Related papers: Nonlinear SPDEs and Maximal Regularity: An Extende…
In light of the question of finite-time blow-up vs. global well-posedness of solutions to problems involving nonlinear partial differential equations, we provide several cautionary examples which indicate that modifications to the boundary…
We present an application of the theory of stochastic processes to model and categorize non-equilibrium physical phenomena. The concepts of uniformly continuous probability measures and modular evolution lead to a systematic hierarchical…
For a nonlinear diffusion equation on graphs whose nonlinearity violates the Lipschitz condition, we prove short-time solution existence and characterize global well-posedness by establishing sufficient criteria for blow-up phenomena and…
This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for a semilinear convection-diffusion problem which may exhibit blow-up in finite time. More specifically,…
We establish higher regularity properties of solutions to fully nonlinear elliptic equations at interior critical points. The key novelty of our estimates lies in the fact that they yield smoothness properties that go beyond the inherent…
The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage…
This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in [22] to domains of polyhedral type. In particular, we study the smoothness in the specific scale…
A framework to establish response theory for a class of nonlinear stochastic partial differential equations (SPDEs) is provided. More specifically, it is shown that for a certain class of observables, the averages of those observables…
In this paper, we develop the well-posedness theory and uncover the noise-regularization effect on scattering for the stochastic Zakharov system in dimensions $d \geq 4$ and beyond the energy space. Our focus is particularly directed at the…
In the studies of the Navier-Stokes (NS) regularity problem, it has become increasingly clear that a more realistic path to improved a priori bounds is to try to break away from the scaling of the energy-level estimates in the realm of the…
These lectures present the analysis of stability and control of long time behavior of PDE models described by nonlinear evolutions of hyperbolic type. Specific examples of the models under consideration include: (i) nonlinear systems of…
A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension $d\in\mathbb{N}$. It is a regularised and inertial version of the…
Sufficient conditions for the invariance of evolution problems governed by perturbations of (possibly nonlinear) $m$-accretive operators are provided. The conditions for the invariance with respect to sublevel sets of a constraint…
This paper investigates the well-posedness and small-noise asymptotics of a class of stochastic partial differential equations defined on a bounded domain of $\mathbb{R}^d$, where the diffusion coefficient depends nonlinearly and…
We study the focusing mass-critical rough nonlinear Schroedinger equations, where the stochastic integration is taken in the sense of controlled rough path. We obtain the global well-posedness if the mass of initial data is below that of…
A parameter estimation problem for a class of semilinear stochastic evolution equations is considered. Conditions for consistency and asymptotic normality are given in terms of growth and continuity properties of the nonlinear part.…
We develop a sharp maximal regularity theory for the resolvent and evolution Stokes equations with no-slip boundary conditions, focusing on bounded domains of low regularity. Our framework covers the full scales of Besov and Sobolev spaces,…
We consider the two-dimensional stationary Navier--Stokes equations on the whole plane $\mathbb{R}^2$. In the higher-dimensional cases $\mathbb{R}^n$ with $n \geqslant 3$, the well-posedness and ill-posedness in scaling critical spaces are…
In this paper, we investigate abstract time-fractional evolution equations with nonlinear perturbations. We construct solutions of Lipschitz perturbation problems in arbitrary large time interval independent of the Lipschitz constants. We…
In this paper we focus on nonlinear SPDEs with singularities included in both drift and noise coefficients, for which the Gelfand-triple argument developed for (local) monotone SPDEs turns out to be invalid. We propose a general framework…