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On Weak-Strong Uniqueness for Stochastic Equations of Incompressible Fluid Flow

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Abstract

We introduce a concept of dissipative measure-valued martingale solution to the stochastic Euler equations describing the motion of an inviscid incompressible fluid. These solutions are characterized by a parametrized Young measure and a concentration defect measure in the total energy balance. Moreover, they are weak in the probabilistic sense i.e., the underlying probability space and the driving Wiener process are intrinsic parts of the solution. We first exhibit the relative energy inequality for the incompressible Euler equations driven by a multiplicative noise and then demonstrate the pathwise weak-strong uniqueness principle. Finally, we also provide a sufficient condition, á la Prodi (Ann Mat Pura Appl 48:173–182, 1959) and Serrin (in: Nonlinear problems, University of Wisconsin Press, Madison, Wisconsin, pp 69–98, 1963), for the uniqueness of weak martingale solutions to the stochastic Navier–Stokes system in the class of finite energy weak martingale solutions.

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Notes

  1. i.e. H is measurable with respect to first variable and continuous with respect to second variable.

  2.  Here \(\langle {\mathcal {V}}^{\omega }_{t,x}; f(\mathbf{u}) \rangle :=\int _{\mathbb {R}^3}f(\mathbf{u})d{\mathcal {V}}^{\omega }_{t,x}(\mathbf{u})\), for any measurable function f.

References

  1. Attouch, H., Buttazzo, G., Gérard, M.: Variational Analysis in Sobolev and BV Spaces. SIAM, Philadelphia (2014). https://doi.org/10.1137/1.9781611973488

    Book  MATH  Google Scholar 

  2. Bhauryal, N., Koley, U., Vallet, G.: The Cauchy problem for a fractional conservation laws driven by Lévy noise. Stoch. Process. Their Appl. 130(9), 5310–5365 (2020). https://doi.org/10.1016/j.spa.2020.03.009

    Article  MATH  Google Scholar 

  3. Bhauryal, N., Koley, U., Vallet, G.: A fractional degenerate parabolic-hyperbolic Cauchy problem with noise. J. Differ. Equ. 284, 433–521 (2021). https://doi.org/10.1016/j.jde.2021.02.061

    Article  MathSciNet  MATH  ADS  Google Scholar 

  4. Biswas, I.H., Koley, U., Majee, A.K.: Continuous dependence estimate for conservation laws with Lévy noise. J. Differ. Equ. 259, 4683–4706 (2015)

    Article  ADS  Google Scholar 

  5. Breit, D., Moyo, T.C.: Dissipative solutions to the stochastic Euler equations. arXiv:2008.09517

  6. Breit, D., Feiresl, E., Hofmanova, M.: Stochastically forced compressible fluid flows. In: De Gruyter Series in Applied and Numerical Mathematics. De Gruyter, Berlin/Munich/Boston (2018)

  7. Breit, D., Feireisl, E., Hofmanová, M.: Stochastically forced compressible fluid flows. In: De Gruyter Series in Applied and Numerical Mathematics. De Gruyter, Berlin/Munich/Boston (2018)

  8. Brenier, Y., De Lellis, C., Székelyhidi, L., Jr.: Weak-strong uniqueness for measure-valued solutions. Commun. Math. Phys. 305(2), 351–361 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  9. Buckmaster, T., Vicol, V.: Convex integration and phenomenologies in turbulence. EMS Surv. Math. Sci. 6(1/2), 173–263 (2019)

    MathSciNet  MATH  Google Scholar 

  10. Buckmaster, T., Vicol, V.: Non-uniqueness of weak solutions to the Navier–Stokes equation. Ann. Math. 189(1), 101–144 (2019)

    Article  MathSciNet  Google Scholar 

  11. Chaudhary, A., Koley, U.: A convergent finite volume scheme for stochastic compressible barotropic Euler equations. Submitted

  12. Chaudhary, A., Koley, U.: Convergence of a spectral method for the stochastic incompressible Euler equations. Submitted

  13. Chiodaroli, E., Kreml, O., Mácha, V., Schwarzacher, S.: Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data. arXiv:1812.09917v1 (2019)

  14. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, p. 1999. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  15. Debussche, A., Glatt-Holtz, N., Temam, R.: Local martingale and pathwise solutions for an abstract fluids model. Physica D 240(14–15), 1123–1144 (1999)

    MathSciNet  MATH  ADS  Google Scholar 

  16. De Lellis, C., Székelyhidi, L., Jr.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225–260 (2010)

    Article  MathSciNet  Google Scholar 

  17. De Lellis, C., Székelyhidi, L., Jr.: The \(h\)-principle and the equations of fluid dynamics. Bull. Am. Math. Soc. (N.S.) 49(3), 347–375 (2012)

    Article  MathSciNet  Google Scholar 

  18. DiPerna, R.J.: Measure valued solutions to conservation laws. Arch. Ration. Mech. Anal. 88(3), 223–270 (1985)

    Article  MathSciNet  Google Scholar 

  19. Diperna, R.J., Majda, A.J.: Oscillations and concentrations in weak solution of the incompressible fluid equations. Commun. Math. Phys. 108(4), 667–689 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  20. Feireisl, E., Gwiazda, P., Świerczewska-Gwiazda, A., Wiedemann, E.: Dissipative measure-valued solutions to the compressible Navier–Stokes system. Calc. Var. Partial Differ. Equ. 55(6), 141 (2016)

    Article  MathSciNet  Google Scholar 

  21. Feireisl, E., Lukáčová-Medviová, M., Mizerová, H.: Convergence of finite volume schemes for the Euler equations via dissipative measure-valued solutions. Found. Comput. Math. 20(4), 923–966 (2020)

    Article  MathSciNet  Google Scholar 

  22. Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 102, 367–391 (1995)

    Article  MathSciNet  Google Scholar 

  23. Glatt-Holtz, N.E., Vicol, V.C.: Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise. Ann. Probab. 42(1), 80–145 (2014)

    Article  MathSciNet  Google Scholar 

  24. Hofmanova, M., Koley, U., Sarkar, U.: Measure-valued solutions to the stochastic compressible Euler equations and incompressible limits. arXiv:2012.07391 (2020)

  25. Hofmanová, M., Zhu, R., Zhu, X.: On ill- and well-posedness of dissipative martingale solutions to stochastic 3D Euler equations. arXiv:2009.09552 (2020)

  26. Jakubowski, A.: The almost sure Skorokhod representation for subsequences in nonmetric spaces. Theory Probab. Appl. 42(1), 164–174 (1998)

    Article  MathSciNet  Google Scholar 

  27. Kim, J.U.: Measure valued solutions to the stochastic Euler equations in \({\cal{R}}^d\). Stoch PDE: Anal Comput. 3, 531–569 (2015)

    Article  Google Scholar 

  28. Koley, U., Majee, A.K., Vallet, G.: A finite difference scheme for conservation laws driven by Lévy noise. IMA J. Numer. Anal. 38(2), 998–1050 (2018)

    Article  MathSciNet  Google Scholar 

  29. Koley, U., Majee, A.K., Vallet, G.: Continuous dependence estimate for a degenerate parabolic-hyperbolic equation with Lévy noise. Stoch. Partial Differ. Equ. Anal. Comput. 5(2), 145–191 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Lions, P. L.: Mathematical topics in fluid mechanics, volume 1, incompressible models. In: Oxford Lecture Series in Mathematics and Its Applications, vol. 3. Clarendom Press, Oxford (1996)

  31. Mausuda, K.: Weak solutions of Navier–Stokes equations. Tôhoku Math. J. 36, 623–646 (1984)

    MathSciNet  Google Scholar 

  32. Motyl, E.: Stochastic Navier–Stokes equations driven by Lévy noise in unbounded 3d domains. Potential Anal. 38(3), 863–912 (2012)

    MATH  Google Scholar 

  33. Prodi, G.: Un teorema di unicità per le euqazioni di Navier–Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959)

    Article  MathSciNet  Google Scholar 

  34. Scheffer, V.: An inviscid flow with compact support in space–time. J. Geom. Anal. 3(4), 343–401 (1993)

    Article  MathSciNet  Google Scholar 

  35. Serrin, J.: The initial value problem for the Navier–Stokes equations. In: Nonlinear Problems, pp. 69–98, University of Wisconsin Press. Madison, Wisconsin (1963)

  36. Shnirelman, A.: On the non-uniqueness of weak solution of the Euler equation. Commun. Pure. Appl. Math. 50(12), 1261–1286 (1997)

    Article  MathSciNet  Google Scholar 

  37. Skorohod, A.V.: Limit theorems for stochastic processes. Teor. Veroyatnost. i Primenen 1, 289–319 (1956)

    MathSciNet  Google Scholar 

  38. Székelyhidi, L., Jr., Wiedemann, E.: Young measures generated by ideal incompressible fluid flows. Arch. Ration. Mech. Anal. 206(1), 333–366 (2012)

    Article  MathSciNet  Google Scholar 

  39. Wiedemann, E.: Weak-strong uniqueness in fluid dynamics. In: Partial Differential Equations in Fluid Mechanics, Volume 452 of London Mathematical Society Lecture Note series, pp. 289–326. Cambridge University Press, Cambridge (2018)

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Acknowledgements

U.K. acknowledges the support of the Department of Atomic Energy, Government of India, under Project No. 12-R &D-TFR-5.01-0520, India SERB Matrics Grant MTR/2017/000002, and DST-SERB SJF Grant DST/SJF/MS/2021/44.

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  1. Centre for Applicable Mathematics, Tata Institute of Fundamental Research, GKVK Post Office, P.O. Box 6503,  Bangalore, 560065, India

    Abhishek Chaudhary & Ujjwal Koley

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  1. Abhishek Chaudhary
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Correspondence to Ujjwal Koley.

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Communicated by F. Flandoli.

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Chaudhary, A., Koley, U. On Weak-Strong Uniqueness for Stochastic Equations of Incompressible Fluid Flow. J. Math. Fluid Mech. 24, 62 (2022). https://doi.org/10.1007/s00021-022-00699-y

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