Flow Representation of the Navier-Stokes Equations in Weighted Sobolev Spaces

Sirisubtawee, Sekson; Manitcharoen, Naowarat; Saksurakan, Chukiat

doi:10.28924/2291-8639-23-2025-27

Research Article

Flow Representation of the Navier-Stokes Equations in Weighted Sobolev Spaces

Sekson Sirisubtawee
Naowarat Manitcharoen
Chukiat Saksurakan*

* Corresponding author

Abstract

Using Constantin-Iyer representation also known more generally as Euler-Lagrangian approach, we prove the local existence of the Navier-Stokes equations in weighted Sobolev spaces with external forcing on R^d, for any dimension d and p such that p>d≥2.

How to Cite

S. Sirisubtawee, N. Manitcharoen, C. Saksurakan, Flow Representation of the Navier-Stokes Equations in Weighted Sobolev Spaces, Int. J. Anal. Appl. 23 (2025), 27. https://doi.org/10.28924/2291-8639-23-2025-27

BibTeX

@article{Sirisubtawee2025,
  author  = {Sirisubtawee, Sekson and Manitcharoen, Naowarat and Saksurakan, Chukiat},
  title   = {Flow Representation of the Navier-Stokes Equations in Weighted Sobolev Spaces},
  journal = {Int. J. Anal. Appl.},
  volume  = {23},
  year    = {2025},
  pages   = {27},
  doi     = {10.28924/2291-8639-23-2025-27}
}

References

[1] V. Arnold, Sur la Géométrie Différentielle des Groupes de Lie de Dimension Infinie et ses Applications à l’Hydrodynamique des Fluides Parfaits, Ann. Inst. Fourier 16 (1966), 319–361. https://doi.org/10.5802/aif.233.
[2] D.G. Ebin, J. Marsden, Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, Ann. Math. 92 (1970), 102-163. https://doi.org/10.2307/1970699.
[3] D.G. Ebin, A Concise Presentation of the Euler Equations of Hydrodynamics, Commun. Partial Differ. Equ. 9 (1984), 539–559. https://doi.org/10.1080/03605308408820341.
[4] R. Mikulevicius, G. Valiukevicius, On Stochastic Euler Equation in Rd, Electron. J. Probab. 5 (2000), 1-20. https://doi.org/10.1214/EJP.v5-62.
[5] M. Maurelli, K. Modin, A. Schmeding, Incompressible Euler Equations with Stochastic Forcing: A Geometric Approach, Stoch. Process. Their Appl. 159 (2023), 101–148. https://doi.org/10.1016/j.spa.2023.01.011.
[6] P. Constantin, G. Iyer, A Stochastic Lagrangian Representation of the Three-dimensional Incompressible NavierStokes Equations, Commun. Pure Appl. Math. 61 (2008), 330–345. https://doi.org/10.1002/cpa.20192.
[7] G. Iyer, A Stochastic Perturbation of Inviscid Flows, Commun. Math. Phys. 266 (2006), 631–645. https://doi.org/10.1007/s00220-006-0058-5.
[8] B.C. Pooley, J.C. Robinson, An Eulerian–Lagrangian Form for the Euler Equations in Sobolev Spaces, J. Math. Fluid Mech. 18 (2016), 783–794. https://doi.org/10.1007/s00021-016-0271-8.
[9] C. Olivera, J.D. Londoño, Euler–Lagrangian Approach to Stochastic Euler Equations in Sobolev Spaces, J. Math. Fluid Mech. 25 (2023), 61. https://doi.org/10.1007/s00021-023-00808-5.
[10] F. Flandoli, D. Luo, Euler-Lagrangian Approach to 3D Stochastic Euler Equations, J. Geom. Mech. 11 (2019), 153–165. https://doi.org/10.3934/jgm.2019008.
[11] M. Cantor, S.S. Chern, Spaces of Functions With Asymptotic Conditions on Rn, Indiana Univ. Math. J. 24 (1975), 897-902. https://www.jstor.org/stable/24890952.
[12] M. Cantor, Perfect Fluid Flows over Rn with Asymptotic Conditions, J. Funct. Anal. 18 (1975), 73-84.
[13] L. Stupelis, Navier-Stokes Equations in Irregular Domains, Springer, Dordrecht, 1995.
[14] L.C. Evans, Partial Differential Equations, American Mathematical Society, 1998.
[15] R. Mikulevicius, On the Cauchy Problem for Stochastic Stokes Equations, SIAM J. Math. Anal. 34 (2002), 121–141. https://doi.org/10.1137/S0036141001390312.
[16] R. Mikulevicius, B.L. Rozovskii, Stochastic Navier–Stokes Equations for Turbulent Flows, SIAM J. Math. Anal. 35 (2004), 1250–1310. https://doi.org/10.1137/S0036141002409167.
[17] R.A. Adams, J.J.F. Fournier, Sobolev Spaces, Elsevier, 2003.
[18] J.M. Leahy, R. Mikulevicius, On Some Properties of Space Inverses of Stochastic Flows, Stoch. Partial Differ. Equ. Anal. Comput. 3 (2015), 445–478. https://doi.org/10.1007/s40072-015-0056-8.
[19] H. Cartan, Differntial Calculus, Hermann, 1971.
[20] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998.
[21] E. Stein, Harmonic Analysis, Princeton University Press, 1993.