Large field problem in coercive singular PDEs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917148116385792 |
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| author | Chevyrev, Ilya Gubinelli, Massimiliano |
| author_facet | Chevyrev, Ilya Gubinelli, Massimiliano |
| contents | We derive a priori estimates for singular differential equations of the form \[ \mathcal{L} ϕ= P(ϕ,\nablaϕ) + f(ϕ,\nablaϕ)ξ\] where $P$ is a polynomial, $f$ is a sufficiently well-behaved function, and $ξ$ is an irregular distribution such that the equation is subcritical. The differential operator $\mathcal L$ is either a derivative in time, in which case we interpret the equation using rough path theory, or a heat operator, in which case we interpret the equation using regularity structures. Our only assumption on $P$ is that solutions with $ξ=0$ exhibit coercivity. Our estimates are local in space and time, and independent of boundary conditions.
One of our main results is an abstract estimate that allows one to pass from a local coercivity property to a global one using scaling, for a large class of equations. This allows us to reduce the problem of deriving a priori estimates to the case when $ξ$ is small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20716 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large field problem in coercive singular PDEs Chevyrev, Ilya Gubinelli, Massimiliano Analysis of PDEs Probability 60H17 We derive a priori estimates for singular differential equations of the form \[ \mathcal{L} ϕ= P(ϕ,\nablaϕ) + f(ϕ,\nablaϕ)ξ\] where $P$ is a polynomial, $f$ is a sufficiently well-behaved function, and $ξ$ is an irregular distribution such that the equation is subcritical. The differential operator $\mathcal L$ is either a derivative in time, in which case we interpret the equation using rough path theory, or a heat operator, in which case we interpret the equation using regularity structures. Our only assumption on $P$ is that solutions with $ξ=0$ exhibit coercivity. Our estimates are local in space and time, and independent of boundary conditions. One of our main results is an abstract estimate that allows one to pass from a local coercivity property to a global one using scaling, for a large class of equations. This allows us to reduce the problem of deriving a priori estimates to the case when $ξ$ is small. |
| title | Large field problem in coercive singular PDEs |
| topic | Analysis of PDEs Probability 60H17 |
| url | https://arxiv.org/abs/2510.20716 |