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arxiv: 2504.14676 · v3 · submitted 2025-04-20 · 🧮 math.AP

Strong well-posedness for a stochastic fluid-rigid body system via stochastic maximal regularity

Felix Brandt , Arnab Roy This is my paper

Pith reviewed 2026-05-22 19:06 UTC · model grok-4.3

classification 🧮 math.AP
keywords stochastic fluid-structure interactionstrong well-posednessmaximal regularityH infinity calculusrigid bodyincompressible Navier-Stokesadditive noisetransport noise
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The pith

A rigid ball immersed in a stochastic incompressible fluid has local strong solutions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a mathematical framework for the motion of a rigid ball inside an incompressible Newtonian fluid that experiences both additive random forces and transport-type noise. It proves that this coupled system admits local strong solutions in three dimensions by merging stochastic maximal L^p-regularity theory with a decoupling technique applied to the fluid-structure operator. The central technical step is establishing that this operator possesses a bounded H^∞-calculus. The result also includes blow-up criteria that limit the maximal existence time of solutions. This supplies the first rigorous strong-solution theory for stochastic fluid-structure interaction problems.

Core claim

We prove local strong well-posedness for the stochastic fluid-rigid body system by showing that the associated fluid-structure operator admits a bounded H^∞-calculus, which together with a decoupling argument allows stochastic maximal L^p-regularity to be applied and yields unique strong solutions up to a possible blow-up time.

What carries the argument

The fluid-structure operator equipped with a bounded H^∞-calculus, which decouples the rigid-body and fluid components so that stochastic maximal regularity theory can be invoked directly on the coupled system.

If this is right

  • Strong solutions exist on a positive time interval whose length depends on the initial data and noise intensity.
  • Blow-up criteria are obtained that characterize the possible loss of regularity at the end of the existence interval.
  • The same maximal-regularity approach covers both additive noise in the equations and transport noise in the fluid velocity.
  • The framework applies to any rigid ball of positive mass and moment of inertia immersed in a viscous incompressible fluid.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the H^∞-calculus bound can be made uniform in time, the local solutions might extend to global ones under small-data assumptions.
  • The decoupling technique could be adapted to other stochastic fluid-structure models such as elastic bodies or multiple particles.
  • The well-posedness result supplies a theoretical foundation for testing numerical schemes that simulate noisy immersed rigid bodies.

Load-bearing premise

The fluid-structure operator must admit a bounded H^∞-calculus.

What would settle it

An explicit initial datum or noise realization for which the fluid-structure operator fails to have a bounded H^∞-calculus and the coupled system loses uniqueness or existence of strong solutions in the expected function spaces.

read the original abstract

We develop a rigorous analytical framework for a coupled stochastic fluid-rigid body system in $\mathbb{R}^3$. The model describes the motion of a rigid ball immersed in an incompressible Newtonian fluid subjected to both additive noise in the fluid and body equations and transport-type noise in the fluid equation. We establish local strong well-posedness of the resulting system by combining stochastic maximal $\mathrm{L}^p$-regularity theory with a decoupling approach for the associated fluid-structure operator. A key step is to prove the boundedness of the $\mathcal{H}^\infty$-calculus for this operator. In addition, we provide blow-up criteria for the maximal existence time of solutions. To our knowledge, this is the first rigorous treatment of strong solutions of stochastic fluid-structure interactions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript develops a rigorous analytical framework for a coupled stochastic fluid-rigid body system in R^3, describing the motion of a rigid ball immersed in an incompressible Newtonian fluid with additive noise in the fluid and body equations together with transport-type noise in the fluid equation. Local strong well-posedness is established by combining stochastic maximal L^p-regularity theory with a decoupling approach for the associated fluid-structure operator; a key step is the proof that this operator admits a bounded H^∞-calculus. Blow-up criteria for the maximal existence time are also derived, and the work is presented as the first rigorous treatment of strong solutions for stochastic fluid-structure interactions.

Significance. If the central claims hold, the result would constitute a meaningful advance in stochastic PDE theory for fluid-structure problems. It extends deterministic fluid-rigid body analyses to a stochastic setting that includes both additive and transport noise, while employing established tools of stochastic maximal regularity and operator theory in a coherent way. The provision of blow-up criteria and the explicit decoupling strategy add practical value for future extensions or numerical work.

major comments (2)
  1. [§4] §4 (fluid-structure operator and decoupling): the proof that the decoupled operator admits a bounded H^∞-calculus is load-bearing for the application of stochastic maximal regularity; the sectoriality constants and the angle of the sector must be shown to be independent of the noise intensity and the ball radius to guarantee the subsequent L^p-estimates remain uniform.
  2. [Theorem 5.2] Theorem 5.2 (local well-posedness): the fixed-point argument for the mild solution relies on the maximal regularity estimate obtained after decoupling; it is not immediately clear how the transport noise term is absorbed into the linear operator without introducing additional commutator terms that could affect the contraction constant.
minor comments (3)
  1. [§2] The notation for the stochastic integrals (Itô vs. Stratonovich) and the precise definition of the transport noise should be stated once in the preliminaries and used consistently thereafter.
  2. [Introduction] Figure 1 (schematic of the geometry) would benefit from an explicit label for the fluid domain and the rigid ball radius to match the notation in the equations.
  3. [§6] A short remark comparing the obtained blow-up criterion with the corresponding deterministic criterion would help readers assess the effect of the noise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We appreciate the positive assessment and the recommendation for minor revision. Below we address the major comments point by point, providing clarifications and indicating the revisions we have made to strengthen the presentation.

read point-by-point responses

  1. Referee: [§4] §4 (fluid-structure operator and decoupling): the proof that the decoupled operator admits a bounded H^∞-calculus is load-bearing for the application of stochastic maximal regularity; the sectoriality constants and the angle of the sector must be shown to be independent of the noise intensity and the ball radius to guarantee the subsequent L^p-estimates remain uniform.

    Authors: We agree that uniformity of the sectoriality constants and angle is essential for the subsequent estimates. In the proof of the bounded H^∞-calculus for the decoupled operator (Section 4), the sectoriality angle is determined by the angle of the underlying Stokes operator on the exterior domain, which depends only on the viscosity coefficient and is independent of the noise intensity. The constants are likewise independent of the ball radius by a scaling argument that reduces the problem to the unit ball without altering the sectoriality parameters. We have added an explicit remark after the statement of the H^∞-calculus result to record this independence and to reference the relevant scaling lemma. revision: yes

  2. Referee: [Theorem 5.2] Theorem 5.2 (local well-posedness): the fixed-point argument for the mild solution relies on the maximal regularity estimate obtained after decoupling; it is not immediately clear how the transport noise term is absorbed into the linear operator without introducing additional commutator terms that could affect the contraction constant.

    Authors: The transport noise is absorbed directly into the principal part of the decoupled linear fluid-structure operator by augmenting the symbol with the transport vector field. The resulting commutator terms that arise when passing to the mild formulation are of lower order in the maximal L^p-regularity space and are estimated by a standard perturbation argument. Their operator norm is controlled by the L^∞ norm of the transport coefficient times a factor that vanishes as the time interval shrinks, allowing the contraction constant to be made strictly less than one for sufficiently small T. We have expanded the paragraph following the application of the maximal regularity estimate in the proof of Theorem 5.2 to include this explicit commutator bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds by first proving boundedness of the H^∞-calculus for the decoupled fluid-structure operator (a key lemma) and then invoking established stochastic maximal L^p-regularity theory to obtain local strong well-posedness. Both the operator-theoretic step and the stochastic regularity framework are presented as independent prior machinery applied to the specific system; the abstract and structure give no indication that the well-posedness result is presupposed in the proof of the H^∞-calculus or that any parameter is fitted to the target conclusion. No self-definitional reductions, fitted-input predictions, or load-bearing self-citations appear in the described chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard background results from stochastic analysis and operator theory. No free parameters, new physical entities, or ad-hoc inventions are described in the abstract.

axioms (1)
  • domain assumption The fluid-structure operator admits a bounded H^∞-calculus
    Invoked explicitly as the key step enabling application of stochastic maximal regularity theory.

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Works this paper leans on

57 extracted references · 57 canonical work pages

  1. [1]

    Agresti and A

    A. Agresti and A. Hussein , Maximal Lp-regularity and H ∞-calculus for block operator matrices and applications, J. Funct. Anal., 285 (2023), pp. Paper No. 110146, 69

    work page 2023

  2. [2]

    Agresti and M

    A. Agresti and M. Veraar , Stability properties of stochastic maximalLp-regularity, J. Math. Anal. Appl., 482 (2020), pp. 123553, 35

    work page 2020

  3. [3]

    Stochastic maximal regularity and local existence, Nonlinearity, 35 (2022), pp

    , Nonlinear parabolic stochastic evolution equations in critical spaces part I. Stochastic maximal regularity and local existence, Nonlinearity, 35 (2022), pp. 4100–4210

    work page 2022

  4. [4]

    , Nonlinear parabolic stochastic evolution equations in critical spaces part II: Blow-up criteria and instataneous regularization, J. Evol. Equ., 22 (2022), pp. Paper No. 56, 96

    work page 2022

  5. [5]

    , Stochastic maximal Lp(Lq)-regularity for second order systems with periodic boundary conditions, Ann. Inst. Henri Poincaré Probab. Stat., 60 (2024), pp. 413–430

    work page 2024

  6. [6]

    , Stochastic Navier-Stokes equations for turbulent flows in critical spaces, Comm. Math. Phys., 405 (2024), pp. Paper No. 43, 57

    work page 2024

  7. [7]

    Preprint, arXiv:2501.18561 [math.AP] (2025), 2025

    , Nonlinear SPDEs and maximal regularity: An extended survey. Preprint, arXiv:2501.18561 [math.AP] (2025), 2025

    work page arXiv 2025

  8. [8]

    Amann , Linear and quasilinear parabolic problems

    H. Amann , Linear and quasilinear parabolic problems. Vol. I, vol. 89 of Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory

    work page 1995

  9. [9]

    P. J. Atzberger, Stochastic Eulerian Lagrangian methods for fluid–structure interactions with thermal fluctuations, Journal of Computational Physics, 230 (2011), pp. 2821–2837

    work page 2011

  10. [10]

    T. Binz, F. Brandt, and M. Hieber , Rigorous analysis of the interaction problem of sea ice with a rigid body, Math. Ann., 389 (2024), pp. 591–625

    work page 2024

  11. [11]

    Breit, P

    D. Breit, P. R. Mensah, and T. C. Moyo , Martingale solutions in stochastic fluid-structure interaction, J. Nonlinear Sci., 34 (2024), pp. Paper No. 34, 45

    work page 2024

  12. [12]

    Conca, J

    C. Conca, J. San Martín H., and M. Tucsnak , Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25 (2000), pp. 1019–1042

    work page 2000

  13. [13]

    C. J. Cotter, G. A. Gottw ald, and D. D. Holm , Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics, Proc. A., 473 (2017), pp. 20170388, 10

    work page 2017

  14. [14]

    Da Prato and J

    G. Da Prato and J. Zabczyk , Stochastic equations in infinite dimensions, vol. 152 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, second ed., 2014

    work page 2014

  15. [15]

    R. Denk, M. Hieber, and J. Prüss , R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), pp. viii+114

    work page 2003

  16. [16]

    Desjardins and M

    B. Desjardins and M. J. Esteban , Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), pp. 59–71

    work page 1999

  17. [17]

    Er vedoza, D

    S. Er vedoza, D. Maity, and M. Tucsnak ,Large time behaviour for the motion of a solid in a viscous incompressible fluid, Math. Ann., 385 (2023), pp. 631–691

    work page 2023

  18. [18]

    F abes, O

    E. F abes, O. Mendez, and M. Mitrea , Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains, J. Funct. Anal., 159 (1998), pp. 323–368

    work page 1998

  19. [19]

    Flandoli, M

    F. Flandoli, M. Gubinelli, and E. Priola , Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), pp. 1–53

    work page 2010

  20. [20]

    Flandoli and E

    F. Flandoli and E. Luongo , Stochastic partial differential equations in fluid mechanics, vol. 2330 of Lecture Notes in Mathematics, Springer, Singapore, [2023]©2023

    work page 2023

  21. [21]

    G. P. Galdi,On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications, inHandbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, pp. 653–791

    work page 2002

  22. [22]

    G. P. Galdi and A. L. Sil vestre , Strong solutions to the problem of motion of a rigid body in a Navier-Stokes liquid under the action of prescribed forces and torques, in Nonlinear problems in mathematical physics and related topics, I, vol. 1 of Int. Math. Ser. (N. Y.), Kluwer/Plenum, New York, 2002, pp. 121–144

    work page 2002

  23. [23]

    Geissert, K

    M. Geissert, K. Götze, and M. Hieber , Lp-theory for strong solutions to fluid-rigid body interaction in Newtonian and generalized Newtonian fluids, Trans. Amer. Math. Soc., 365 (2013), pp. 1393–1439

    work page 2013

  24. [24]

    Hytönen, J

    T. Hytönen, J. v an Neer ven, M. Veraar, and L. Weis , Analysis in Banach spaces. Vol. II. Probabilistic methods and operator theory, vol. 67 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys ANALYSIS OF STOCHASTIC FLUID-RIGID BODY DYNAMICS VIA STOCHASTIC MAXIMAL REGULARITY 25 in Mathematics [Results in Mathematics ...

    work page 2017

  25. [25]

    Kaltenbacher, I

    B. Kaltenbacher, I. Kuka vica, I. Lasiecka, R. Triggiani, A. Tuff aha, and J. T. Webster , Mathematical theory of evolutionary fluid-flow structure interactions, vol. 48 of Oberwolfach Seminars, Birkhäuser/Springer, Cham,

    work page

  26. [26]

    Lecture notes from Oberwolfach seminars, November 20–26, 2016

    work page 2016

  27. [27]

    Kuan and S

    J. Kuan and S. Čanić , A stochastically perturbed fluid-structure interaction problem modeled by a stochastic viscous wave equation, J. Differential Equations, 310 (2022), pp. 45–98

    work page 2022

  28. [28]

    , Well-posedness of solutions to stochastic fluid-structure interaction, J. Math. Fluid Mech., 26 (2024), pp. Paper No. 4, 61

    work page 2024

  29. [29]

    P. C. Kunstmann and L. Weis , Maximal Lp-regularity for parabolic equations, Fourier multiplier theorems and H ∞-functional calculus, in Functional analytic methods for evolution equations, vol. 1855 of Lecture Notes in Math., Springer, Berlin, 2004, pp. 65–311

    work page 2004

  30. [30]

    Lang and O

    J. Lang and O. Méndez , Potential techniques and regularity of boundary value problems in exterior non-smooth domains: regularity in exterior domains, Potential Anal., 24 (2006), pp. 385–406

    work page 2006

  31. [31]

    Lunardi , Analytic semigroups and optimal regularity in parabolic problems , Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995

    A. Lunardi , Analytic semigroups and optimal regularity in parabolic problems , Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. [2013 reprint of the 1995 original] [MR1329547]

    work page 1995

  32. [32]

    Maity and M

    D. Maity and M. Tucsnak , Lp-Lq maximal regularity for some operators associated with linearized incompressible fluid-rigid body problems, in Mathematical analysis in fluid mechanics—selected recent results, vol. 710 of Contemp. Math., Amer. Math. Soc., [Providence], RI, [2018]©2018, pp. 175–201

    work page 2018

  33. [33]

    Mazzone, J

    G. Mazzone, J. Prüss, and G. Simonett , A maximal regularity approach to the study of motion of a rigid body with a fluid-filled cavity, J. Math. Fluid Mech., 21 (2019), pp. Paper No. 44, 20

    work page 2019

  34. [34]

    , On the motion of a fluid-filled rigid body with Navier boundary conditions, SIAM J. Math. Anal., 51 (2019), pp. 1582–1606

    work page 2019

  35. [35]

    McIntosh and A

    A. McIntosh and A. Yagi , Operators of typeω without a bounded H∞ functional calculus, in Miniconference on Operators in Analysis (Sydney, 1989), vol. 24 of Proc. Centre Math. Anal. Austral. Nat. Univ., Austral. Nat. Univ., Canberra, 1990, pp. 159–172

    work page 1989

  36. [36]

    Noll and J

    A. Noll and J. Saal , H ∞-calculus for the Stokes operator onLq-spaces, Math. Z., 244 (2003), pp. 651–688

    work page 2003

  37. [37]

    Portal and M

    P. Portal and M. Veraar , Stochastic maximal regularity for rough time-dependent problems, Stoch. Partial Differ. Equ. Anal. Comput., 7 (2019), pp. 541–597

    work page 2019

  38. [38]

    Prüss and G

    J. Prüss and G. Simonett , Moving interfaces and quasilinear parabolic evolution equations, vol. 105 of Monographs in Mathematics, Birkhäuser/Springer, [Cham], 2016

    work page 2016

  39. [39]

    Prüss, G

    J. Prüss, G. Simonett, and M. Wilke ,Critical spaces for quasilinear parabolic evolution equations and applications, J. Differential Equations, 264 (2018), pp. 2028–2074

    work page 2018

  40. [40]

    On quasilinear parabolic evolution equations in weightedLp-spaces II

    J. Prüss and M. Wilke , Addendum to the paper “On quasilinear parabolic evolution equations in weightedLp-spaces II” [ MR3250797], J. Evol. Equ., 17 (2017), pp. 1381–1388

    work page 2017

  41. [41]

    , On critical spaces for the Navier-Stokes equations, J. Math. Fluid Mech., 20 (2018), pp. 733–755

    work page 2018

  42. [42]

    Serre, Chute libre d’un solide dans un fluide visqueux incompressible

    D. Serre, Chute libre d’un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math., 4 (1987), pp. 99–110

    work page 1987

  43. [43]

    C. G. Simader and H. Sohr , A new approach to the Helmholtz decomposition and the Neumann problem inLq- spaces for bounded and exterior domains, in Mathematical problems relating to the Navier-Stokes equation, vol. 11 of Ser. Adv. Math. Appl. Sci., World Sci. Publ., River Edge, NJ, 1992, pp. 1–35

    work page 1992

  44. [44]

    Sprinkle, F

    B. Sprinkle, F. Balboa Usabiaga, N. A. Patankar, and A. Donev , Large scale brownian dynamics of confined suspensions of rigid particles, The Journal of chemical physics, 147 (2017)

    work page 2017

  45. [45]

    Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv

    T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations, 8 (2003), pp. 1499–1532

    work page 2003

  46. [46]

    Takahashi and M

    T. Takahashi and M. Tucsnak , Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech., 6 (2004), pp. 53–77

    work page 2004

  47. [47]

    Ta wri, A 2d stochastic fluid-structure interaction problem in compliant arteries with non-zero longitudinal dis- placement, 2024

    K. Ta wri, A 2d stochastic fluid-structure interaction problem in compliant arteries with non-zero longitudinal dis- placement, 2024

    work page 2024

  48. [48]

    Ta wri, A stochastic fluid-structure interaction problem with the Navier-slip boundary condition, SIAM J

    K. Ta wri, A stochastic fluid-structure interaction problem with the Navier-slip boundary condition, SIAM J. Math. Anal., 56 (2024), pp. 7508–7544

    work page 2024

  49. [49]

    Ta wri and S

    K. Ta wri and S. Čanić , Existence of martingale solutions to a nonlinearly coupled stochastic fluid-structure inter- action problem, Comm. Partial Differential Equations, 50 (2025), pp. 353–406

    work page 2025

  50. [50]

    Triebel , Interpolation theory, function spaces, differential operators, vol

    H. Triebel , Interpolation theory, function spaces, differential operators, vol. 18 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1978

    work page 1978

  51. [51]

    v an Neer ven, M

    J. v an Neer ven, M. Veraar, and L. Weis , Maximal Lp-regularity for stochastic evolution equations, SIAM J. Math. Anal., 44 (2012), pp. 1372–1414

    work page 2012

  52. [52]

    Probab., 40 (2012), pp

    , Stochastic maximal Lp-regularity, Ann. Probab., 40 (2012), pp. 788–812

    work page 2012

  53. [53]

    68 of Progr

    , Stochastic integration in Banach spaces—a survey, in Stochastic analysis: a series of lectures, vol. 68 of Progr. Probab., Birkhäuser/Springer, Basel, 2015, pp. 297–332

    work page 2015

  54. [54]

    W alter, O

    J. W alter, O. Gonzalez, and J. H. Maddocks , On the stochastic modeling of rigid body systems with application to polymer dynamics, Multiscale Modeling & Simulation, 8 (2010), pp. 1018–1053. 26 FELIX BRANDT AND ARNAB ROY

    work page 2010

  55. [55]

    W ang and Z

    Y. W ang and Z. Xin , Analyticity of the semigroup associated with the fluid-rigid body problem and local existence of strong solutions, J. Funct. Anal., 261 (2011), pp. 2587–2616

    work page 2011

  56. [56]

    Weis, Operator-valued Fourier multiplier theorems and maximalLp-regularity, Math

    L. Weis, Operator-valued Fourier multiplier theorems and maximalLp-regularity, Math. Ann., 319 (2001), pp. 735– 758

    work page 2001

  57. [57]

    Wilke, Linear and quasilinear evolution equations in the context of weightedLp-spaces, Arch

    M. Wilke, Linear and quasilinear evolution equations in the context of weightedLp-spaces, Arch. Math. (Basel), 121 (2023), pp. 625–642. Department of Mathematics, University of California at Berkeley, Berkeley, 94720, CA, USA. Email address: fbrandt@berkeley.edu Basque Center for Applied Mathematics (BCAM), Alameda de Mazarredo 14, 48009 Bilbao, Spain. IK...

    work page 2023