On Le Jan-Sznitman’s stochastic approach to the Navier-Stokes equations

Dascaliuc, Radu; Pham, Tuan; Thomann, Enrique

doi:10.1090/tran/8974

On Le Jan-Sznitman’s stochastic approach to the Navier-Stokes equations
HTML articles powered by AMS MathViewer

by Radu Dascaliuc, Tuan N. Pham and Enrique Thomann;

Trans. Amer. Math. Soc. 377 (2024), 2335-2365

DOI: https://doi.org/10.1090/tran/8974

Published electronically: February 9, 2024

HTML | PDF | Request permission

Abstract:

The paper explores the symbiotic relation between the Navier-Stokes equations and the associated stochastic cascades. Specifically, we examine how some well-known existence and uniqueness results for the Navier-Stokes equations can inform about the probabilistic features of the associated stochastic cascades, and how some probabilistic features of the stochastic cascades can, in turn, inform about the existence and uniqueness (or the lack thereof) of solutions. Our method of incorporating the stochastic explosion gives a simpler and more natural method to construct the solution compared to the original construction by Le Jan and Sznitman. This new stochastic construction is then used to show the finite-time blowup, non-existence of minimal blowup data, and non-uniqueness of the initial value problem for the Montgomery-Smith equation. We exploit symmetry properties inherent in our construction to give a simple proof of the global well-posedness results for small initial data in scale-critical Fourier-Besov spaces. We also obtain the pointwise convergence of the Picard iteration associated with the Fourier-transformed Navier-Stokes equations.

References

Hajer Bahouri, Jean-Yves Chemin, and Raphaël Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. MR 2768550, DOI 10.1007/978-3-642-16830-7
Rabi N. Bhattacharya, Larry Chen, Scott Dobson, Ronald B. Guenther, Chris Orum, Mina Ossiander, Enrique Thomann, and Edward C. Waymire, Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations, Trans. Amer. Math. Soc. 355 (2003), no. 12, 5003–5040. MR 1997593, DOI 10.1090/S0002-9947-03-03246-X
Rabi N. Bhattacharya and Edward C. Waymire, Stochastic processes with applications, Classics in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. Reprint of the 1990 original [ MR1054645]. MR 3396216, DOI 10.1137/1.9780898718997.ch1
D. Blömker, M. Romito, and R. Tribe, A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees, Ann. Inst. H. Poincaré Probab. Statist. 43 (2007), no. 2, 175–192 (English, with English and French summaries). MR 2303118, DOI 10.1016/j.anihpb.2006.02.001
Marco Cannone and Grzegorz Karch, Smooth or singular solutions to the Navier-Stokes system?, J. Differential Equations 197 (2004), no. 2, 247–274., DOI 10.1016/j.jde.2003.10.003
Marco Cannone and Gang Wu, Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces, Nonlinear Anal. 75 (2012), no. 9, 3754–3760., DOI 10.1016/j.na.2012.01.029
Kung-Ching Chang, Wei Yue Ding, and Rugang Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geom. 36 (1992), no. 2, 507–515. MR 1180392
Radu Dascaliuc, Nicholas Michalowski, Enrique Thomann, and Edward C. Waymire, Symmetry breaking and uniqueness for the incompressible Navier-Stokes equations, Chaos 25 (2015), no. 7, 075402, 16. MR 3405857, DOI 10.1063/1.4913236
Radu Dascaliuc, Tuan N. Pham, Enrique Thomann, and Edward C. Waymire, Doubly stochastic Yule cascades (Part I): The explosion problem in the time-reversible case, J. Funct. Anal. 284 (2023), no. 1, Paper No. 109722, 25. MR 4492598, DOI 10.1016/j.jfa.2022.109722
Radu Dascaliuc, Tuan N Pham, Enrique Thomann, and Edward C Waymire, Doubly Stochastic Yule Cascades (Part II): The explosion problem in the non-reversible case, Ann. Inst. H. Poincaré Probab. Statist. (to appear).
Radu Dascaliuc, Enrique A. Thomann, and Edward C. Waymire, Stochastic explosion and non-uniqueness for $\alpha$-Riccati equation, J. Math. Anal. Appl. 476 (2019), no. 1, 53–85. MR 3944418, DOI 10.1016/j.jmaa.2018.11.064
E. B. Fabes, B. F. Jones, and N. M. Rivière, The initial value problem for the Navier-Stokes equations with data in $L^{p}$, Arch. Rational Mech. Anal. 45 (1972), 222–240. MR 316915, DOI 10.1007/BF00281533
Gerald B. Folland, Real analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. Modern techniques and their applications; A Wiley-Interscience Publication. MR 767633
Isabelle Gallagher, Gabriel S. Koch, and Fabrice Planchon, A profile decomposition approach to the $L^\infty _t(L^3_x)$ Navier-Stokes regularity criterion, Math. Ann. 355 (2013), no. 4, 1527–1559. MR 3037023, DOI 10.1007/s00208-012-0830-0
C. S. Herz, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms, J. Math. Mech. 18 (1968/69), 283–323.
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, RI, 1957. rev. ed. MR 89373
Hao Jia and Vladimír Šverák, Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions, Invent. Math. 196 (2014), no. 1, 233–265. MR 3179576, DOI 10.1007/s00222-013-0468-x
Tosio Kato, Strong $L^{p}$-solutions of the Navier-Stokes equation in $\textbf {R}^{m}$, with applications to weak solutions, Math. Z. 187 (1984), no. 4, 471–480. MR 760047, DOI 10.1007/BF01174182
Herbert Koch and Daniel Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. 157 (2001), no. 1, 22–35. MR 1808843, DOI 10.1006/aima.2000.1937
PawełKonieczny and Tsuyoshi Yoneda, On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations, J. Differential Equations 250 (2011), no. 10, 3859–3873. MR 2774071, DOI 10.1016/j.jde.2011.01.003
Marek Kuczma, An introduction to the theory of functional equations and inequalities, 2nd ed., Birkhäuser Verlag, Basel, 2009. Cauchy’s equation and Jensen’s inequality; Edited and with a preface by Attila Gilányi. MR 2467621, DOI 10.1007/978-3-7643-8749-5
Y. Le Jan and A. S. Sznitman, Stochastic cascades and $3$-dimensional Navier-Stokes equations, Probab. Theory Related Fields 109 (1997), no. 3, 343–366. MR 1481125, DOI 10.1007/s004400050135
Zhen Lei and Fanghua Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math. 64 (2011), no. 9, 1297–1304. MR 2839302, DOI 10.1002/cpa.20361
P. G. Lemarié-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR 1938147, DOI 10.1201/9781420035674
Pierre Gilles Lemarié-Rieusset, The Navier-Stokes problem in the 21st century, CRC Press, Boca Raton, FL, 2016. MR 3469428, DOI 10.1201/b19556
Jean Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248 (French). MR 1555394, DOI 10.1007/BF02547354
Jingyue Li and Xiaoxin Zheng, The well-posedness of the incompressible magnetohydro dynamic equations in the framework of Fourier-Herz space, J. Differential Equations 263 (2017), no. 6, 3419–3459. MR 3659367, DOI 10.1016/j.jde.2017.04.027
H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math. 28 (1975), no. 3, 323–331. MR 400428, DOI 10.1002/cpa.3160280302
Stephen Montgomery-Smith, Finite time blow up for a Navier-Stokes like equation, Proc. Amer. Math. Soc. 129 (2001), no. 10, 3025–3029. MR 1840108, DOI 10.1090/S0002-9939-01-06062-2
John Christopher Orum, Stochastic cascades and 2D Fourier Navier-Stokes equations, Lectures on Multiscale and Multiplicative Processes in Fluid Flows, 2002, pp. 99–107.
John Christopher Orum, Branching processes and partial differential equations, ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–Oregon State University. MR 2706194
John Christospher Orum and Mina Ossiander, Exponent bounds for a convolution inequality in Euclidean space with applications to the Navier-Stokes equations, Proc. Amer. Math. Soc. 141 (2013), no. 11, 3883–3897., DOI 10.1090/S0002-9939-2013-11662-X
Tuan Pham, Topics in the regularity theory of the Navier-Stokes equations, Ph.D. Thesis, University of Minnesota, 2018, http://hdl.handle.net/11299/201125.
P. Poláčik and V. Šverák, Zeros of complex caloric functions and singularities of complex viscous Burgers equation, J. Reine Angew. Math. 616 (2008), 205–217. MR 2369491, DOI 10.1515/CRELLE.2008.022
Eugénie Poulon, About the behavior of regular Navier-Stokes solutions near the blow up, Bull. Soc. Math. France 146 (2018), no. 2, 355–390. MR 3933879, DOI 10.24033/bsmf.2760
Pierre Raphaël and Jeremie Szeftel, Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc. 24 (2011), no. 2, 471–546. MR 2748399, DOI 10.1090/S0894-0347-2010-00688-1
W. Rusin and V. Šverák, Minimal initial data for potential Navier-Stokes singularities, J. Funct. Anal. 260 (2011), no. 3, 879–891. MR 2737400, DOI 10.1016/j.jfa.2010.09.009
Yohei Tsutsui, The Navier-Stokes equations and weak Herz spaces, Adv. Differential Equations 16 (2011), no. 11-12, 1049–1085. MR 2858524
R. Vilela Mendes, Stochastic solutions of some nonlinear partial differential equations, Stochastics 81 (2009), no. 3-4, 279–297. MR 2549488, DOI 10.1080/17442500903080389
Weiliang Xiao, Jiecheng Chen, Dashan Fan, and Xuhuan Zhou, Global well-posedness and long time decay of fractional Navier-Stokes equations in Fourier-Besov spaces, Abstr. Appl. Anal. , posted on (2014), Art. ID 463639, 11. MR 3246334, DOI 10.1155/2014/463639