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Hölder regularity of the densities for the Navier-Stokes equations with noise. (English) Zbl 1352.60093
Summary: We prove that the densities of the finite dimensional projections of weak solutions of the Navier-Stokes equations driven by Gaussian noise are bounded and Hölder continuous, thus improving the results of A. Debussche and the author [Probab. Theory Relat. Fields 158, No. 3–4, 575–596 (2014; Zbl 1452.76193)]. The proof is based on analytical estimates on a conditioned Fokker-Planck equation solved by the density, that has a “non-local” term that takes into account the influence of the rest of the infinite dimensional dynamics over the finite subspace under observation.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G30 Continuity and singularity of induced measures
35R60 PDEs with randomness, stochastic partial differential equations
35Q84 Fokker-Planck equations
35Q30 Navier-Stokes equations
Full Text: DOI
[1] Bally, V., Caramellino, L, .: Convergence and regularity of probability laws by using an interpolation method (2014). arXiv:1409.3118 [math.AP] · Zbl 1377.60066
[2] Debussche, A.: Ergodicity results for the stochastic Navier-Stokes equations: an introduction, Topics in mathematical fluid mechanics, Lecture Notes in Math., vol. 2073. Springer, Heidelberg, 2013. Lectures given at the C.I.M.E.-E.M.S. Summer School in applied mathematics held in Cetraro, September 6-11, 2010, Edited by Franco Flandoli and Hugo Beirao da Veiga, pp. 23-108 (2013) · Zbl 1301.35086
[3] Debussche, A; Fournier, N, Existence of densities for stable-like driven SDE’s with Hölder continuous coefficients, J. Funct. Anal., 264, 1757-1778, (2013) · Zbl 1272.60032
[4] Marco, S, Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions, Ann. Appl. Probab., 21, 1282-1321, (2011) · Zbl 1246.60082
[5] Debussche, A; Odasso, C, Markov solutions for the 3D stochastic Navier-Stokes equations with state dependent noise, J. Evol. Equ., 6, 305-324, (2006) · Zbl 1110.35110
[6] Da, G; Debussche, A, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl. (9), 82, 877-947, (2003) · Zbl 1109.60047
[7] Da Prato, Giuseppe, Zabczyk, Jerzy: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992) · Zbl 0761.60052
[8] Debussche, A; Romito, M, Existence of densities for the 3D Navier-Stokes equations driven by Gaussian noise, Probab. Theory Relat. Fields, 158, 575-596, (2014) · Zbl 1452.76193
[9] Flandoli, F.: An introduction to 3D stochastic fluid dynamics, SPDE in hydrodynamic: recent progress and prospects. In: Prato, G.D., Röckner, M. (eds.), Lecture Notes in Math., vol. 1942. Springer, Berlin, 2008, Lectures given at the C.I.M.E. Summer School held in Cetraro, August 29-September 3, 2005, pp. 51-150 (2008)
[10] Fournier, N, Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition, Ann. Appl. Probab., 25, 860-897, (2015) · Zbl 1322.82013
[11] Fournier, N; Printems, J, Absolute continuity for some one-dimensional processes, Bernoulli, 16, 343-360, (2010) · Zbl 1248.60062
[12] Flandoli, F; Romito, M, Markov selections and their regularity for the three-dimensional stochastic Navier-Stokes equations, C. R. Math. Acad. Sci. Paris, 343, 47-50, (2006) · Zbl 1098.60059
[13] Flandoli, F., Romito, M.: Regularity of transition semigroups associated to a 3D stochastic Navier-Stokes equation, stochastic differential equations: theory and applications. In: Baxendale, P.H., Lototski, S.V. (eds.) Interdiscip. Math. Sci., vol. 2, pp. 263-280. World Sci. Publ, Hackensack (2007) · Zbl 1136.35112
[14] Flandoli, F; Romito, M, Markov selections for the 3D stochastic Navier-Stokes equations, Probab. Theory Relat. Fields, 140, 407-458, (2008) · Zbl 1133.76016
[15] Hayashi, M; Kohatsu-Higa, A; Yûki, G, Local Hölder continuity property of the densities of solutions of SDEs with singular coefficients, J. Theor. Probab., 26, 1117-1134, (2013) · Zbl 1294.60081
[16] Hayashi, M; Kohatsu-Higa, A; Yûki, G, Hölder continuity property of the densities of SDEs with singular drift coefficients, Electron. J. Probab., 19, 22, (2014) · Zbl 1310.60081
[17] Kohatsu-Higa, A; Tanaka, A, A Malliavin calculus method to study densities of additive functionals of SDE’s with irregular drifts, Ann. Inst. Henri Poincaré Probab. Stat., 48, 871-883, (2012) · Zbl 1248.60058
[18] Jonathan, C, Mattingly and étienne pardoux, Malliavin calculus for the stochastic 2D Navier-Stokes equation, Commun. Pure Appl. Math., 59, 1742-1790, (2006) · Zbl 1113.60058
[19] Nualart, D.: The Malliavin Calculus and Related Topics. Probability and its Applications (New York), 2nd edn. Springer, Berlin (2006) · Zbl 1099.60003
[20] Odasso, C, Exponential mixing for the 3D stochastic Navier-Stokes equations, Commun. Math. Phys., 270, 109-139, (2007) · Zbl 1122.60059
[21] Romito, M, Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise, J. Stat. Phys., 114, 155-177, (2004) · Zbl 1060.76027
[22] Romito, M, Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise, J. Stat. Phys., 131, 415-444, (2008) · Zbl 1150.82027
[23] Romito, M.: Densities for the Navier-Stokes equations with noise, 2013, Lecture notes for the “Winter school on stochastic analysis and control of fluid flow”, School of Mathematics of the Indian Institute of Science Education and Research, Thiruvananthapuram (India) ( 2013) · Zbl 1452.76193
[24] Romito, M.: Unconditional existence of densities for the Navier-Stokes equations with noise, Mathematical analysis of viscous incompressible fluid, RIMS Kôkyûroku, vol. 1905, pp. 5-17. Kyoto University (2014) · Zbl 1248.60062
[25] Romito, M.: Some probabilistic topics in the Navier-Stokes equations. In: Recent Progress in the Theory of the Euler and Navier-Stokes Equations. London MathematicalSociety Lecture Notes Series, vol. 430, Cambridge University Press, pp. 175-228 (2016) · Zbl 1362.35215
[26] Romito, M.: Time regularity of the densities for the Navier-Stokes equations with noise. J. Evol. Equ. (2016), (to appear) · Zbl 1360.76218
[27] Romito, M; Lihu, X, Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noise, Stoch. Process. Appl., 121, 673-700, (2011) · Zbl 1369.76048
[28] Sanz-Solé, M; Süß, A, Absolute continuity for SPDEs with irregular fundamental solution, Electron. Commun. Probab., 20, 11, (2015) · Zbl 1321.60138
[29] Sanz-Solé, M., Süß, A.: Non elliptic SPDEs and ambit fields: existence of densities, (2015). arXiv:1502.02386 · Zbl 0763.46025
[30] Temam, R.: Navier-Stokes equations and nonlinear functional analysis, 2nd edn. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1995) · Zbl 0833.35110
[31] Triebel, H, Theory of function spaces, No. 78, (1983), Basel · Zbl 0546.46028
[32] Triebel, H, Theory of function spaces. II, (1992), Basel · Zbl 0763.46025
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