Heat semigroup and singular PDEs

By Ismaël F. Bailleul, Frédéric Bernicot. 2015

61 citations recorded by the source corpus.

We provide in this work a semigroup approach to the study of singular PDEs, in the line of the paracontrolled approach developed recently by Gubinelli, Imkeller and Perkowski. Starting from a heat semigroup, we develop a functional calculus and introduce a paraproduct based on the semigroup, for which commutator estimates and Schauder estimates are proved, together with their paracontrolled extensions. This machinery allows us to investigate singular PDEs in potentially unbounded Riemannian manifolds under mild geometric conditions. As an illustration, we study the generalized parabolic Anderson model equation and prove, under mild geometric conditions, its well-posed character in Holders spaces, in small time on a potentially unbounded 2-dimensional Riemannian manifold, for an equation driven by a weighted noise, and for all times for the linear parabolic Anderson model equation in 2-dimensional unbounded manifolds. This machinery can be extended to an even more singular setting and deal with Sobolev scales of spaces rather than Holder spaces.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/057c509aa558295e46a31ec1c501ca0c050cf582

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In Applied Mathematics

1. Ismaël F. Bailleul
2. Frédéric Bernicot

In Computational Theory and Mathematics

1. Frédéric Bernicot
2. Ismaël F. Bailleul

In Mathematical Physics

1. Ismaël F. Bailleul
2. Frédéric Bernicot

Citations in the corpus

• HIGH ORDER PARACONTROLLED CALCULUS (2016, methodology): “› L8pMq ; this norm does not depend on the integer a ą |α| 2 , and the two spaces H α and Cα coincide and have equivalent norms when 0 ă α ă 1 – see for instance [1].”