%0 Generic %1 bedrossian2018lagrangian %A Bedrossian, Jacob %A Blumenthal, Alex %A Punshon-Smith, Samuel %D 2018 %K navier-stokes %T Lagrangian chaos and scalar advection in stochastic fluid mechanics %U http://arxiv.org/abs/1809.06484 %X We study the Lagrangian flow associated to velocity fields arising from various models of fluid mechanics subject to white-in-time, $H^s$-in-space stochastic forcing in a periodic box. We prove that in many circumstances, these flows are chaotic, that is, the top Lyapunov exponent is strictly positive. Our main results are for the Navier-Stokes equations on $T^2$ and the hyper-viscous regularized Navier-Stokes equations on $T^3$ (at arbitrary Reynolds number and hyper-viscosity parameters), subject to forcing which is non-degenerate at high frequencies. As an application, we study statistically stationary solutions to the passive scalar advection-diffusion equation driven by these velocities and subjected to random sources. The chaotic Lagrangian dynamics are used to prove a version of anomalous dissipation in the limit of vanishing diffusivity, which in turn, implies that the scalar satisfies Yaglom's law of scalar turbulence -- the analogue of the Kolmogorov 4/5 law. Key features of our study are the use of tools from ergodic theory and random dynamical systems, namely the Multiplicative Ergodic Theorem and a version of Furstenberg's Criterion, combined with hypoellipticity via Malliavin calculus and approximate control arguments.

@misc{bedrossian2018lagrangian, abstract = {We study the Lagrangian flow associated to velocity fields arising from various models of fluid mechanics subject to white-in-time, $H^s$-in-space stochastic forcing in a periodic box. We prove that in many circumstances, these flows are chaotic, that is, the top Lyapunov exponent is strictly positive. Our main results are for the Navier-Stokes equations on $\mathbb T^2$ and the hyper-viscous regularized Navier-Stokes equations on $\mathbb T^3$ (at arbitrary Reynolds number and hyper-viscosity parameters), subject to forcing which is non-degenerate at high frequencies. As an application, we study statistically stationary solutions to the passive scalar advection-diffusion equation driven by these velocities and subjected to random sources. The chaotic Lagrangian dynamics are used to prove a version of anomalous dissipation in the limit of vanishing diffusivity, which in turn, implies that the scalar satisfies Yaglom's law of scalar turbulence -- the analogue of the Kolmogorov 4/5 law. Key features of our study are the use of tools from ergodic theory and random dynamical systems, namely the Multiplicative Ergodic Theorem and a version of Furstenberg's Criterion, combined with hypoellipticity via Malliavin calculus and approximate control arguments.}, added-at = {2019-12-14T18:20:27.000+0100}, author = {Bedrossian, Jacob and Blumenthal, Alex and Punshon-Smith, Samuel}, biburl = {https://www.bibsonomy.org/bibtex/228ce781f648555d55f32ee134533d9ff/sarvilpargi}, description = {Lagrangian chaos and scalar advection in stochastic fluid mechanics}, interhash = {f33470769c419b1686d28ada4e8d98db}, intrahash = {28ce781f648555d55f32ee134533d9ff}, keywords = {navier-stokes}, note = {cite arxiv:1809.06484Comment: 72 pages}, timestamp = {2019-12-14T18:20:27.000+0100}, title = {Lagrangian chaos and scalar advection in stochastic fluid mechanics}, url = {http://arxiv.org/abs/1809.06484}, year = 2018 }