Abstract
In this article we study a system of nonlinear non-parabolic stochastic evolution equations driven by Lévy noise type. This system describes the motion of second grade fluids driven by random force. Global existence of a martingale solution is proved under general conditions on the noise. Since the coefficient of the noise does not satisfy a Lipschitz property, we could not prove any pathwise uniqueness result. We note that this is the first work dealing with a stochastic model for non-Newtonian fluids excited by external forces of Lévy noise type.
References
Albeverio, S., Brzezńiak, Z., Wu, J.L.: Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients. J. Math. Anal. Appl. 371(1), 309–322 (2010)
Bensoussan, A.: Stochastic Navier–Stokes equations. Acta Appl. Math. 38, 267–304 (1995)
Bensoussan, A., Temam, R.: Equations stochastiques du type Navier–Stokes. J. Funct. Anal. 13, 195–222 (1973)
Bernard, J.M.: Weak and classical solutions of equations of motion for second grade fluids. Commun. Appl. Nonlinear Anal 5, 1–32 (1998)
Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. Wiley, New York (1999)
Brookes, M.: The matrix reference manual. [online] http://www.ee.imperial.ac.uk/hp/staff/dmb/matrix/intro.html (2011). Accessed 27 July 2012
Brzeźniak, Z., Hausenblas, E.: Maximal regularity for stochastic convolutions driven by Lévy processes. Probab. Theory Relat. Fields 145(3–4), 615–637 (2009)
Brzeźniak, Z., Hausenblas, E.: Martingale solutions for Stochastic equation of reaction diffusion type driven by Levy noise or Poisson random measure. Preprint (2009). arXiv:1010.5933
Brzeźniak, Z., Hausenblas, E.: Uniqueness in law of the Itô integral with respect to Lévy Noise. In: Seminar on Stochastic Analysis, Random Fields and Applications VI Progress in Probability, vol. 63, part 1, pp. 37–57 (2011)
Brzeźniak, Z., Zabczyk, J.: Regularity of Ornstein–Uhlenbeck processes driven by a Lévy white noise. Potential Anal. 32(2), 153–188 (2010)
Brzeźniak, Z., Goldys, B., Imkeller, P., Peszat, S., Priola E., Zabczyk, J.: Time irregularity of generalized Ornstein–Uhlenbeck processes. C. R. Math. Acad. Sci. Paris 348(5–6), 273–276 (2010)
Busuioc, A.V.: On second grade fluids with vanishing viscosity. C. R. Acad. Paris, Série I 328(12), 1241–1246 (1999)
Busuioc, A.V., Ratiu, T.S.: The second grade fluid and averaged Euler equations with Navier-slip boundary conditions. Nonlinearity 16, 1119–1149 (2003)
Chen, S., Foias, C., Titi, E., Wynne, S.: A connection between the Camassa–Holm equations and turbulent flows in channels and pipes. Phys. Fluids 11, 2343–2353 (1999)
Chueshov, I., Millet, A.: Stochastic 2D hydrodynamical type systems: well posedness and large deviations. Appl. Math. Optim. 61(3), 379–420 (2010)
Cioranescu, D., Girault, V.: Weak and classical solutions of a family of second grade fluids. Int. J. Non-Linear Mech. 32(2), 317–335 (1997)
Cioranescu, D., Ouazar, E.H.: Existence and uniqueness for fluids of second grade. In: Nonlinear Partial Differential Equations, vol. 109, pp. 178–197. Collège de France Seminar, Pitman (1984)
Cioranescu, D., Ouazar, E.H.: Existence et unicité pour les fluides de grade deux. C. R. Acad. Sci. Paris, Série I 298(13), 285–287 (1984)
Deugoue, G., Sango, M.: On the stochastic 3D Navier–Stokes-α model of fluids turbulence. Abstr. Appl. Anal. Art. ID 723236, 27 pp. (2009)
Deugoué, G., Razafimandimby, P.A., Sango, M.: On the 3-D stochastic magnetohydrodynamic-α model. Stoch. Process. Their Appl. 122, 2211–2248 (2012)
Dong, Z., Zhai, J.: Martingale solutions and Markov selection of stochastic 3D Navier–Stokes equations with jump. J. Differ. Equ. 250, 2737–2778 (2011)
Dunn, J.E., Fosdick, R.L.: Thermodynamics, stability and boundedness of fluids of complexity two and fluids of second grade. Arch. Ration. Mech. Anal. 56, 191–252 (1974)
Dunn, J.E., Rajagopal, K.R.: Fluids of differntial type: Critical review and thermodynamic analysis. Int. J. Eng. Sci. 33, 668–729 (1995)
Ethier, S., Kurtz, T.: Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1986)
Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 102, 367–391 (1995)
Foias, C., Holm, D.D., Titi, E.S.: The Navier–Stokes-alpha model of fluid turbulence. Advances in nonlinear mathematics and science. Physica D 152–153, 505–519 (2001)
Fosdick, R.L., Rajagopal, K.R.: Anomalous features in the model of second grade fluids. Arch. Ration. Mech. Anal. 70, 145–152 (1978)
Fosdick, R.L., Truesdell, C.: Universal flows in the simplest theories of fluids. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 4(4), 323–341 (1977)
Gyöngy, I., Krylov, N.V.: On stochastics equations with respect to semimartingales. II. Itô formula in Banach spaces. Stochastics 6, 153–173 (1981/1982)
Holm, D.D., Marsden, J.E., Ratiu, T.: The Euler–Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137, 1–81 (1998)
Holm, D.D., Marsden, J.E., Ratiu, T.: Euler–Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 349, 4173–4177 (1998)
Iftimie, D.: Remarques sur la limite α→0 pour les fluides de grade 2. C. R. Acad. Sci. Paris, Ser. I 334, 83–86 (2002)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. In: North-Holland Mathematical Library, vol. 24, 2nd edn. North-Holland, Amsterdam (1989)
Imkeller, P., Pavlyukevich, I.: First exit times of SDEs driven by stable Lévy processes. Stoch. Process. Their Appl. 116, 611–642 (2006)
Jacod, J. Shiryaev, A.: Limit theorems for stochastic processes. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, 2nd edn. Springer, Berlin (2003)
Joffe, A., Métivier, M.: Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. Appl. Probab. 18, 20–65 (1986)
Kallenberg, O.: Foundations of modern probability. In: Probability and its Applications (New York). Springer, New York (1997)
Kushner, H.J.: Numerical Methods for Controlled Stochastic Delay Systems. Birkhäuser, Boston (2008)
Métivier, M.: Stochastic Partial Differential Equations in Infinite Dimensional Spaces. Scuola Normale Superiore, Pisa (1988)
Mikulevicius, R., Rozovskii, B.L.: Martingale problems for stochastic PDE’s. In: Stochastic Partial Differential Equations: Six Perspectives. Math. Surveys Monogr., vol. 64, pp. 243–325. Amer. Math. Soc., Providence (1999)
Mueller, C.: The heat equation with Lévy noise. Stoch. Process. Their Appl. 74, 67–82 (1998)
Mytnik, L.: Stochastic partial differential equation driven by stable noise. Probab. Theory Relat. Fields 123, 157–201 (2002)
Noll, W., Truesdell, C.: The nonlinear field theory of mechanics. In: Handbuch der Physik, vol. III. Springer, Berlin (1975)
Parthasarathy, K.R.: Probability measures on metric spaces. In: Probability and Mathematical Statistics, vol. 3. Academic, New York (1967)
Peszat, S., Zabczyk, J.: Stochastic partial differential equations with Levy Noise. An evolution equation approach. In: Encyclopedia of Mathematics and its Applications, vol. 113. Cambridge University Press (2007)
Rajagopal, K.R., Truesdell, C.A.: An introduction to the mechanics of fluids. In: Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston (2000)
Razafimandimby, P.A: Some mathematical problems in the dynamics of stochastic second-grade fluids. Ph.D. Thesis, University of Pretoria (2010)
Razafimandimby, P.A., Sango, M.: Weak solutions of a stochastic model for two-dimensional second grade fluids. Boundary Value Problems 2010, Article ID 636140, 47 pp. (2010). doi:10.1155/2010/636140
Razafimandimby, P.A., Sango, M.: Asymptotic behavior of solutions of stochastic evolution equations for second grade fluids. C R. Math. Acad. Sci. Paris 348, 787–790 (2010)
Razafimandimby, P.A., Sango, M.: Strong solution for a stochastic model of two-dimensional second grade fluids: existence, uniqueness and asymptotic behavior. Nonlinear Anal. 75, 4251–4270 (2012)
Rivlin, R.S.: The relation between the flow of non-Newtonian fluids and turbulent Newtonian fluids. Q. Appl. Math. 15, 212–215 (1957).
Sango, M.: Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing. Discrete Contin. Dyn. Syst. Ser. B 7(4), 885–905 (2007)
Sango, M.: Magnetohydrodynamic turbulent flows: existence results. Phys. D 239(12), 912–923 (2010)
Shkoller, S.: Geometry and curvature of diffeomorphism groups with H 1 metric and hydrodynamics. J. Funct. Anal. 160, 337–365 (1998)
Shkoller, S.: Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations. Appl. Math. Lett. 14, 539–543 (2001)
Situ, R.: Theory of stochstic differential equations with jumps and applications. In: Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York (2005)
Solonnikov, V.A.: On general boundary problems for systems which are elliptic in the sense of A. Douglis and L. Nirenberg. II. Proc. Steklov Inst. Math. 92, 269–339 (1968)
Temam, R.: Navier–Stokes Equations. North-Holland (1979)
Truesdell, C.A.: A first course in rational continuum mechanics. Vol. 1. General concepts. In: Pure and Applied Mathematics, vol. 71, 2nd edn. Academic Press, Boston (1991)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hausenblas, E., Razafimandimby, P.A. & Sango, M. Martingale Solution to Equations for Differential Type Fluids of Grade Two Driven by Random Force of Lévy Type. Potential Anal 38, 1291–1331 (2013). https://doi.org/10.1007/s11118-012-9316-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-012-9316-7
Keywords
- Second grade fluid
- Lévy noise
- Stochastic partial differential equations
- Poisson random measure
- Non-Newtonian fluids