by NNSFC (12301180) and Research Center for Nonlinear Analysis at The Hong Kong Polytechnic University. The major part of this work was carried out when Panpan Ren and Hao Tang were supported by the Alexander von Humboldt Foundation.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare that they have no conflict of interest and data sharing is not applicable to this article since no datasets were generated or analyzed during the current study.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Ren, P., Tang, H. & Wang, FY. Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations. Potential Anal (2024). https://doi.org/10.1007/s11118-023-10113-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11118-023-10113-5
Keywords
- Distribution-Path Dependent Nonlinear SPDEs
- Stochastic transport type equation
- Stochastic Camassa-Holm type equation