We conider the evolution of an advection-diffusion equation coupled to a Stokes flow in an evolving perforated domain. This models for instance a partially desolved, partially mineralized substance. For simplicity, the solid phase is assumed to consist of the union of balls with centers that do not evolve in time. The evolution of their radii is coupled to the advection-diffusion equation through a mass exchange term at the interface. The complete microscopic system is therefore a parabolic-elliptic free boundary problem.

We study the homogenization limit $\epsilon \to 0$ when the number of the particles is of order $\epsilon^3$ and their size is of order $\epsilon^\alpha$. We prove (optimal) quantitative convergence results to the solution of an effective limit system both in the critical case $\alpha =3$ and in the superciritical case $\alpha \in (1,3)$.
In the critical case, the effective fluid velocity is governed by the Brinkman equation. The effective advection-diffusion includes a new exchange term that is coupled to a transport equation for the evolution of the solid phase. In the supercritical case, the effective fluid velocity is governed by the Brinkman equation and the exchange term between dissolved and mineralized substance does not change in the homogenization limit.

We provide a pathwise construction of the solutions of stochastic partial differential
equations with nonlinear multiplicative boundary noise and quasilinear systems. To this
aim we combine functional analytic tools with rough path theory. We also discuss the
long-time behavior of such systems using the theory of random dynamical systems. This
talk is based on joint works with M. Ghani Varzaneh, A. Hocquet and T. Seitz.

We study the properties of the Laplace and Stokes operator on a 2D wedge domain $G$, subject to suitable boundary conditions. Through a transformation, the problem is transformed to a strip domain $\Omega$. On this transformed domain, we can apply the operator sum method for non-commuting operators to establish a bounded $\mathcal{H}^{\infty}$-calculus for the transformed operator operators in $L^p$. Using the Euclidean structures and their implied characterization of $\mathcal{H}^{\infty}-$calculus, we transfer this property back to the original wedge domain.

In this work, we study Riesz transforms associated to the generalized Stokes operator $A$ given by

\begin{align*}

A u = f \quad \Leftrightarrow \quad \left\{ \begin{aligned} - \operatorname{div} (\mu \nabla u) + \nabla \phi &= f && \text{in } \mathbb{R}^d, \\ \operatorname{div}(u) &= 0 && \text{in } \mathbb{R}^d. \end{aligned} \right.

Besides ellipticity of $\mu$, we only assume that the coefficients are bounded and measurable. We show that the associated Riesz transform $\nabla A^{- 1/2}$ is bounded on $\mathrm{L}^2 (\mathbb{R}^d)$ which is an extension of the resolution of Kato's square root problem for elliptic operators in divergence form to generalized Stokes operators. In addition, we study lower and upper bounds of the Riesz transforms in $\mathrm{L}^p (\mathbb{R}^d)$, \textit{i.e.},

\begin{align*}

\| \nabla A^{- 1/2} u \|_{\mathrm{L}^p} \leq C \| u \|_{\mathrm{L}^p} \quad \text{as well as} \quad \| \nabla A^{- 1/2} u \|_{\mathrm{L}^p} \geq c \| u \|_{\mathrm{L}^p}

\end{align*}

and provide ranges of $p$ for which such an upper or lower bound hold in general. This part can be seen as an extension of results in the monograph of Auscher.

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This is joint research with Luca Haardt.