Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Martingale Suitable Weak Solutions of $3$-D Stochastic Navier-Stokes Equations with Vorticity Bounds (2402.11482v3)

Published 18 Feb 2024 in math.PR, math-ph, math.AP, and math.MP

Abstract: In this paper, we construct martingale suitable weak solutions for $3$-dimensional incompressible stochastic Navier-Stokes equations with generally non-linear noise. In deterministic setting, as widely known, suitable weak solutions'' are Leray-Hopf weak solutions enjoying two different types of local energy inequalities (LEIs). In stochastic setting, we apply the idea ofmartingale solution", avoid transforming to random system, and show new stochastic versions of the two local energy inequalities. In particular, in additive and linear multiplicative noise case, OU-processes and the exponential formulas DO NOT play a role in our formulation of LEIs. This is different to \cite{FR02,Rom10} where the additive noise case is dealt. Also, we successfully apply the concept of a.e. super-martingale'' to describe this local energy behavior. To relate the well-knowndissipative weak solutions" come up with in \cite{DR00}, we derive a local energy equality and extend the concept onto stochastic setting naturally. For further regularity of solutions, we are able to bound the $L\infty\big([0,T];L1(\Omega\times\mathbb T3)\big)$ norm of the vorticity and $L{\frac{4}{3+\delta}}\big(\Omega\times[0,T]\times\mathbb T3\big)$ norm of the gradient of the vorticity, in case that the initial vorticity is a finite regular signed measure.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (45)
  1. Construction of suitable weak solutions of the navier–stokes equations. Contemporary Mathematics, 49:1–18, 2007.
  2. H. Beirão da Veiga. On the construction of suitable weak solutions to the navier–stokes equations via a general approximation theorem. J. Math. Pures Appl., 64:321–334, 1985.
  3. H. Beirão da Veiga. On the suitable weak solutions to the navier–stokes equa- tions in the whole space. J. Math. Pures Appl., 64:77–68, 1985.
  4. L. C. Berselli and S. Spirito. Weak solutions to the navier–stokes equations con- structed by semi-discretization are suitable. Contemporary Mathematics, 666(13):85–97, 2016.
  5. L. C. Berselli and S. Spirito. On the construction of suitable weak solutions to the 3d navier–stokes equations in a bounded domain by an artificial compressibility method. Commun. Contemp. Math., 20(01):1650064(1–16), 2018.
  6. Sharp non-uniqueness of solutions to stochastic navier-stokes equations. SIAM J. Math Anal., page to appear, 2023.
  7. Partial regularity of suitable weak solutions of the navier-stokes equations. Communicationson Pure and Applied Mathematics, XXXV:771–831, 1982.
  8. Sharp nonuniqueness for the Navier-Stokes equations. Invent. Math., 229:987–1054, 05 2022.
  9. Peter Constantin. Navier-stokes equations and area of interfaces. Commun. Math. Phys., 129:241–266, 1990.
  10. G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1992.
  11. L3,∞subscript𝐿3{L}_{3,\infty}italic_L start_POSTSUBSCRIPT 3 , ∞ end_POSTSUBSCRIPT-solutions of Navier-Stokes equations and backward uniqueness. Uspekhi Mat. Nauk, 58(2(350)):3–44, 2003.
  12. Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Relat. Fields, 102:367–391, 1995.
  13. The initial value problem for the Navier-Stokes equations with data in Lpsuperscript𝐿𝑝{L}^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT. Arch. Rational Mech. Anal., 45:222–240, 1972.
  14. Unicité dans L3⁢(ℝ3)superscript𝐿3superscriptℝ3{L}^{3}(\mathbb{R}^{3})italic_L start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) et d’autres espaces fonctionnels limites pour Navier-Stokes. Rev. Mat. Iberoamericana, 16(3):605–667, 2000.
  15. Partial regularity for the stochastic navier-stokes equations. Trans. Amer. Math. Soc., 354(6):2207–2241, 02 2002.
  16. Markov selections for the 3D stochastic Navier–Stokes equations. Probability Theory and Related Fields, 140(3-4):407–458, 2008.
  17. D. Gatarek. A note on nonlinear stochastic equations in hilbert space. Statist. and Probab. Lett., 17:387–394, 1993.
  18. D. Gatarek and B. Goldys. On weak solutions of stochastic equations in hilbert spaces. Stochastic, 46:41–51, 1994.
  19. J.-L. Guermond. Finite-element-based faedo–galerkin weak solutions to the navier–stokes equations in the three-dimensional torus are suitable. J. Math. Pures Appl., 85:451–464, 2006.
  20. J.-L. Guermond. Faedo–galerkin weak solutions of the navier–stokes equa- tions with dirichlet boundary conditions are suitable. J. Math. Pures Appl., 88:87–106, 2007.
  21. E. Hopf. Über die anfangswertaufgabe für die hydrodynamischen grundgleichungen. erhard schmidt zu seinem 75. geburtstag gewidmet. Math. Nachr., 4(1-6):213–231, 1951.
  22. On ill- and well-posedness of dissipative martingale solutions to stochastic 3d euler equations. Communications on Pure and Applied Mathematics, 75(11):2446–2510, Nov. 2022.
  23. Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier–Stokes equations: existence and non-uniqueness. Ann. Probab., 51(2):524–579, March 2023.
  24. Non-uniqueness in law of stochastic 3D Navier–Stokes equations. J. Eur. Math. Soc., page to appear, Jun. 2023.
  25. Tosio Kato. Strong Lpsuperscript𝐿𝑝{L}^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-solutions of the Navier-Stokes equation in 𝐑𝐦superscript𝐑𝐦\bf{R}^{m}bold_R start_POSTSUPERSCRIPT bold_m end_POSTSUPERSCRIPT, with applications to weak solutions. Math. Z., 187(4):471–480, 1984.
  26. R. V. Kohn. Partial regularity and the navier–stokes equations. Lecture Notes in Num. Appl. Anal., 5:101–118, 1982.
  27. I. Kukavica and Y. Pei. An estimate on the parabolic fractal dimension of the singular set for solutions of the navier–stokes system. Nonlinearity, 25:2775– 2783, 2012.
  28. I. Kukavica. The fractal dimension of the singular set for solutions of the navier–stokes system. Nonlinearity, 22:2889–2900, 2009.
  29. I. Kukavica. Partial regularity results for solutions of the Navier–Stokes system. Number 121–145 in Partial differential equations and fluid mechanics. Cambridge University Press, Cambridge, 2009.
  30. O. A. Ladyženskaja. Uniqueness and smoothness of generalized solutions of Navier-Stokes equations. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 5:169–185, 1967.
  31. J. Leray. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math, 63(1):193–248, 1934.
  32. Fanghua Lin. A new proof of the caffarelli-kohn-nirenberg theorem. Communications on Pure and Applied Mathematics, LI:0241–0257, 1998.
  33. P.-L. Lions. Mathematical topics in fluid mechanics, volume Volume 1: Incompressible models. Oxford University Press, Oxford, 1994.
  34. P.-L. Lions and N. Masmoudi. Uniqueness of mild solutions of the Navier-Stokes system in LNsuperscript𝐿𝑁{L}^{N}italic_L start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT. Comm. Partial Differential Equations, 26(11-12):2211–2226, 2001.
  35. Stochastic Partial Differential Equations: An Introduction. Universtext. Springer Science and Business Media, 2015.
  36. On partial regularity of suitable weak solutions to the three-dimensional navier–stokes equations. J. Math. Fluid Mech., 1:356–387, 1999.
  37. Mefivier. Stochastic partial differential equations in infinite dimensional spaces. Quaderni Scuola Normale Superiore, Pisa,, 1988.
  38. G. Prodi. Un teorema di unicità per le equazioni di navier-stokes. Ann. Mat. Pura ed Appl., 48(1):173–182, 1959.
  39. M. Romito. Existence of martingale and stationary suitable weak solutions for a stochastic navier-stokes system. Stochastics: An International Journal of Probability and Stochastics Processes, 82(3):327–337, 06 2010.
  40. Vladimir Scheffer. Hausdorff measure and the navier-stokes equations. Commun. math. Phys., 55:97–112, 1977.
  41. Vladimir Scheffer. The navier-stokes equations on a bounded domain. Commun.Math. Phys., 73:1–42, 1980.
  42. Vladimir Scheffer. Nearly one dimensional singularities o f solutions to the navier-stokes inequality. Commun.Math. Phys., 110:525–551, 1987.
  43. J. Serrin. On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rational Mech. Anal., 9:187–195, 1962.
  44. A. Vasseur. A new proof of partial regularity of solutions to navier–stokes equations. Nonlinear Diff. Equ. Appl., 14:753–785, 2007.
  45. Viot. Solutions faibtes d’equations aux derivees partielles stochastiques non lineaires. These de Doctorat, Paris VI, 1976.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets