The Incompressible Navier--Stokes--Fourier System with Thermal Noise
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911548112371712 |
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| author | Gess, Benjamin Sauerbrey, Max Wu, Zhengyan |
| author_facet | Gess, Benjamin Sauerbrey, Max Wu, Zhengyan |
| contents | We establish a solution theory for the incompressible Navier--Stokes--Fourier system with thermal noise, posed on the three-dimensional torus. While in the incompressible deterministic setting the equation for the velocity can be solved independently of the temperature, the inclusion of the effects of thermal fluctuations by means of the GENERIC framework leads to a nonlinear gradient noise term, which couples the dynamics of both variables. Therefore, the analysis poses new challenges, which are absent in the deterministic incompressible Navier--Stokes--Fourier equations. In particular, the a priori estimates used in the deterministic setting are not readily generalizable, the noise introduces strongly nonlinear gradient terms and the total energy lacks convexity.
These challenges are overcome in the present work by a novel variable transformation, and novel entropy dissipation estimates. Thereby, the existence of global-in-time weak solutions for $L_x^2$ initial data, the existence of local-in-time strong solutions for regular initial data, and weak-strong uniqueness are obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26307 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Incompressible Navier--Stokes--Fourier System with Thermal Noise Gess, Benjamin Sauerbrey, Max Wu, Zhengyan Probability Analysis of PDEs We establish a solution theory for the incompressible Navier--Stokes--Fourier system with thermal noise, posed on the three-dimensional torus. While in the incompressible deterministic setting the equation for the velocity can be solved independently of the temperature, the inclusion of the effects of thermal fluctuations by means of the GENERIC framework leads to a nonlinear gradient noise term, which couples the dynamics of both variables. Therefore, the analysis poses new challenges, which are absent in the deterministic incompressible Navier--Stokes--Fourier equations. In particular, the a priori estimates used in the deterministic setting are not readily generalizable, the noise introduces strongly nonlinear gradient terms and the total energy lacks convexity. These challenges are overcome in the present work by a novel variable transformation, and novel entropy dissipation estimates. Thereby, the existence of global-in-time weak solutions for $L_x^2$ initial data, the existence of local-in-time strong solutions for regular initial data, and weak-strong uniqueness are obtained. |
| title | The Incompressible Navier--Stokes--Fourier System with Thermal Noise |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2603.26307 |