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PreprintCC BY 4.0arXiv

Gaussian measures under the Navier-Stokes and heat flows are mutually singular

  • Niko Nikov
  • Leonardo Tolomeo
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Abstract

We consider the incompressible Navier-Stokes equations on the torus 𝕋d, d ≥ 2, with translation invariant Gaussian random initial data. We show that, for the Gaussian measure formally given by μ0 ∼ exp(−1/2 | u |Ḣα2) on divergence-free fields, its nonlinear evolution νt and its linear evolution μt are mutually singular for almost every time. Moreover, the same holds for the nonlinear evolution of an arbitrary initial measure ν0. This is in sharp contrast to the available results for Hamiltonian PDEs, where Gaussian measures are commonly quasi-invariant. We achieve this by showing that there exists a set 𝒢t such that μt(𝒢t) = 1, but for any smooth solution of Navier-Stokes, the set of times t such that u(t) ∈ 𝒢t is null, which we deduce from a quantitative convexity bound for frequency-localised versions of the (typically infinite) quantity | e−t Δ u(t) |L22.

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Cite

BibTeXRIS
BibTeX
@misc{nikov2026gaussian,
  title = {{Gaussian measures under the Navier-Stokes and heat flows are mutually singular}},
  author = {Niko Nikov and Leonardo Tolomeo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23591},
  eprint = {2610.23591},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23591},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Gaussian measures under the Navier-Stokes and heat flows are mutually singular
AU  - Niko Nikov
AU  - Leonardo Tolomeo
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23591
UR  - https://arxiv.org/abs/2610.23591
AB  - We consider the incompressible Navier-Stokes equations on the torus 𝕋^d, d ≥ 2, with translation invariant Gaussian random initial data. We show that, for the Gaussian measure formally given by μ_0 ∼ exp(−1/2 | u |_Ḣ^α^2) on divergence-free fields, its nonlinear evolution ν_t and its linear evolution μ_t are mutually singular for almost every time. Moreover, the same holds for the nonlinear evolution of an arbitrary initial measure ν_0. This is in sharp contrast to the available results for Hamiltonian PDEs, where Gaussian measures are commonly quasi-invariant. We achieve this by showing that there exists a set 𝒢_t such that μ_t(𝒢_t) = 1, but for any smooth solution of Navier-Stokes, the set of times t such that u(t) ∈ 𝒢_t is null, which we deduce from a quantitative convexity bound for frequency-localised versions of the (typically infinite) quantity | e^−t Δ u(t) |_L^2^2.
ER  -