[Paper Review] The stochastic primitive equations with non-isothermal turbulent pressure

Antonio Agresti, Matthias Hieber+2 more|arXiv (Cornell University)|Oct 12, 2022

Stochastic processes and financial applications4 citations

TL;DR

This paper introduces and establishes global well-posedness for the stochastic primitive equations with non-isothermal turbulent pressure and transport noise, derived from the Navier–Stokes equations via stochastic Boussinesq and hydrostatic approximations. The key contribution is proving $ H^1 $-well-posedness in both Itô and Stratonovich formulations, with novel energy estimates and continuous dependence on initial data, even in the isothermal case.

ABSTRACT

In this paper, we introduce and study the primitive equations with $ extit{non}$-isothermal turbulent pressure and transport noise. They are derived from the Navier-Stokes equations by employing stochastic versions of the Boussinesq and the hydrostatic approximations. The temperature dependence of the turbulent pressure can be seen as a consequence of an additive noise acting on the small vertical dynamics. For such a model we prove global well-posedness in $H^1$ where the noise is considered in both the Itô and Stratonovich formulations. Compared to previous variants of the primitive equations, the one considered here presents a more intricate coupling between the velocity field and the temperature. The corresponding analysis is seriously more involved than in the deterministic setting. Finally, the continuous dependence on the initial data and the energy estimates proven here are new, even in the case of isothermal turbulent pressure.

Motivation & Objective

• To model geophysical flows with non-isothermal turbulent pressure arising from thermal fluctuations acting on small vertical dynamics.
• To incorporate transport noise into the primitive equations, motivated by Kraichnan's theory of turbulence and subgrid-scale parameterizations.
• To establish global well-posedness in $ H^1 $ for the resulting stochastic system under both Itô and Stratonovich formulations.
• To provide new energy estimates and continuous dependence on initial data, extending beyond previous deterministic and isothermal stochastic settings.
• To rigorously analyze the enhanced coupling between velocity and temperature due to temperature-dependent turbulent pressure, which increases mathematical complexity.

Proposed method

• Derive the stochastic primitive equations from the Navier–Stokes equations using stochastic versions of the Boussinesq and hydrostatic approximations.
• Model the non-isothermal turbulent pressure as a consequence of additive noise acting on vertical dynamics, introducing temperature dependence in the pressure term.
• Employ transport noise (Kraichanan-type) to represent subgrid-scale stochasticity in velocity and temperature fields.
• Apply the hydrostatic Helmholtz projection to reformulate the system into a coupled system of stochastic PDEs for velocity and temperature.
• Use stochastic maximal regularity and a priori estimates in $ H^1 $-based spaces to control nonlinearities and noise terms.
• Establish energy estimates and continuous dependence via detailed $ L^2 $-based estimates in time and space, including martingale and gradient noise terms.

Experimental results

Research questions

• RQ1 How does the inclusion of temperature-dependent turbulent pressure affect the well-posedness and regularity of the stochastic primitive equations?
• RQ2 Can global $ H^1 $-well-posedness be established for the stochastic primitive equations with non-isothermal turbulent pressure under both Itô and Stratonovich noise formulations?
• RQ3 What is the impact of transport noise on the coupling between velocity and temperature fields in geophysical flow models?
• RQ4 How do energy estimates and continuous dependence on initial data differ in the non-isothermal case compared to the isothermal or deterministic settings?
• RQ5 To what extent does the non-isothermal pressure term introduce new analytical challenges beyond those in standard stochastic primitive equations?

Key findings

• Global well-posedness in $ H^1 $ is established for the stochastic primitive equations with non-isothermal turbulent pressure in both Itô and Stratonovich formulations.
• The energy estimates derived in this work are new even in the isothermal case, providing stronger a priori control on solutions.
• Continuous dependence on initial data is proven, extending the regularity and stability analysis beyond previous results.
• The system exhibits a more intricate coupling between velocity and temperature due to the temperature dependence of the turbulent pressure, increasing analytical complexity.
• The use of stochastic maximal regularity and detailed $ L^2 $-estimates allows control of nonlinear terms and noise interactions, particularly involving vertical derivatives.
• The Stratonovich formulation is shown to be equivalent to an Itô-type system with additional drift terms arising from the noise's quadratic variation, with global existence preserved.