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Time regularity of the densities for the Navier-Stokes equations with noise. (English) Zbl 1360.76218
Summary: We prove that the density of the law of any finite-dimensional projection of solutions of the Navier-Stokes equations with noise in dimension three is Hölder continuous in time with values in the natural space $$L^{1}$$. When considered with values in Besov spaces, Hölder continuity still holds. The Hölder exponents correspond, up to arbitrarily small corrections, to the expected, at least with the known regularity, diffusive scaling.

##### MSC:
 76M35 Stochastic analysis applied to problems in fluid mechanics 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60G30 Continuity and singularity of induced measures 35Q30 Navier-Stokes equations
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##### References:
 [1] Aronszajn, N.; Smith, K. T., Theory of Bessel potentials I, Ann. Inst. Fourier (Grenoble), 11, 385-475, (1961) · Zbl 0102.32401 [2] Vlad Bally and Lucia Caramellino, Regularity of probability laws by using an interpolation method, 2012, arXiv:1211.0052 [math.AP]. · Zbl 1371.60096 [3] Debussche, Arnaud; Fournier, Nicolas, Existence of densities for stable-like driven SDE’s with Hölder continuous coefficients, J. Funct. Anal., 264, 1757-1778, (2013) · Zbl 1272.60032 [4] De Marco, Stefano, 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] Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. · Zbl 0761.60052 [6] Debussche, Arnaud; Romito, Marco, Existence of densities for the 3D Navier-Stokes equations driven by Gaussian noise, Probab. Theory Related Fields, 158, 575-596, (2014) · Zbl 1452.76193 [7] Franco Flandoli, An introduction to 3D stochastic fluid dynamics, SPDE in hydrodynamic: recent progress and prospects, 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, Edited by Giuseppe Da Prato and Michael Röckner, pp. 51-150. · Zbl 1246.60082 [8] Nicolas Fournier, Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition, 2012, to appear on Ann. Appl. Probab. · Zbl 1322.82013 [9] Fournier, Nicolas; Printems, Jacques, Absolute continuity for some one-dimensional processes, Bernoulli, 16, 343-360, (2010) · Zbl 1248.60062 [10] Masafumi Hayashi, Arturo Kohatsu-Higa, and Gô Yûki, Hölder continuity property of the densities of SDEs with singular drift coefficients, 2013, preprint. · Zbl 1294.60081 [11] Hayashi, Masafumi; Kohatsu-Higa, Arturo; Yûki, Gô, Local Hölder continuity property of the densities of solutions of SDEs with singular coefficients, J. Theoret. Probab., 26, 1117-1134, (2013) · Zbl 1294.60081 [12] Kohatsu-Higa, Arturo; Tanaka, Akihiro, 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 [13] Robert S. Liptser and Albert N. Shiryaev, Statistics of random processes. I, expanded ed., Applications of Mathematics (New York), vol. 5, Springer, Berlin, 2001, General theory, Translated from the 1974 Russian original by A. B. Aries, Stochastic Modelling and Applied Probability. · Zbl 1008.62072 [14] Marco Romito, 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). [15] Marco Romito, Unconditional existence of densities for the Navier-Stokes equations with noise, Mathematical analysis of viscous incompressible fluid, vol. 1905, RIMS, Kyoto University, 2014, pp. 5-17. [16] Sanz-Solé, Marta, Properties of the density for a three-dimensional stochastic wave equation, J. Funct. Anal., 255, 255-281, (2008) · Zbl 1152.60049 [17] Roger Temam, Navier-Stokes equations and nonlinear functional analysis, second ed., CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. · Zbl 0833.35110 [18] Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. · Zbl 0546.46028 [19] Hans Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992. · Zbl 0763.46025
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