In this paper we give growth estimates for ‖Tⁿ‖ for n→∞ 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 such as Hilbert and L^p-spaces, but also intermediate UMD spaces such as non-commutative L^p-spaces and variable Lebesgue spaces.

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 equation) and dissipative systems (e.g., equations in coagulation dynamics). Moreover, we prove global well-posedness for two weakly dissipative systems: Lotka-Volterra equations for d \in \{1, 2, 3, 4\} and the Brusselator for d \in \{1, 2, 3\}. Many of the results are also new without transport noise. The proofs are based on maximal regularity techniques, positivity results, and sharp blow-up criteria developed in our recent works, combined with energy estimates based on It\^ o's formula and stochastic Gronwall inequalities. Key novelties include the introduction of new L^\zeta-coercivity/dissipativity conditions and the development of an L^p(L^q)framework for systems of reaction-diffusion equations, which are needed when treating dimensions d \in \{2, 3\} in the case of cubic or higher order nonlinearities.

In this paper we consider an SPDE where the leading term is a second order operator with periodic boundary conditions, coefficients which are measurable in (t,ω), and Hölder continuous in space. Assuming stochastic parabolicity conditions, we prove L^p((0,T)× Ω,t^κ dt;H^σ,q(T^d))-estimates. The main novelty is that we do not require p = q. Moreover, we allow arbitrary σ ∈ R and weights in time. Such mixed regularity estimates play a crucial role in applications to nonlinear SPDEs which is clear from our previous work. To prove our main results we develop a general perturbation theory for SPDEs. Moreover, we prove a new result on pointwise multiplication in spaces with fractional smoothness.

This paper introduces a new p-dependent coercivity condition through which (Formula presented.) -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 condition, which only yields second moments for the solution. The abstract result is shown to be optimal. Moreover, the results are applied to obtain (Formula presented.) -moments of solutions for several classical SPDEs such as stochastic heat equations with Dirichlet and Neumann boundary conditions, Burgers' equation and the Navier–Stokes equations in two spatial dimensions. Furthermore, we can recover recent results for systems of SPDEs and higher-order SPDEs using our unifying coercivity condition.

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 in the critical case B_q,p^d/q-1 for q∈[2,2d) and p large enough. Moreover, we obtain new regularization results for solutions, and new blow-up criteria which can be seen as a stochastic version of the Serrin blow-up criteria. The latter is used to prove global well-posedness with high probability for small initial data in critical spaces in any dimensions d⩾2. Moreover, for d=2, we obtain new global well-posedness results and regularization phenomena which unify and extend several earlier results.

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn–Hilliard equation, tamed Navier–Stokes equations, and Allen–Cahn equation.

Before addressing this question for the Calderón{Zygmund type operators of the kind studied in Chapter 11, we investigate a number of related objects in a simpler dyadic model. Besides serving as an introduction to some of the key techniques, it turns out that these dyadic operators can be, and will be, also used as building blocks of the proper singular integral operators towards the end of the chapter.

We develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent p ∈ (0, 1]. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic partial differential equations.

In this chapter we address a couple of topics in the theory of H^∞-calculus centering around the question what can be said about an operator of the form A+B when A and B have certain “good” properties such as being (R-)sectorial or admitting a bounded H^∞-calculus.

This chapter presents an in-depth study of several classes of vector-valued function spaces defined by smoothness conditions.

In this chapter, we complement the discussion of three major themes of Fourier analysis that we have studied in the previous Volumes.

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, critical spaces and the proof of higher order regularity of solutions – even in case of non-smooth initial data. Crucial tools are L^p(L^q)-theory, maximal regularity estimates and sharp blow-up criteria. We view the results of this paper as a general toolbox for establishing global well-posedness for a large class of reaction-diffusion systems of practical interest, of which many are completely open. In our follow-up work [8], the results of this paper are applied in the specific cases of the Lotka-Volterra equations and the Brusselator model.

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 improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equations, where uniform trace estimates on the half-line are shown.