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An Introduction to Singular SPDEs

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Book cover Stochastic Partial Differential Equations and Related Fields (SPDERF 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 229))

Abstract

We review recent results on the analysis of singular stochastic partial differential equations in the language of paracontrolled distributions.

Financial support by the DFG via CRC 1060 (MG) and Research Unit FOR 2402 (NP) and is gratefully acknowledged.

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Notes

  1. 1.

    Possibly T is quite small because when setting up the Picard iteration we pick up a superlinear estimate so at this point we cannot exclude the possibility that the solution blows up in finite time. To prove existence for all times we have to make use of the sign of the nonlinearity \(-\varphi ^3\) in (4), see [35].

  2. 2.

    This is not exactly true, we also need some time regularity of f but for simplicity we omit this in the discussion.

  3. 3.

    We pick up a quadratic estimate from the paralinearisation result (6) because it is based on a second order Taylor expansion, and therefore we cannot exclude that the solutions blows up in finite time. But given an a priori bound on the \(L^\infty \) norm of u one can show that \((u,v^\sharp )\) stays bounded in \(C_T\mathscr {C}^{1-} \times C_T \mathscr {C}^{2-}\), see [21], and such an a priori bound can for certain nonlinearities G be derived from the maximum principle, see [12].

  4. 4.

    Strictly speaking it is not possible to invert the Laplace operator and we have to shift it and consider \((1 - \varDelta )^{- 1}\) instead, but for simplicity we ignore this here.

  5. 5.

    Besov spaces on \(\mathbb {R}\) are defined exactly in the same way as on \(\mathbb {T}\) and they have essentially the same properties.

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Authors and Affiliations

  1. Hausdorff Center for Mathematics & Institute for Applied Mathematics, Universität Bonn, Bonn, Germany

    Massimiliano Gubinelli

  2. Institut für Mathematik, Humboldt–Universität zu Berlin, Berlin, Germany

    Nicolas Perkowski

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  1. Massimiliano Gubinelli
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Correspondence to Massimiliano Gubinelli .

Editor information

Editors and Affiliations

  1. Institute for Applied Mathematics, University of Bonn, Bonn, Germany

    Andreas Eberle

  2. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Martin Grothaus

  3. Faculty of Mathematics, Bielefeld University, Bielefeld, Germany

    Walter Hoh

  4. Faculty of Mathematics, Bielefeld University, Bielefeld, Germany

    Moritz Kassmann

  5. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Wilhelm Stannat

  6. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, Korea (Republic of)

    Gerald Trutnau

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Gubinelli, M., Perkowski, N. (2018). An Introduction to Singular SPDEs. In: Eberle, A., Grothaus, M., Hoh, W., Kassmann, M., Stannat, W., Trutnau, G. (eds) Stochastic Partial Differential Equations and Related Fields. SPDERF 2016. Springer Proceedings in Mathematics & Statistics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-319-74929-7_4

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