Nontrivial absolutely continuous part of anomalous dissipation measures in time
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866910029351747584 |
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| author | Johansson, Carl Johan Peter Sorella, Massimo |
| author_facet | Johansson, Carl Johan Peter Sorella, Massimo |
| contents | We positively answer Question 2.2 and Question 2.3 in [Bruè, De Lellis, 2023] in dimension $4$ by building new examples of solutions to the forced $4d$ incompressible Navier-Stokes equations, which exhibit anomalous dissipation, related to the zeroth law of turbulence [K41]. We also prove that the unique smooth solution $v_ν$ of the $4d$ Navier--Stokes equations with time-independent body forces is $L^\infty$-weakly* converging to a solution of the forced Euler equations $v_0$ as the viscosity parameter $ν\to 0$. Furthermore, the sequence $ν|\nabla v_ν|^2$ is weakly* converging (up to subsequences), in the sense of measure, to $μ\in \mathcal{M} ((0,1) \times \mathbb{T}^4)$ and $μ_T = π_{\#} μ$ has a non-trivial absolutely continuous part where $π$ is the projection onto the time variable. Moreover, we also show that $μ$ is close, up to an error measured in $H^{-1}_{t,x}$, to the Duchon--Robert distribution $\mathcal{D}[v_0]$ of the solution to the $4d$ forced Euler equations. Finally, the kinetic energy profile of $v_0$ is smooth in time. Our result relies on a new anomalous dissipation result for the advection--diffusion equation with a divergence free $3d$ autonomous velocity field and the study of the $3+\frac{1}{2} $ dimensional incompressible Navier--Stokes equations. This study motivates some open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_09486 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nontrivial absolutely continuous part of anomalous dissipation measures in time Johansson, Carl Johan Peter Sorella, Massimo Analysis of PDEs Mathematical Physics 76F02, 76D05, 76F10, 35Q31 We positively answer Question 2.2 and Question 2.3 in [Bruè, De Lellis, 2023] in dimension $4$ by building new examples of solutions to the forced $4d$ incompressible Navier-Stokes equations, which exhibit anomalous dissipation, related to the zeroth law of turbulence [K41]. We also prove that the unique smooth solution $v_ν$ of the $4d$ Navier--Stokes equations with time-independent body forces is $L^\infty$-weakly* converging to a solution of the forced Euler equations $v_0$ as the viscosity parameter $ν\to 0$. Furthermore, the sequence $ν|\nabla v_ν|^2$ is weakly* converging (up to subsequences), in the sense of measure, to $μ\in \mathcal{M} ((0,1) \times \mathbb{T}^4)$ and $μ_T = π_{\#} μ$ has a non-trivial absolutely continuous part where $π$ is the projection onto the time variable. Moreover, we also show that $μ$ is close, up to an error measured in $H^{-1}_{t,x}$, to the Duchon--Robert distribution $\mathcal{D}[v_0]$ of the solution to the $4d$ forced Euler equations. Finally, the kinetic energy profile of $v_0$ is smooth in time. Our result relies on a new anomalous dissipation result for the advection--diffusion equation with a divergence free $3d$ autonomous velocity field and the study of the $3+\frac{1}{2} $ dimensional incompressible Navier--Stokes equations. This study motivates some open problems. |
| title | Nontrivial absolutely continuous part of anomalous dissipation measures in time |
| topic | Analysis of PDEs Mathematical Physics 76F02, 76D05, 76F10, 35Q31 |
| url | https://arxiv.org/abs/2303.09486 |