We develop a strategy making extensive use of tent spaces to study parabolic equations with quadratic nonlinearities as for the Navier–Stokes system. We begin with a new proof of the well-known result of Koch and Tataru on the well-posedness of Navier–Stokes equations in $\mathbb{R}^{n}$ with small initial data in $\mathit{BMO}^{-1}(\mathbb{R}^{n})$. We then study another model where neither pointwise kernel bounds nor self-adjointness are available.

In this paper, we consider diffusion, reaction and dissolution of mobile and immobile chemical species present in a porous medium. Inflow–outflow boundary conditions are considered at the outer boundary and the reactions amongst the species are assumed to be reversible which yield highly nonlinear reaction rate terms. The dissolution of immobile species takes place on the surfaces of the solid parts. Modelling of these processes leads to a system of coupled semilinear partial differential equations together with a system of ordinary differential equations (ODEs) with multi-valued right-hand sides. We prove the global existence of a unique positive weak solution of this model using a regularization technique, Schaefer's fixed point theorem and Lyapunov type arguments.

We discuss ℓp-maximal regularity of power-bounded operators and relate the discrete to the continuous time problem for analytic semigroups. We give a complete characterization of operators with ℓ1 and -maximal regularity. We also introduce an unconditional form of Ritt’s condition for power-bounded operators, which plays the role of the existence of an -calculus, and give a complete characterization of this condition in the case of Banach spaces which are L1-spaces, C(K)-spaces or Hilbert spaces.

In this paper we show boundedness of vector-valued Bergman projections on simple connected domains. With this result we show R-sectoriality of the derivative on the Bergman space on C+ and maximal Lp-regularity for an integrodifferential equation with a kernel in the Bergman space.

Maximal regularity for an integro-differential equation with infinite delay on periodic vector-valued Besov spaces is studied. We use Fourier multipliers techniques to characterize periodic solutions solely in terms of spectral properties on the data. We study a resonance case obtaining a compatibility condition which is necessary and sufficient for the existence of periodic solutions.

We study closed extensions $\underset{\scriptscriptstyle-}{A}$ of an elliptic differential operator $A$ on a manifold with conical singularities, acting as an unbounded operator on a weighted ${{L}_{p}}$-space. Under suitable conditions we show that the resolvent ${{\left( \lambda -\underset{\scriptscriptstyle-}{A} \right)}^{-1}}$ exists in a sector of the complex plane and decays like $1/\left| \lambda \right|$ as $\left| \lambda \right|\to \infty $. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of $\underset{\scriptscriptstyle-}{A}$.

As an application we treat the Laplace–Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem $\dot{u}\,-\,\Delta u\,=\,f,$$u\left( 0 \right)\,=\,0$.

Let $1\leq p,q\leq\infty$, $s\in\mathbb{R}$ and let $X$ be a Banach space. We show that the analogue of Marcinkiewicz’s Fourier multiplier theorem on $L^p(\mathbb{T})$ holds for the Besov space $B_{p,q}^s(\mathbb{T};X)$ if and only if $1\ltp\lt\infty$ and $X$ is a UMD-space. Introducing stronger conditions we obtain a periodic Fourier multiplier theorem which is valid without restriction on the indices or the space (which is analogous to Amann’s result (Math. Nachr. 186 (1997), 5–56) on the real line). It is used to characterize maximal regularity of periodic Cauchy problems.

AMS 2000 Mathematics subject classification: Primary 47D06; 42A45