On the well-posedness of (nonlinear) rough continuity equations
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arXiv
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| Format: | Preprint |
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2025
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| author | Galeati, Lucio Leahy, James-Michael Nilssen, Torstein |
| author_facet | Galeati, Lucio Leahy, James-Michael Nilssen, Torstein |
| contents | Motivated by applications to fluid dynamics, we study rough differential equations (RDEs) and rough partial differential equations (RPDEs) with non-Lipschitz drifts. We prove well-posedness and existence of a flow for RDEs with Osgood drifts, as well as well-posedness of weak $L^p$-valued solutions to linear rough continuity and transport equations on $\mathbb{R}^d$ under DiPerna--Lions regularity conditions; a combination of the two then yields flow representation formula for linear RPDEs. We apply these results to obtain existence, uniqueness and continuous dependence for $L^1\cap L^\infty$-valued solutions to a general class of nonlinear continuity equations. In particular, our framework covers the $2$D Euler equations in vorticity form with rough transport noise, providing a rough analogue of Yudovich's theorem. As a consequence, we construct an associated continuous random dynamical system, when the driving noise is a fractional Brownian motion with Hurst parameter $H \in (1/3,1)$. We further prove weak existence of solutions for initial vorticities in $L^1\cap L^p$, for any $p\in [1,\infty)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_04982 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the well-posedness of (nonlinear) rough continuity equations Galeati, Lucio Leahy, James-Michael Nilssen, Torstein Analysis of PDEs Probability 60L20, 60L50, 60H15, 35R60, 35Q31 Motivated by applications to fluid dynamics, we study rough differential equations (RDEs) and rough partial differential equations (RPDEs) with non-Lipschitz drifts. We prove well-posedness and existence of a flow for RDEs with Osgood drifts, as well as well-posedness of weak $L^p$-valued solutions to linear rough continuity and transport equations on $\mathbb{R}^d$ under DiPerna--Lions regularity conditions; a combination of the two then yields flow representation formula for linear RPDEs. We apply these results to obtain existence, uniqueness and continuous dependence for $L^1\cap L^\infty$-valued solutions to a general class of nonlinear continuity equations. In particular, our framework covers the $2$D Euler equations in vorticity form with rough transport noise, providing a rough analogue of Yudovich's theorem. As a consequence, we construct an associated continuous random dynamical system, when the driving noise is a fractional Brownian motion with Hurst parameter $H \in (1/3,1)$. We further prove weak existence of solutions for initial vorticities in $L^1\cap L^p$, for any $p\in [1,\infty)$. |
| title | On the well-posedness of (nonlinear) rough continuity equations |
| topic | Analysis of PDEs Probability 60L20, 60L50, 60H15, 35R60, 35Q31 |
| url | https://arxiv.org/abs/2502.04982 |