No Arabic abstract
First we introduce the Bailleul-Hoshinos result [4], which links the theory of regularity structures and the paracontrolled calculus. As an application of their result, we give another algebraic proof of the multicomponent commutator estimate [3], which is a generalized version of the Gubinelli-Imkeller-Perkowskis commutator estimate [11, Lemma 2.4].
We extend the results in [6] to Besov spaces $B_{p,q}^alpha$ with $p,qin[1,infty]$ and $0<alpha<1$.
We start in this work the study of the relation between the theory of regularity structures and paracontrolled calculus. We give a paracontrolled representation of the reconstruction operator and provide a natural parametrization of the space of admissible models.
We prove a general equivalence statement between the notions of models and modelled distributions over a regularity structure, and paracontrolled systems indexed by the regularity structure. This takes in particular the form of a parametrisation of the set of models over a regularity structure by the set of reference functions used in the paracontrolled representation of these objects. A number of consequences are emphasized. The construction of a modelled distribution from a paracontrolled system is explicit, and takes a particularly simple form in the case of the regularity structures introduced by Bruned, Hairer and Zambotti for the study of singular stochastic partial differential equations.
We give a short essentially self-contained treatment of the fundamental analytic and algebraic features of regularity structures and its applications to the study of singular stochastic PDEs.
In this paper, we establish an optimal dual version of trace estimate involving angular regularity. Based on this estimate, we get the generalized Morawetz estimates and weighted Strichartz estimates for the solutions to a large class of evolution equations, including the wave and Schr{o}dinger equation. As applications, we prove the Strauss conjecture with a kind of mild rough data for $2le nle 4$, and a result of global well-posedness with small data for some nonlinear Schr{o}dinger equation with $L^2$-subcritical nonlinearity.