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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.

MSC:
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
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