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Random initial conditions for semi-linear PDEs

Part of: Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  29 January 2019

Dirk Blömker ,
Giuseppe Cannizzaro  and
Marco Romito
Dirk Blömker
Affiliation:
Institüt fur Mathematik, Universität Augsburg, D-86135 Augsburg, Germany (dirk.bloemker@math.uni-augsburg.de)
Giuseppe Cannizzaro
Affiliation:
Imperial College London, Department of Mathematics, 180 Queen's Gate, LondonSW7 2AZ, UK (g.cannizzaro@imperial.ac.uk)
Marco Romito
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, I–56127Pisa, Italia (marco.romito@unipi.it); URL: http://people.dm.unipi.it/romito
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Abstract

We analyse the effect of random initial conditions on the local well-posedness of semi-linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework.

In particular, in some cases, stochastic initial conditions extend the validity of the fixed-point argument to larger spaces than deterministic initial conditions would allow, but in general, it is never possible to go beyond the threshold that is predicted by critical scaling, as in our general class of equations we are not exploiting any special structure present in the equation.

We also give a specific example where the level of regularity for the fixed-point argument reached by random initial conditions is not yet critical, but it is already sharp in the sense that we find infinitely many random initial conditions of slightly lower regularity, where there is no solution at all. Thus criticality cannot be reached even by random initial conditions.

The existence and uniqueness in a critical space is always delicate, but we can consider the Burgers equation in logarithmically sub-critical spaces, where existence and uniqueness hold, and again random initial conditions allow to extend the validity to spaces of lower regularity which are still logarithmically sub-critical.

Keywords

Random initial conditionsemilinar PDEs

MSC classification

Primary: 35R60: Partial differential equations with randomness, stochastic partial differential equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1533 - 1565
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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References

1Árpád, B., Oh, T. and Pocovnicu, O.. Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ3, https://arxiv.org/abs/1709.01910 arXiv:1709.01910, 2017.Google Scholar
2Blömker, D. and Romito, M.. Local existence and uniqueness in the largest critical space for a surface growth model. NoDEA Nonlinear Differential Equations Appl. 19 (2012), 365–381.10.1007/s00030-011-0133-2CrossRefGoogle Scholar
3Blömker, D. and Romito, M.. Stochastic PDEs and Lack of Regularity. Jahresber. Dtsch. Math.-Ver. 117 (2015), 233–286.10.1365/s13291-015-0123-0CrossRefGoogle Scholar
4Bourgain, J.. Periodic nonlinear Schrödinger equation and invariant measures. Comm. Math. Phys. 166 (1994), 1–26.10.1007/BF02099299CrossRefGoogle Scholar
5Bourgain, J.. Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. Comm. Math. Phys. 176 (1996), 421–445.10.1007/BF02099556CrossRefGoogle Scholar
6Burq, N. and Tzvetkov, N.. Random data Cauchy theory for supercritical wave equations I. Local theory. Invent. Math. 173 (2008a), 449–475.CrossRefGoogle Scholar
7Burq, N. and Tzvetkov, N.. Random data Cauchy theory for supercritical wave equations II. A global existence result. Invent. Math. 173 (2008b), 477–496.10.1007/s00222-008-0123-0CrossRefGoogle Scholar
8Da Prato, G. and Debussche, A.. Two-dimensional Navier-Stokes equations driven by a space-time white noise. J. Funct. Anal. 196 (2002), 180–210.10.1006/jfan.2002.3919CrossRefGoogle Scholar
9Da Prato, G. and Debussche, A.. Strong solutions to the stochastic quantization equations. Ann. Probab. 31 (2003), 1900–1916.Google Scholar
10Da Prato, G., Debussche, A., Tubaro, L.. A modified Kardar-Parisi-Zhang model. Electron. Comm. Probab. 12 (2007), 442–453.10.1214/ECP.v12-1333CrossRefGoogle Scholar
11Dispersive Wiki contributors, Critical, Dispersive PDE Wiki https://dispersivewiki.org/DispersiveWiki/index.php?title=Critical&oldid=4367 [accessed January 16, 2019].Google Scholar
12Duplantier, B. and Sheffield, S.. Liouville quantum gravity and KPZ. Invent. Math. 185 (2011), 333–393.CrossRefGoogle Scholar
13Gubinelli, M., Imkeller, P. and Perkowski, N.. Paracontrolled distributions and singular PDEs. Forum Math. Pi 3 (2015), e6, 75.10.1017/fmp.2015.2CrossRefGoogle Scholar
14Hairer, M.. Solving the KPZ equation. Ann. of Math. (2) 178 (2013), 559–664.10.4007/annals.2013.178.2.4CrossRefGoogle Scholar
15Hairer, M.. A theory of regularity structures. Invent. Math. 198 (2014), 269–504.10.1007/s00222-014-0505-4CrossRefGoogle Scholar
16Kahane, J.-P.. Some random series of functions, 2nd edn, Cambridge Studies in Advanced Mathematics, vol. 5 (Cambridge: Cambridge University Press, 1985.Google Scholar
17Mourrat, J.-C., Weber, H. and Xu, W.. Construction of diagrams for pedestrians. From particle systems to partial differential equations, Springer Proc. Math. Stat., vol. 209, pp. 1–46 (Cham: Springer, 2017.Google Scholar
18Nahmod, A. R., Pavlović, N. and Staffilani, G.. Almost sure existence of global weak solutions for supercritical Navier-Stokes equations. SIAM J. Math. Anal. 45 (2013), 3431–3452.10.1137/120882184CrossRefGoogle Scholar
19Nelson, E.. The free Markoff field. J. Functional Analysis 12 (1973), 211–227.10.1016/0022-1236(73)90025-6CrossRefGoogle Scholar
20Oh, T. and Tzvetkov, N.. Quasi-invariant Gaussian measures for the two-dimensional defocusing cubic nonlinear wave equation, https://arxiv.org/abs/1703.10718 arXiv:1703.10718, 2017.Google Scholar
21Simon, B.. The P(ϕ)2 Euclidean (quantum) field theory. Princeton Series in Physics (Princeton, N.J., Princeton University Press, 1974).Google Scholar
22Tzvetkov, N.. On Hamiltonian partial differential equations with random initial conditions, Lecture Notes for the CIME-EMS summer school in applied mathematics Singular random dynamics, notes available at https://arxiv.org/abs/1704.01191 arXiv:1704.01191, 2016.Google Scholar