Existence of densities for the 3D Navier–Stokes equations driven by Gaussian noise

Abstract

We prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic Navier-Stokes equations in dimension 3. In particular, under very mild assumptions on the noise, we prove that finite dimensional projections of the solutions have densities with respect to the Lebesgue measure which have some smoothness when measured in a Besov space. This is proved thanks to a new argument inspired by an idea introduced in [18]

Hölder regularity of the densities for the Navier–Stokes equations with noise

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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 Debussche and Romito [8]. 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.

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“…This settles the need of methods to prove existence and regularity of the density that do not rely on this calculus, as done in [8]. For other works in this direction, see for instance [1,4,[15][16][17].…”

“…The results of [8] ensure that the density of the projection of solution at some fixed time on some finite dimensional sub-space is in the Besov space B 1− 1,∞ . Roughly speaking, this says that densities (almost) have integrable derivative (see Sect.…”

“…Time regularity of the density has been proved in [26]. The method introduced in [8] is simple but effective and has been already used in other problems (see for instance [3,10,28,29]). …”

“…This makes our Fokker-Planck equation slightly non-standard. We re-derive in this framework the results of [8] (see Proposition 4.1), we then prove the core result of the paper, namely boundedness of the densities, in Proposition 4.4, and finally the Hölder regularity.…”