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Complex Burgers Equation: A Probabilistic Perspective

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Sojourns in Probability Theory and Statistical Physics - I

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 298))

Abstract

In 1997 Yves LeJan and Alain-Sol Sznitman provided a probabilistic gateway in the form of a stochastic cascade model for the treatment of 3d incompressible, Navier–Stokes equations in free space. The equations themselves are noteworthy for the inherent mathematical challenges that they pose to proving existence, uniqueness and regularity of solutions. The main goal of the present article is to illustrate and explore the LeJan–Sznitman cascade in the context of a simpler quasi-linear pde, namely the complex Burgers equation. In addition to providing some unexpected results about these equations, consideration of mean-field models suggests analysis of branching random walks having naturally imposed time delays.

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Notes

  1. 1.

    The reference to “doodle” is deliberate. Once, when asked how he discovered his central limit theorem for associated random variables given in [43], Chuck replied that he had expected for some time that a central limit theorem should be possible for ferromagnetic Ising models at high temperatures as a consequence of correlation decay: “Then, one day I was doodling with the FKG inequalities and out popped just the right correlation inequalities for the characteristic function of the magnetization”. This would prove that central limit theorem.

  2. 2.

    The mean-field models for the Navier–Stokes equation, on the other hand, involve parameters \(\beta >1\) as well; see [15] in this regard.

  3. 3.

    An unfortunate typo occurs in the Appendix to [13] in which the explosion event should be denoted \([\zeta <\infty ]\), not \([\zeta =\infty ]\).

  4. 4.

    Another closely related Markov evolution that takes place in the sequence space \(\ell _1\) is given in [14].

  5. 5.

    The explosion problem for the self-similar LJS-cascade has been resolved in [16], where it has been shown that indeed explosion occurs.

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Acknowledgments

This work was partially supported by grants DMS-1408947, DMS-1408939, DMS-1211413, and DMS-1516487 from the National Science Foundation.

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Authors and Affiliations

  1. Department of Mathematics, Oregon State University, Corvallis, OR, 97331, USA

    Radu Dascaliuc, Enrique Thomann & Edward C. Waymire

  2. Department of Mathematics, New Mexico State University, Las Cruces, NM, 88003, USA

    Nicholas Michalowski

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  1. Radu Dascaliuc
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Correspondence to Edward C. Waymire .

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  1. NYU Shanghai, Shanghai, China

    Vladas Sidoravicius

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Dascaliuc, R., Michalowski, N., Thomann, E., Waymire, E.C. (2019). Complex Burgers Equation: A Probabilistic Perspective. In: Sidoravicius, V. (eds) Sojourns in Probability Theory and Statistical Physics - I. Springer Proceedings in Mathematics & Statistics, vol 298. Springer, Singapore. https://doi.org/10.1007/978-981-15-0294-1_6

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