A large deviation principle for fluids of third grade

References

• A. Bensoussan, Stochastic Navier–Stokes equations, Acta. Appl. Math. 38(3) (1995), pp. 267–304.
   Web of Science ®Google Scholar
• H. Bessaih and A. Millet, Large deviations and the zero viscosity limit for 2D stochastic NavierStokes equations with free boundary, SIAM J. Math. Anal. 44(3) (2012), pp. 1861–1893.
   Web of Science ®Google Scholar
• H.I. Breckner, Approximation and optimal control of the stochastic Navier–Stokes equations, Ph.D. Thesis, Halle (Saale), 1999.
   Google Scholar
• Z. Brzeniak, B. Goldys, and T. Jegaraj, Large deviations and transitions between equilibria for stochastic Landaulifshitzgilbert equation, Arch. Ration. Mech. Anal. 226 (2) (2017), pp. 497–558.
   Web of Science ®Google Scholar
• Z. Brzeniak, E. Hausenblas, and P.A. Razafimandimby, Stochastic reaction–diffusion equations driven by jump processes, Potential Anal. 49 (1) (2018), pp. 131–201.
   Web of Science ®Google Scholar
• A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist. 20(1) (2000), pp. 39–61.
   Google Scholar
• A.V. Busuioc and D. Iftimie, global existence and uniqueness of solutions for the equations of third grade fluids, Int. J. Non Linear Mech. 39 (1) (2004), pp. 1–12.
   Web of Science ®Google Scholar
• A.V. Busuioc and D. Iftimie, A non-Newtonian fluid with Navier boundary conditions, J. Dyn. Differ. Equ. 18(2) (2006), pp. 357–379.
   Web of Science ®Google Scholar
• A.V. Busuioc and T.S. Ratiu, The second grade fluid and averaged Euler equations with Navier-slip boundary conditions, Nonlinearity 16 (3) (2003), pp. 1119–1149.
   Web of Science ®Google Scholar
• N.V. Chemetov and F. Cipriano, Boundary layer problem: Navier–Stokes equations and Euler equations, Nonlinear Anal. Real World Appl. 14(6) (2013), pp. 2091–2104.
   Web of Science ®Google Scholar
• N.V. Chemetov and F. Cipriano, The inviscid limit for the Navier–Stokes equations with slip condition on permeable walls, J. Nonlinear Sci. 23(5) (2013), pp. 731–750.
   Web of Science ®Google Scholar
• N.V. Chemetov and F. Cipriano, Well-posedness of stochastic second grade fluids, J. Math. Anal. Appl. 454 (2) (2017), pp. 585–616.
   Web of Science ®Google Scholar
• N.V. Chemetov and F. Cipriano, Optimal control for two-dimensional stochastic second grade fluids, Stochastic Processes Appl. 128(8) (2018), pp. 2710–2749.
   Web of Science ®Google Scholar
• I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviations, Appl. Math. Optim. 61(3) (2010), pp. 379–420.
   Web of Science ®Google Scholar
• F. Cipriano, On the asymptotic behaviour and stochastic stabilization of second grade fluids, Stochastics 91(7) (2019), pp. 1020–1040.
   Google Scholar
• F. Cipriano and T. Costa, A large deviations principle for stochastic flows of viscous fluids, J. Differ. Equ. 264 (2019), pp. 5070–5108.
   Web of Science ®Google Scholar
• F. Cipriano and D. Pereira, On the existence of optimal and ϵ-optimal feedback controls for stochastic second grade fluids, J. Math. Anal. Appl. 475 (2) (2019), pp. 1956–1977.
   Web of Science ®Google Scholar
• F. Cipriano, F. Didier, and S. Guerra, Well-posedness of stochastic third grade fluid equation, J. Differ. Equ. 285 (2021), pp. 496–535.
   Web of Science ®Google Scholar
• T. Clopeau, A. Mikelic, and R. Robert, On the vanishing viscosity limit for the 2D incompressible Navier–Stokes equations with the friction type boundary conditions, Nonlinearity 11 (6) (1998), pp. 1625–1636.
   Web of Science ®Google Scholar
• A. Dembo and A. Zeitouni, Large Deviations Techniques and Applications, Springer, Berlin, 1998.
   Google Scholar
• L. Desvillettes and C. Villani, On a variant of Korn's inequality arising in statistical mechanics, ESAIM Control Optim. Calc. Var. 8 (2002), pp. 603–619.
   Web of Science ®Google Scholar
• P. Dupuis and R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, John Wiley & Sons, Inc, 1997.
   Google Scholar
• R.L. Fosdick and K.R. Rajagopal, Thermodynamics and stability of fuids of third grade, Proc. Roy. Soc. Lond. Ser. A 339 (1980), pp. 351–377.
   Web of Science ®Google Scholar
• M.I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, New York, 1984.
   Google Scholar
• T. Grafke and E. Vanden-Eijnden, Numerical computation of rare events via large deviation theory, Chaos 29 (6) (2019), pp. 063118.
   Google Scholar
• L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa. 20 (1966), pp. 733–737.
   Google Scholar
• A. Rasheed, A. Kausar, A. Wahab, and T. Akbar, Stabilized approximation of steady flow of third grade fluid in presence of partial slip, Results Phys. 7 (2017), pp. 3181–3189.
   Web of Science ®Google Scholar
• P.A. Razafimandimby, Viscosity limit and deviations principles for a grade-two fluid driven by multiplicative noise, Ann. Mat. Pura Appl. 197(5) (2018), pp. 1547–1583.
   Web of Science ®Google Scholar
• P.A. Razafimandimby and M. Sango, Asymptotic behaviour of solutions of stochastic evolution equations for second grade fluids, C. R. Acad. Sci. Paris Ser. I. 348(13-14) (2010), pp. 787–790.
   Google Scholar
• P.A. Razafimandimby and M. Sango, Weak solutions of a stochastic model for two-dimensional second grade fluids, Bound. Value Probl. 2010, (2010), Article number 636140, pp. 1–47.(electr. version).
   Web of Science ®Google Scholar
• P.A. Razafimandimby and M. Sango, Strong solution for a stochastic model of two-dimensional second grade fluids: existence, uniqueness and asymptotic behavior, Nonlinear Anal. Theory Methods Appl. 75 (11) (2012), pp. 4251–4270.
   Web of Science ®Google Scholar
• R.S. Rivlin and J.L. Ericksen, Stress-deformation relations for isotropic materials, Arch. Ration. Mech. Anal. 4 (1955), pp. 323–425.
   Web of Science ®Google Scholar
• M. Röckner and T.S. Zhang, Large deviations for stochastic tamed 3D NavierStokes equations, Appl. Math. Optim. 61(2) (2010), pp. 267–285.
   Web of Science ®Google Scholar
• S. Shang, J. Zhai, and T. Zhang, Strong solutions for a stochastic model of two-dimensional second grade fluids driven by Levy noise, J. Math. Anal. Appl. 471(1-2) (2019), pp. 126–146.
   Web of Science ®Google Scholar
• J. Simon, Compact sets in the space Lp(0,T;B), Ann. Mat. Pura. Appl. (IV) CXLVI (1987), pp. 65–86.
   Google Scholar
• S.S. Sritharan and P. Sundar, Large deviations for the two-dimensional NavierStokes equations with multiplicative noise, Stoch. Process. Appl. 116(11) (2006), pp. 1636–1659.
   Web of Science ®Google Scholar
• S.R.S. Varadhan, Large Deviations and Applications, SIAM Philadelphia, Pennsylvania, 1984.
   Google Scholar
• J. Zhai and T. Zhang, Large deviations for stochastic models of two-dimensional second grade fluids, Appl. Math. Optim. 75(3) (2017), pp. 471–498.
   Web of Science ®Google Scholar
• J. Zhai, T. Zhang, and W. Zheng, Moderate deviations for stochastic models of two-dimensional second grade fluids, Stoch. Dyn. 18(03) (2018), pp. 1850026.
   Google Scholar