Mark Veraar
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View article: A stochastic flow approach to De Giorgi-Nash-Moser estimates for SPDEs with smooth transport noise
A stochastic flow approach to De Giorgi-Nash-Moser estimates for SPDEs with smooth transport noise Open
The celebrated De Giorgi-Nash-Moser theory ensures that solutions to uniformly elliptic or parabolic PDEs are bounded and Hölder continuous, even with merely bounded measurable coefficients. For parabolic SPDEs with transport noise, bounde…
View article: A Note on the failure of temporal regularity for stochastic PDEs
A Note on the failure of temporal regularity for stochastic PDEs Open
We consider solutions to linear parabolic SPDEs of the form \[ \mathrm{d} u(t) + A u(t)\, \mathrm{d} t = g(t)\, \mathrm{d} β, \qquad u(0)=0, \] where $A$ is a positive, invertible, and self-adjoint operator on a Hilbert space $X$, $β$ is a…
View article: Nonlinear SPDEs and Maximal Regularity: An Extended Survey
Nonlinear SPDEs and Maximal Regularity: An Extended Survey Open
View article: Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains
Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains Open
We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded $H^{\infty}$-functional calculus on weighted Sobolev spaces, where the weights are powers of the dis…
View article: An extrapolation result in the variational setting: improved regularity, compactness, and applications to quasilinear systems
An extrapolation result in the variational setting: improved regularity, compactness, and applications to quasilinear systems Open
In this paper we consider the variational setting for SPDE on a Gelfand triple $$(V, H, V^*)$$ . Under the standard conditions on a linear coercive pair ( A , B ), and a symmetry condition on A we manage to extrapolate the class…
View article: Discrete stochastic maximal regularity
Discrete stochastic maximal regularity Open
In this paper, we investigate discrete regularity estimates for a broad class of temporal numerical schemes for parabolic stochastic evolution equations. We provide a characterization of discrete stochastic maximal $\ell^p$-regularity in t…
View article: Functional calculus on weighted Sobolev spaces for the Laplacian on the half-space
Functional calculus on weighted Sobolev spaces for the Laplacian on the half-space Open
View article: Nonlinear SPDEs and Maximal Regularity: An Extended Survey
Nonlinear SPDEs and Maximal Regularity: An Extended Survey Open
In this survey, we provide an in-depth exposition of our recent results on the well-posedness theory for stochastic evolution equations, employing maximal regularity techniques. The core of our approach is an abstract notion of critical sp…
View article: An extended variational setting for critical SPDEs with L{é}vy noise
An extended variational setting for critical SPDEs with L{é}vy noise Open
The critical variational setting was recently introduced and shown to be applicable to many important SPDEs not covered by the classical variational setting. In this paper, we extend the critical variational setting in several ways. We int…
View article: Higher order moments for SPDE with monotone nonlinearities*
Higher order moments for SPDE with monotone nonlinearities* Open
This paper introduces a new p-dependent coercivity condition through which Lp-moments for solutions can be obtained for a large class of SPDEs in the variational framework. If p = 2, our condition reduces to the classical coercivity condit…
View article: Pathwise uniform convergence of time discretization schemes for SPDEs
Pathwise uniform convergence of time discretization schemes for SPDEs Open
In this paper we prove convergence rates for time discretization schemes for semilinear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly continuous…
View article: Improved polynomial decay for unbounded semigroups
Improved polynomial decay for unbounded semigroups Open
We obtain polynomial decay rates for $C_{0}$-semigroups, assuming that the resolvent grows polynomially at infinity in the complex right half-plane. Our results do not require the semigroup to be uniformly bounded, and for unbounded semigr…
View article: Reaction-Diffusion Equations with Transport Noise and Critical Superlinear Diffusion: Global Well-Posedness of Weakly Dissipative Systems
Reaction-Diffusion Equations with Transport Noise and Critical Superlinear Diffusion: Global Well-Posedness of Weakly Dissipative Systems Open
In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the d-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g., the Allen-Cahn eq…
View article: Counterexamples to maximal regularity for operators in divergence form
Counterexamples to maximal regularity for operators in divergence form Open
In this paper, we present counterexamples to maximal $$L^p$$ -regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory tha…
View article: Functional calculus on weighted Sobolev spaces for the Laplacian on the half-space
Functional calculus on weighted Sobolev spaces for the Laplacian on the half-space Open
In this paper, we consider the Laplace operator on the half-space with Dirichlet and Neumann boundary conditions. We prove that this operator admits a bounded $H^\infty$-calculus on Sobolev spaces with power weights measuring the distance …
View article: Strongly Kreiss bounded operators in UMD Banach spaces
Strongly Kreiss bounded operators in UMD Banach spaces Open
In this paper we give growth estimates for $$\Vert T^n\Vert $$ for $$n\rightarrow \infty $$ in the case T is a strongly Kreiss bounded operator on a $${{\,\textrm{UMD}\,}}$$ Banach space X . In several special cases we pro…
View article: Temporal approximation of stochastic evolution equations with irregular nonlinearities
Temporal approximation of stochastic evolution equations with irregular nonlinearities Open
View article: Large Deviations for Stochastic Evolution Equations in the Critical Variational Setting
Large Deviations for Stochastic Evolution Equations in the Critical Variational Setting Open
Using the weak convergence approach, we prove the large deviation principle (LDP) for solutions to quasilinear stochastic evolution equations with small Gaussian noise in the critical variational setting, a recently developed general varia…
View article: The critical variational setting for stochastic evolution equations
The critical variational setting for stochastic evolution equations Open
View article: Stochastic Navier–Stokes Equations for Turbulent Flows in Critical Spaces
Stochastic Navier–Stokes Equations for Turbulent Flows in Critical Spaces Open
In this paper we study the stochastic Navier–Stokes equations on the d -dimensional torus with transport noise, which arise in the study of turbulent flows. Under very weak smoothness assumptions on the data we prove local well-posedness i…
View article: Counterexamples to maximal regularity for operators in divergence form
Counterexamples to maximal regularity for operators in divergence form Open
In this paper, we present counterexamples to maximal $L^p$-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions' theory that such …
View article: An extrapolation result in the variational setting: improved regularity, compactness, and applications to quasilinear systems
An extrapolation result in the variational setting: improved regularity, compactness, and applications to quasilinear systems Open
In this paper we consider the variational setting for SPDE on a Gelfand triple $(V, H, V^*)$. Under the standard conditions on a linear coercive pair $(A,B)$, and a symmetry condition on $A$ we manage to extrapolate the classical $L^2$-est…
View article: Strongly Kreiss Bounded Operators in UMD Banach Spaces
Strongly Kreiss Bounded Operators in UMD Banach Spaces Open
In this paper we give growth estimates for $\|T^n\|$ for $n\to \infty$ in the case $T$ is a strongly Kreiss bounded operator on a UMD Banach space $X$. In several special cases we provide explicit growth rates. This includes known cases su…
View article: Temporal approximation of stochastic evolution equations with irregular nonlinearities
Temporal approximation of stochastic evolution equations with irregular nonlinearities Open
In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on $2$-sm…
View article: Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity
Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity Open
In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise,…
View article: Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs
Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs Open
In this paper, we prove convergence rates for time discretisation schemes for semi-linear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly continuo…
View article: Reaction-diffusion equations with transport noise and critical superlinear diffusion: Global well-posedness of weakly dissipative systems
Reaction-diffusion equations with transport noise and critical superlinear diffusion: Global well-posedness of weakly dissipative systems Open
In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the $d$-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g. the Allen-Cahn e…
View article: On the trace embedding and its applications to evolution equations
On the trace embedding and its applications to evolution equations Open
In this paper, we consider traces at initial times for functions with mixed time‐space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improveme…
View article: Reaction-diffusion equations with transport noise and critical superlinear diffusion: local well-posedness and positivity
Reaction-diffusion equations with transport noise and critical superlinear diffusion: local well-posedness and positivity Open
In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise,…
View article: Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence*
Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence* Open
In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases…