Ilya Chevyrev and Massimiliano Gubinelli
We derive a priori estimates for singular differential equations of the form $\mathcal{L} \phi = P(\phi,\nabla\phi) + f(\phi,\nabla\phi)\xi$ where $P$ is a polynomial, $f$ is a sufficiently well-behaved function, and $\xi$ 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 $\xi=0$ 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 $\xi$ is small.
Ilya Chevyrev and Massimiliano Gubinelli. Large field problem in coercive singular PDEs. Papers of Massimiliano Gubinelli, 2025, version 2. https://mgubi.org/papers/paper/large-field-problem/
Licence: Creative Commons Attribution 4.0 International
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