We consider the barotropic Navier--Stokes system driven by a physically well-motivated transport noise in both a continuity as well as momentum equation. We focus on three different situations: (i) the noise is smooth in time and the equations are understood as in the sense of the classical weak deterministic theory, (ii) the noise is rough in time and we interpret the equations in the framework of rough paths with unbounded rough drivers, and (iii) we have a Brownian noise of Stratonovich type and study the existence of martingale solutions. The first situation serves as an approximation for (ii) and (iii), while (ii) and (iii) are motivated by recent results on the incompressible Navier--Stokes system concerning the physical modeling as well as regularization by noise.


  1. compressible fluids
  2. stochastic Navier--Stokes system
  3. transport noise

MSC codes

  1. 60H15
  2. 35R60
  3. 76N10
  4. 35Q35

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D. Alonso-Orán, A. Bethencourt de León, D. D. Holm, and S. Takao, Modelling the climate and weather of a 2D Lagrangian-averaged Euler-Boussinesq equation with transport noise, J. Stat. Phys., 179 (2020), pp. 1267--1303.
M. Arnaudon and A. B. Cruzeiro, Stochastic Lagrangian flows and the Navier-Stokes equations, in Stochastic Analysis: A Series of Lectures, Progr. Probab. 68, Birkhäuser/Springer, Basel, 2015, pp. 55--75.
A. Aw and M. Rascle, Resurrection of second order models of traffic flow, SIAM J. Appl. Math., 60 (2000), pp. 916--938.
I. Bailleul and M. Gubinelli, Unbounded rough drivers, Ann. Fac. Sci. Toulouse Math. (6), 26 (2017).
F. Berthelin, P. Degond, M. Delitala, and M. Rascle, A model for the formation and evolution of traffic jams, Arch. Ration. Mech. Anal., 187 (2008), pp. 185--220.
F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, M. Rascle, and J. Royer, A traffic-flow model with constraints for the modeling of traffic jams, Math. Models Methods Appl. Sci., 18 (2008), pp. 1269--1298.
D. Breit and E. Feireisl, Stochastic Navier--Stokes--Fourier equations, Indiana Univ. Math. J., 69 (2020), pp. 911--975.
D. Breit, E. Feireisl, and M. Hofmanová, Stochastically forced compressible fluid flows, De Gruyter Ser. Appl. Numer. Math., De Gruyter, Berlin, 2018.
D. Breit and M. Hofmanová, Stochastic Navier-Stokes equations for compressible fluids, Indiana Univ. Math. J., 65 (2016), pp. 1183--1250.
H. Brenner, Navier-Stokes revisited, Phys. A, 349 (2005), pp. 60--132.
H. Brenner, Fluid mechanics revisited, Phys. A, 349 (2006), pp. 190--224.
X. Chen, A. B. Cruzeiro, and T. S. Ratiu, Stochastic Variational Principles for Dissipative Equations with Advected Quantities, preprint, arXiv:1506.05024v2, 2018.
A. B. Cruzeiro and R. Lassalle, On the stochastic least action principle for the Navier-Stokes equation, in Stochastic Analysis and Applications 2014, Springer Proc. Math. Stat. 100, Springer, New York, 2014, pp. 163--184.
P. Degond, P. Minakowski, L. Navoret, and E. Zatorska, Finite volume approximations of the Euler system with variable congestion, Comput. & Fluids, 169 (2018), pp. 23--39.
A. Deya, M. Gubinelli, M. Hofmanová, and S. Tindel, A priori estimates for rough PDEs with application to rough conservation laws, J. Funct. Anal., 276 (2019), pp. 3577--3645.
T. D. Drivas and D. D. Holm, Circulation and energy theorem preserving stochastic fluids, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), pp. 2776--2814.
R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), pp. 511--547.
E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.
E. Feireisl, A. Novotný, and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid. Mech., 3 (2001), pp. 358--392.
F. Flandoli, M. Gubinelli, and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), pp. 1--53.
F. Flandoli, M. Hofmanová, D. Luo, and T. Nilssen, Global well-posedness of the \textup3D Navier--Stokes equations perturbed by a deterministic vector field, Ann. Appl. Probab., to appear; also available online from arXiv:2004.07528.
F. Flandoli and D. Luo, High mode transport noise improves vorticity blow-up control in \textup3D Navier--Stokes equations, Probab. Theory Related Fields, 180 (2021), pp. 309--363.
P. K. Friz and M. Hairer, A Course on Rough Paths: With an Introduction to Regularity Structures, Universitext, Springer, New York, 2014.
P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge Stud. Adv. Math. 120, Cambridge University Press, Cambridge, 2010.
M. Hofmanová, J.-M. Leahy, and T. Nilssen, On the Navier--Stokes equation perturbed by rough transport noise, J. Evol. Equ., 19 (2018), pp. 203--247.
M. Hofmanová, J.-M. Leahy, and T. Nilssen, On a rough perturbation of the Navier-Stokes system and its vorticity formulation, Ann. Appl. Probab., 31 (2021), pp. 736--777.
D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. A., 471 (2015), 20140963.
D. D. Holm, Stochastic modelling in fluid dynamics: Itô versus Stratonovich, Proc. A., 476 (2020), 20190812.
A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, Theory Probab. Appl., 42 (1998), pp. 167--174 (in English).
P. L. Lions, Mathematical Topics in Fluid Mechanics: Volume 2. Compressible Models, Oxford Lecture Ser. Math. Appl. 10, Clarendon Press, Oxford, 1998.
T. J. Lyons, M. Caruana, and T. Lévy, Differential Equations Driven by Rough Paths, Lecture Notes in Math. 1908, Springer, New York, 2007.
R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), pp. 1250--1310.
B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, in Mathematics of Complexity and Dynamical Systems, R. Meyers, ed., Springer, New York, 2012.
S. A. Smith, Random perturbations of viscous, compressible fluids: Global existence of weak solutions, SIAM J. Math. Anal., 49 (2017), pp. 4521--4578.

Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 4465 - 4494
ISSN (online): 1095-7154


Submitted: 13 December 2021
Accepted: 5 April 2022
Published online: 21 July 2022


  1. compressible fluids
  2. stochastic Navier--Stokes system
  3. transport noise

MSC codes

  1. 60H15
  2. 35R60
  3. 76N10
  4. 35Q35



Funding Information

Institute of Mathematics of the Academy of Sciences of the Czech Republic : RVO:67985840
Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : SFB1283
H2020 European Research Council https://doi.org/10.13039/100010663 : 949981
Grantová Agentura České Republiky https://doi.org/10.13039/501100001824 : 21-02411S
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/V000586/1

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