Related papers: Paracontrolled calculus for quasilinear singular P…
We introduce a non-linear paracontrolled calculus and use it to renormalise a class of singular SPDEs including certain quasilinear variants of the periodic two dimensional parabolic Anderson model.
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…
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…
We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm some singular partial differential equations with the same efficiency as regularity structures. This work deals with…
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…
We derive existence results and first order necessary optimality conditions for optimal control problems governed by quasilinear parabolic PDEs with a class of first order nonlinearities that include for instance quadratic gradient terms.…
This paper is concerned with quasi-linear parabolic equations driven by an additive forcing $\xi \in C^{\alpha-2}$, in the full sub-critical regime $\alpha \in (0,1)$. We are inspired by Hairer's regularity structures, however we work with…
We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global…
We study in this short note a counterpart to the quasilinear generalized parabolic Anderson model (gPAM) on the 2-dimensional torus where the coefficients are nonlocal functionals of the solution. Under a positivity assumption on the…
We introduce an approach to study certain singular PDEs which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems like…
We present in this note a local in time well-posedness result for the singular $2$-dimensional quasilinear generalized parabolic Anderson model equation $$ \partial_t u - a(u)\Delta u = g(u)\xi $$ The key idea of our approach is a simple…
Classical results of second order parabolic quasi-linear equations always require that the nonlinear terms are controlled by a power of the unknown functions and their first derivatives. We improve the previous results. More precisely, in…
This paper is devoted to studying a multi-objective control problem for a class of multi-dimensional quasi-linear parabolic equations. The considered system is driven by a leader control and two follower controls. For each leader control, a…
In this paper, we propose quasilinearization methods that convert nonlocal fully-nonlinear parabolic systems into the nonlocal quasilinear parabolic systems. The nonlocal parabolic systems serve as important mathematical tools for modelling…
The paper concerns the theory of parabolic equations on a broad class of closed subsets of Euclidean space possessing a kind of tangent structure. A necessary framework for considering evolutionary problems is developed, and fundamental…
This paper studies the feedback stabilization of abstract Cauchy problems with unbounded output operators by finite-dimensional controllers. Both necessary conditions and sufficient conditions for feedback stabilizability are presented. The…
In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based…
We consider parabolic PDEs associated with fractional type operators drifted by non-linear singular first order terms. When the drift enjoys some boundedness properties in appropriate Lebesgue and Besov spaces, we establish by exploiting a…
In this paper we wonder whether a quasilinear system of PDEs of first order admits Hamiltonian formulation with local and nonlocal operators. By using the theory of differential coverings, we find differential-geometric conditions necessary…
We study linear nonautonomous parabolic systems with dynamic boundary conditions. Next, we apply these results to show a theorem of local existence and uniqueness of a classical solution to a second order quasilinear system with nonlinear…