Symmetry breaking and uniqueness for the incompressible Navier-Stokes equations

Dascaliuc, Radu; Michalowski, Nicholas; Thomann, Enrique; Waymire, Edward C.

doi:10.1063/1.4913236

Cited by 10 publications

(19 citation statements)

References 33 publications

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“…In particular, a basic question about the branching structure becomes that of stochastic explosion: does the stochastic cascade generate infinitely many branches by a finite time t > 0? An answer to this question directly affects existence and uniqueness properties of solutions to (1.1); see [12,17]. The purpose of this paper is twofold: first, to identify a general stochastic structure which is flexible enough to accommodate a variety of similar models, and second, to analyze the issue of explosion.…”

Section: Background Motivation and Definition Of Doubly Stochastic Yu...mentioning

confidence: 99%

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Doubly Stochastic Yule Cascades (Part I): The explosion problem in the time-reversible case

Dascaliuc¹,

Pham²,

Thomann³

et al. 2021

Preprint

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Motivated by the probabilistic methods for nonlinear differential equations introduced by McKean (1975) for the Kolmogorov-Petrovski-Piskunov (KPP) equation, and by Le Jan and Sznitman (1997) for the incompressible Navier-Stokes equations, we identify a new class of stochastic cascade models, referred to as doubly stochastic Yule cascades. We establish nonexplosion criteria under the assumption that the randomization of Yule intensities from generation to generation is by an ergodic time-reversible Markov chain. In addition to the cascade models that arise in the analysis of certain deterministic nonlinear differential equations, this model includes the multiplicative branching random walks, the branching Markov processes, and the stochastic generalizations of the percolation and/or cell aging models introduced by Aldous and Shields (1988) and independently by Athreya (1985).

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Section: Background Motivation and Definition Of Doubly Stochastic Yu...mentioning

confidence: 99%

“…This informal thinking lead to a previous erroneous proof in[12, Prop. 5.1 in the Appendix], although the assertion remains valid as shown in the present paper.…”

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confidence: 99%

Doubly Stochastic Yule Cascades (Part I): The explosion problem in the time-reversible case

Dascaliuc¹,

Pham²,

Thomann³

et al. 2021

Preprint

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“…Considerations of evolutionary processes, to be referred to as delayed Yule processes, arise somewhat naturally in the probabilistic analysis of quasi-linear evolution equations such as incompressible Navier-Stokes equations, and complex Burgers equation by probabilistic methods originating with Le Jan and Sznitman [4]. In particular, considerations of non-uniqueness and/or explosion problems in [1] for this framework prompted the present considerations. However this paper has a purely probabilistic focus and does not depend on such motivations.…”

Section: Introductionmentioning

confidence: 99%

A Delayed Yule Process

Dascaliuc

Michalowski

Thomann

et al. 2017

Proc. Amer. Math. Soc.

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In now classic work, David Kendall (1966) recognized that the Yule process and Poisson process could be related by a (random) time change. Furthermore, he showed that the Yule population size rescaled by its mean has an almost sure exponentially distributed limit as t → ∞. In this note we introduce a class of coupled delayed Yule processes parameterized by 0 < α ≤ 1 that includes the Poisson process at α = 1/2. Moreover we extend Kendall's limit theorem to include a larger class of positive martingales derived from functionals that gauge the population genealogy. A somewhat surprising connection with the Holley-Liggett smoothing transformation also emerges in this context. Specifically, the latter is exploited to uniquely characterize the moment generating functions of distributions of the limit martingales, generalizing Kendall's mean one exponential limit.

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“…Dascaliuc et al (2015) report that while the three dimensional incompressible fluid flow is mathematically described by Navier-Stokes equations, it is still unknown whether smooth initial data lead to the existence of unique smooth solutions that are valid for all times. Toward the solution of this problem, they provide a new mathematical approach that establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades.…”

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confidence: 99%

“…The set of three papers by Dascaliuc et al (2015), Ercan and Kavvas (2015), and Haltas and Ulusoy (2015) explores various scaling issues in nonlinear hydrodynamics. Dascaliuc et al (2015) report that while the three dimensional incompressible fluid flow is mathematically described by Navier-Stokes equations, it is still unknown whether smooth initial data lead to the existence of unique smooth solutions that are valid for all times.…”

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confidence: 99%