Nonlinear Fokker-Planck equations as smooth Hilbertian gradient flows
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866929759661850624 |
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| author | Barbu, Viorel Röckner, Michael |
| author_facet | Barbu, Viorel Röckner, Michael |
| contents | Under suitable assumptions on $β:\mathbb{R}\!\to\!\mathbb{R}, \,D:\mathbb{R}^d\!\to\!\mathbb{R}^d$ and $b:\mathbb{R}^d\!\to\!\mathbb{R}$, the nonlinear Fokker-Planck equation $u_t-Δβ(u)+{\rm div}(Db(u)u)=0$, in $(0,\infty)\times\mathbb{R}^d$ where $D=-\nablaΦ$, can be identified as a smooth gradient flow $\frac{d^+}{dt}\,u(t)+\nabla E_{u(t)}=0$, $\forall t>0$. Here, $E:\mathcal{P}^*\cap L^\infty(\mathbb{R}^d)\to\mathbb{R}$ is the energy function associated to the equation, where $\mathcal{P}^*$ is a certain convex subset of the space of probability densities. $\mathcal{P}^*$ is invariant under the flow and $\nabla E_u$ is the gradient of $E$, that is, the tangent vector field to $\mathcal{P}$ at $u$ defined by $\left<\nabla E_u,z_u\right>_u={\rm diff}\,E_u\cdot z_u$ for all vector fields $z_u$ on $\mathcal{P}^*$, where $\left<\cdot,\cdot\right>_u$ is a scalar product on a suitable tangent space $\mathcal{T}_u(\mathcal{P}^*)\subset\mathcal{D}'(\mathbb{R}^d)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10906 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear Fokker-Planck equations as smooth Hilbertian gradient flows Barbu, Viorel Röckner, Michael Probability Analysis of PDEs 60H15, 47H05, 47J05 Under suitable assumptions on $β:\mathbb{R}\!\to\!\mathbb{R}, \,D:\mathbb{R}^d\!\to\!\mathbb{R}^d$ and $b:\mathbb{R}^d\!\to\!\mathbb{R}$, the nonlinear Fokker-Planck equation $u_t-Δβ(u)+{\rm div}(Db(u)u)=0$, in $(0,\infty)\times\mathbb{R}^d$ where $D=-\nablaΦ$, can be identified as a smooth gradient flow $\frac{d^+}{dt}\,u(t)+\nabla E_{u(t)}=0$, $\forall t>0$. Here, $E:\mathcal{P}^*\cap L^\infty(\mathbb{R}^d)\to\mathbb{R}$ is the energy function associated to the equation, where $\mathcal{P}^*$ is a certain convex subset of the space of probability densities. $\mathcal{P}^*$ is invariant under the flow and $\nabla E_u$ is the gradient of $E$, that is, the tangent vector field to $\mathcal{P}$ at $u$ defined by $\left<\nabla E_u,z_u\right>_u={\rm diff}\,E_u\cdot z_u$ for all vector fields $z_u$ on $\mathcal{P}^*$, where $\left<\cdot,\cdot\right>_u$ is a scalar product on a suitable tangent space $\mathcal{T}_u(\mathcal{P}^*)\subset\mathcal{D}'(\mathbb{R}^d)$. |
| title | Nonlinear Fokker-Planck equations as smooth Hilbertian gradient flows |
| topic | Probability Analysis of PDEs 60H15, 47H05, 47J05 |
| url | https://arxiv.org/abs/2503.10906 |