Comptes Rendus
Équations aux dérivées partielles, Probabilités
On the two-dimensional singular stochastic viscous nonlinear wave equations
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1227-1248.

We study the stochastic viscous nonlinear wave equations (SvNLW) on 𝕋 2 , forced by a fractional derivative of the space-time white noise ξ. In particular, we consider SvNLW with the singular additive forcing D 1 2 ξ such that solutions are expected to be merely distributions. By introducing an appropriate renormalization, we prove local well-posedness of SvNLW. By establishing an energy bound via a Yudovich-type argument, we also prove pathwise global well-posedness of the defocusing cubic SvNLW. Lastly, in the defocusing case, we prove almost sure global well-posedness of SvNLW with respect to certain Gaussian random initial data.

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DOI : 10.5802/crmath.377
Classification : 35L71, 60H15
Ruoyuan Liu 1 ; Tadahiro Oh 1

1 School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Ruoyuan Liu; Tadahiro Oh. On the two-dimensional singular stochastic viscous nonlinear wave equations. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1227-1248. doi : 10.5802/crmath.377. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.377/

[1] Sergio Albeverio; Zbigniew Haba; Francesco Russo Trivial solutions for a nonlinear two-space dimensional wave equation perturbed by space-time white noise, Stochastics Stochastics Rep., Volume 56 (1996) no. 1-2, pp. 127-160 | DOI | Zbl

[2] Nikolay Barashkov; Massimiliano Gubinelli A variational method for Φ 3 4 , Duke Math. J., Volume 169 (2020) no. 17, pp. 3339-3415

[3] Árpád Bényi; Tadahiro Oh; Oana Pocovnicu Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, Excursions in harmonic analysis, Volume 4 (Applied and Numerical Harmonic Analysis), Birkhäuser/Springer, 2015, pp. 3-25 | DOI | Zbl

[4] Jean Bourgain Periodic nonlinear Schrödinger equation and invariant measures, Commun. Math. Phys., Volume 166 (1994) no. 1, pp. 1-26 | DOI | Zbl

[5] Jean Bourgain Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys., Volume 176 (1996) no. 2, pp. 421-445 | DOI | Zbl

[6] Jean Bourgain Nonlinear Schrödinger equations, Hyperbolic equations and frequency interactions (IAS/Park City Mathematics Series), Volume 5, American Mathematical Society, 1999, pp. 3-157 | DOI | Zbl

[7] Haïm Brézis; Thierry Gallouet Nonlinear Schrödinger evolution equations, Nonlinear Anal., Theory Methods Appl., Volume 4 (1980) no. 4, pp. 677-681 | DOI | Zbl

[8] Bjoern Bringmann Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics (to appear in J. Eur. Math. Soc.)

[9] David C. Brydges; Gordon Slade Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation, Commun. Math. Phys., Volume 182 (1996) no. 2, pp. 485-504 | DOI | Zbl

[10] Nicolas Burq; Nikolay Tzvetkov Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc., Volume 16 (2014) no. 1, pp. 1-30 | DOI | MR | Zbl

[11] Robert H. Cameron; William T. Martin Transformations of Wiener integrals under translations, Ann. Math., Volume 45 (1944), pp. 386-396 | DOI | MR | Zbl

[12] Giuseppe Da Prato; Arnaud Debussche Strong solutions to the stochastic quantization equations, Ann. Probab., Volume 31 (2003) no. 4, pp. 1900-1916 | MR | Zbl

[13] Giuseppe Da Prato; Luciano Tubaro Wick powers in stochastic PDEs: an introduction, 2006 (Technical Report UTM, 39 pp.) | Zbl

[14] Peter K. Friz; Nicolas B. Victoir Multidimensional stochastic processes as rough paths. Theory and applications, Cambridge Studies in Advanced Mathematics, 120, Cambridge University Press, 2010, xiv+656 pages | DOI | Numdam

[15] James Glimm; Arthur Jaffe Quantum physics. A functional integral point of view, Springer, 1987, xxii+535 pages

[16] Leonard Gross Abstract Wiener spaces, Proc. 5th Berkeley Sym. Math. Stat. Prob. 2, 1965, pp. 31-42

[17] Massimiliano Gubinelli; Herbert Koch; Tadahiro Oh Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity (to appear in J. Eur. Math. Soc.)

[18] Massimiliano Gubinelli; Herbert Koch; Tadahiro Oh Renormalization of the two-dimensional stochastic nonlinear wave equations, Trans. Am. Math. Soc., Volume 370 (2018) no. 10, pp. 7335-7359 | DOI | MR | Zbl

[19] Massimiliano Gubinelli; Herbert Koch; Tadahiro Oh; Leonardo Tolomeo Global dynamics for the two-dimensional stochastic nonlinear wave equations, Int. Math. Res. Not., Volume 2021 (2021), rnab084, 46 pages | DOI | Zbl

[20] Martin Hairer A theory of regularity structures, Invent. Math., Volume 198 (2014) no. 2, pp. 269-504 | DOI | MR | Zbl

[21] Martin Hairer; Marc Daniel Ryser; Hendrik Weber Triviality of the 2D stochastic Allen-Cahn equation, Electron. J. Probab., Volume 17 (2012), 39, 14 pages | MR | Zbl

[22] Jeffrey Kuan; Sunčica Čanić Deterministic ill-posedness and probabilistic well-posedness of the viscous nonlinear wave equation describing fluid-structure interaction, Trans. Am. Math. Soc., Volume 374 (2021) no. 8, pp. 5925-5994 | DOI | MR | Zbl

[23] Jeffrey Kuan; Sunčica Čanić A stochastically perturbed fluid-structure interaction problem modeled by a stochastic viscous wave equation, J. Differ. Equations, Volume 310 (2022), pp. 45-98 | DOI | MR | Zbl

[24] Jeffrey Kuan; Tadahiro Oh; Sunčica Čanić Probabilistic global well-posedness for a viscous nonlinear wave equation modeling fluid-structure interaction, Appl. Anal., Volume 101 (2022) no. 12, pp. 4349-4373 | DOI | MR | Zbl

[25] Hui-Hsiung Kuo Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics, 463, Springer, 1975

[26] Mickaël Latocca Almost sure existence of global solutions for supercritical semilinear wave equations, J. Differ. Equations, Volume 273 (2021), pp. 83-121 | DOI | MR | Zbl

[27] Ruoyuan Liu Global well-posedness of the two-dimensional random viscous nonlinear wave equations, 2022 (https://arxiv.org/abs/2203.15393)

[28] Henry P. McKean Statistical mechanics of nonlinear wave equations. IV. Cubic Schrödinger, Commun. Math. Phys., Volume 168 (1995) no. 3, pp. 479-491 erratum in ibid. 173 (1995), no. 3, p. 675 | DOI | Zbl

[29] Jean-Christophe Mourrat; Hendrik Weber Global well-posedness of the dynamic Φ 4 model in the plane, Ann. Probab., Volume 45 (2017) no. 4, pp. 2398-2476 | MR | Zbl

[30] Jean-Christophe Mourrat; Hendrik Weber; Weijun Xu Construction of Φ 3 4 diagrams for pedestrians, From particle systems to partial differential equations (Springer Monographs in Mathematics), Volume 209, Springer, 2017, pp. 1-46 | MR | Zbl

[31] Tadahiro Oh; Mamoru Okamoto Comparing the stochastic nonlinear wave and heat equations: a case study, Electron. J. Probab., Volume 26 (2021), 9, 44 pages | MR | Zbl

[32] Tadahiro Oh; Mamoru Okamoto; Tristan Robert A remark on triviality for the two-dimensional stochastic nonlinear wave equation, Stochastic Processes Appl., Volume 130 (2020) no. 9, pp. 5838-5864 | MR | Zbl

[33] Tadahiro Oh; Mamoru Okamoto; Leonardo Tolomeo Focusing Φ 3 4 -model with a Hartree-type nonlinearity (2020) (https://arxiv.org/abs/2009.03251)

[34] Tadahiro Oh; Mamoru Okamoto; Leonardo Tolomeo Stochastic quantization of the Φ 3 3 -model (2021) (https://arxiv.org/abs/2108.06777)

[35] Tadahiro Oh; Oana Pocovnicu Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on 3 , J. Math. Pures Appl., Volume 105 (2016) no. 3, pp. 342-366 | MR | Zbl

[36] Tadahiro Oh; Jeremy Quastel On Cameron-Martin theorem and almost sure global existence, Proc. Edinb. Math. Soc., Volume 59 (2016) no. 2, pp. 483-501 | MR | Zbl

[37] Tadahiro Oh; Tristan Robert; Philippe Sosoe; Yuzhao Wang On the two-dimensional hyperbolic stochastic sine-Gordon equation, Stoch. Partial Differ. Equ., Anal. Comput., Volume 9 (2021) no. 1, pp. 1-32 | MR | Zbl

[38] Tadahiro Oh; Tristan Robert; Nikolay Tzvetkov Stochastic nonlinear wave dynamics on compact surfaces (2019) (https://arxiv.org/abs/1904.05277)

[39] Tadahiro Oh; Kihoon Seong; Leonardo Tolomeo A remark on Gibbs measures with log-correlated Gaussian fields (2020) (https://arxiv.org/abs/2012.06729)

[40] Tadahiro Oh; Laurent Thomann A Pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations, Stoch. Partial Differ. Equ., Anal. Comput., Volume 6 (2018) no. 3, pp. 397-445 | Zbl

[41] Tadahiro Oh; Laurent Thomann Invariant Gibbs measures for the 2-d defocusing nonlinear wave equations, Ann. Fac. Sci. Toulouse, Math., Volume 29 (2020) no. 1, pp. 1-26 | MR | Zbl

[42] Barry Simon The P(φ) 2 Euclidean (quantum) field theory, Princeton Series in Physics, Princeton University Press, 1974, xx+392 pages

[43] Leonardo Tolomeo Global well-posedness of the two-dimensional stochastic nonlinear wave equation on an unbounded domain, Ann. Probab., Volume 49 (2021) no. 3, pp. 1402-1426 | MR | Zbl

[44] William J. Trenberth Global well-posedness for the two-dimensional stochastic complex Ginzburg-Landau equation (2019) (https://arxiv.org/abs/1911.09246)

[45] V. Yudovich Non-stationary flows of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz., Volume 3 (1963), pp. 1032-1066 | MR

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