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On the Probabilistic Cauchy Theory for Nonlinear Dispersive PDEs

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Abstract

In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.

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Notes

  1. 1.

    For conciseness, we restrict our attention to the defocusing case in the following.

  2. 2.

    In fact, there are other critical regularities induced by the Galilean invariance for (1.1.1) and the Lorentzian symmetry for (1.1.2) below which the equations are ill-posed; see [25, 42, 51, 56]. We point out, however, that these additional critical regularities are relevant only when the dimension is low and/or the degree p is small. For example, for NLS (1.1.1) with an algebraic nonlinearity (\(p \in 2\mathbb {N}+ 1\)), the critical regularity induced by the Galilean invariance is relevant (i.e., higher than the scaling-critical regularity \(s_\text {crit}\) in (1.1.4)) only for \(d = 1\) and \(p = 3\). For simplicity, we only consider the scaling-critical regularities in the following.

  3. 3.

    Namely, local-in-time existence of unique solutions almost surely with respect to given random initial data.

  4. 4.

    Hereafter, we use Z to denote various normalizing constants so that the resulting measure is a probability measure provided that it makes sense.

  5. 5.

    Here, we added the mass in the exponent to avoid a problem at the zeroth frequency in (1.2.8) below.

  6. 6.

    In the following, we drop the harmless factor of \(2\pi \).

  7. 7.

    When \( d\ge 3\), it is known that the Gibbs measure \(\rho \) can be constructed only for \(d = 3\) and \(p = 3\). In this case, the resulting Gibbs measure \(\rho \) is not absolutely continuous with respect to the Gaussian measure \(\mu _1\). See [2] for the references therein, regarding the construction of the Gibbs measure (the \(\varPhi ^4_3\) measure) in the real-valued setting.

  8. 8.

    In terms of the Besov spaces \(B^\sigma _{p, \infty }\), \(p < \infty \), we see that \(u_0^\omega \) in (1.2.8) lies almost surely in the critical spaces \(B^0_{2, \infty }\). See [4].

  9. 9.

    In the real-valued setting, we also need to impose that \(g_{ -n} = \overline{g_{n}}\) so that, given a real-valued function \(u_0\), the resulting randomization \(u_0^\omega \) remains real-valued. A similar comment applies to the randomization (1.2.13) introduced for functions on \(\mathbb {R}^d\).

  10. 10.

    See (1.2.16) below for the scaling condition on \(\mathbb {R}^d\).

  11. 11.

    One can choose \(C_\varepsilon = \big (\frac{1}{c} \log \frac{C}{\varepsilon }\big )^\frac{3}{2}\).

  12. 12.

    It is also called the unit-scale randomization in [33].

  13. 13.

    For the local-in-time argument, the defocusing/focusing nature of the equation does not play any role.

  14. 14.

    Needless to say, the solution v is random since it depends on the random linear solution \(z^\omega \). For simplicity, however, we suppress the superscript \(\omega \).

  15. 15.

    Note that, due to the spatial integration in (1.3.17), the largest two frequencies of the dyadic pieces must be comparable.

  16. 16.

    In fact, almost sure norm inflation at \((u_0^\omega , u_1^\omega )\) holds.

  17. 17.

    This argument is not limited to nonlinear dispersive PDEs. For instance, see [45] for an application of this argument in studying a stochastic parabolic PDE.

  18. 18.

    See Theorems 1.1 and 1.2 for such nonlinear smoothing in the probabilistic setting.

  19. 19.

    Namely, the local existence time depends only on the norm of initial data.

  20. 20.

    We also mention a recent work by Dodson–Lührmann–Mendelson [34] that appeared after the completion of this note. The main new idea in [34] is to adapt the functional framework for the derivative NLS and Schrödinger maps to study the perturbed NLS (1.3.14).

  21. 21.

    Here, we assume that \(z_3\) has positive regularity. For example, we know that \(z_3\) has spatial regularity at least \(3\alpha - \frac{9}{2} + 1-\varepsilon \) and hence \(\alpha > \frac{7}{6}\) suffices.

  22. 22.

    Recall the following paraproduct decomposition of the product fg of two functions f and g:

    Since the paraproducts and always make sense as distributions, it suffices to give a meaning to the resonant product in a probabilistic manner.

  23. 23.

    Here, the implicit constant depends on the choice of \(\lambda \) in (1.3.29), which needs to be chosen in terms of N. See [14] for more on this issue.

  24. 24.

    Things are not as simple as stated here due to the unboundedness of the linear solution operator on \(L^r\), \(r\ne 2\), for dispersive equations. In the case of the nonlinear heat equation, however, this heuristics can be seen more clearly. Consider the following nonlinear heat equation on \(\mathbb {R}^d\):

    $$\begin{aligned} \partial _tu = \varDelta u - |u|^{p-1}u \end{aligned}$$
    (1.4.1)

    with initial data \(u_0 \in L^2(\mathbb {R}^d)\). In general, (when \(4 < d(p-1)\) for example), we do not know how to construct a solution with initial data in \(L^2(\mathbb {R}^d)\). By randomizing the initial data \(u_0\) as in (1.2.13), we see that the randomized initial data \(u_0^\omega \) lies almost surely in \(L^r(\mathbb {R}^d)\) for any finite \(r\ge 2\). Then, by taking \(r > \frac{d(p-1)}{2}\), we can apply the deterministic subcritical local well-posedness result in [17] to conclude (rather trivial) almost sure local well-posedness of (1.4.1) with respect to the Wiener randomization \(u_0^\omega \). This is an instance of “making the problem subcritical” by randomization.

  25. 25.

    At this point, we do not know how to apply the theory of regularity structures to study dispersive PDEs, partly because we do not know how to lift the Duhamel integral operator for dispersive PDEs to regularity structures.

  26. 26.

    This is the so-called stochastic convolution.

  27. 27.

    For example, for the subcritical SQE on \(\mathbb {T}^3\), the second-order iterate (an analogue of \(z_3\) in (1.3.21)) gains one derivative as compared to the stochastic convolution.

  28. 28.

    At least on \(\mathbb {T}^d\). On \(\mathbb {R}^d\), there is a limitation on the gain of integrability. See [30, 68].

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Acknowledgements

Á. B. is partially supported by a grant from the Simons Foundation (No. 246024). T. O. was supported by the European Research Council (grant no. 637995 “ProbDynDispEq”). The authors would like to thank Justin Forlano for careful proofreading.

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Authors and Affiliations

  1. Department of Mathematics, Western Washington University, 516 High Street, Bellingham, WA, 98225, USA

    Árpád Bényi

  2. 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, UK

    Tadahiro Oh

  3. Department of Mathematics, Heriot-Watt University and The Maxwell Institute for the Mathematical Sciences, Edinburgh, EH14 4AS, UK

    Oana Pocovnicu

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  1. Árpád Bényi
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  1. Dipartimento di Matematica “G. Peano”, University of Torino, Turin, Italy

    Paolo Boggiatto

  2. Dipartimento di Matematica “G. Peano”, University of Torino, Turin, Italy

    Elena Cordero

  3. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Maurice de Gosson

  4. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Hans G. Feichtinger

  5. Department of Mathematical Sciences, Polytechnic University of Torino, Turin, Italy

    Fabio Nicola

  6. Dipartimento di Matematica “G. Peano”, University of Torino, Turin, Italy

    Alessandro Oliaro

  7. Department of Mathematical Sciences, Polytechnic University of Torino, Turin, Italy

    Anita Tabacco

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Bényi, Á., Oh, T., Pocovnicu, O. (2019). On the Probabilistic Cauchy Theory for Nonlinear Dispersive PDEs. In: Boggiatto, P., et al. Landscapes of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-05210-2_1

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