In this joint work with Amandine Aftalion we study the minimisers of an energy functional in two-dimensions describing a rotating two-component condensate. This involves in particular separating a line-energy term and a vortex term which have different orders of magnitude, and requires new estimates for functionals of the Cahn-Hilliard (or Modica-Mortola) type.

# Past Partial Differential Equations Seminar

We prove that for closed surfaces of fixed topological type, orientable or non-orientable, there exists a unit volume metric, smooth away from finitely many conical singularities, that

maximizes the first eigenvalue of the Laplace operator among all unit volume metrics. The key ingredient are several monotonicity results, which have partially been conjectured to hold before. This

is joint work with Henrik Matthiesen.

For liquid films with a thickness in the order of 10¹−10³ molecule layers, classical models of continuum mechanics do not always give a precise description of thin-film evolution: While morphologies of film dewetting are captured by thin-film models, discrepancies arise with respect to time-scales of dewetting.

In this talk, we study stochastic thin-film equations. By multiplicative noise inside an additional convective term, these stochastic partial differential equations differ from their deterministic counterparts, which are fourth-order degenerate parabolic. First, we present some numerical simulations which indicate that the aforementioned discrepancies may be overcome under the influence of noise.

In the main part of the talk, we prove existence of almost surely nonnegative martingale solutions. Combining spatial semi-discretization with appropriate stopping time arguments, arbitrary moments of coupled energy/entropy functionals can be controlled.

Having established Hölder regularity of approximate solutions, the convergence proof is then based on compactness arguments - in particular on Jakubowski’s generalization of Skorokhod’s theorem - weak convergence methods, and recent tools for martingale convergence.

The results have been obtained in collaboration with K. Mecke and M. Rauscher and with J. Fischer, respectively

It is well-known that only a limited number of the fluid flow problems can be solved (or approximated) by the solutions in the explicit form. To derive such solutions, we usually need to start with (over)simplified mathematical models and consider ideal geometries on the flow domains with no distortions introduced. However, in practice, the boundary of the fluid domain can contain various small irregularities (rugosities, dents, etc.) being far from the ideal one. Such problems are challenging from the mathematical point of view and, in most cases, can be treated only numerically. The analytical treatments are rare because introducing the small parameter as the perturbation quantity in the domain boundary forces us to perform tedious change of variables. Having this in mind, our goal is to present recent analytical results on the effects of a slightly perturbed boundary on the fluid flow through a channel filled with a porous medium. We start from a rectangular domain and then perturb the upper part of its boundary by the product of the small parameter $\varepsilon$ and arbitrary smooth function. The porous medium flow is described by the Darcy-Brinkman model which can handle the presence of a boundary on which the no-slip condition for the velocity is imposed. Using asymptotic analysis with respect to $\varepsilon$, we formally derive the effective model in the form of the explicit formulae for the velocity and pressure. The obtained asymptotic approximation clearly shows the nonlocal effects of the small boundary perturbation. The error analysis is also conducted providing the order of accuracy of the asymptotic solution. We will also address the problem of the solute transport through a semi-infinite channel filled with a fluid saturated sparsely packed porous medium. A small perturbation of magnitude $\varepsilon$ is applied on the channel's walls on which the solute particles undergo a first-order chemical reaction. The effective model for solute concentration in the small-Péclet-number-regime is derived using asymptotic analysis with respect to $\varepsilon$. The obtained mathematical model clearly indicates the influence of the porous medium, chemical reaction and boundary distortion on the effective flow.

This is a joint work with Eduard Marušić-Paloka (University of Zagreb).

We consider a couple of problems belonging to Random Geometry, and describe some new analytical challenges they pose for planar PDE's via Beltrami equations. The talk is based on joint work with various people including K. Astala, P. Jones, A. Kupiainen, Steffen Rohde and T. Tao.

R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes showed in their celebrated paper from 1993 that numerous compensated compactness quantities such as Jacobians of mappings in $W^{1,n}(\mathbb{R}^n,\mathbb{R}^n)$ belong the real-variable Hardy space $\mathcal{H}^1(\mathbb{R}^n)$. They proceeded to ask what is the exact range of these nonlinear quantities and in particular whether the Jacobian operator $J$ maps $W^{1,2}(\mathbb{R}^2,\mathbb{R}^2)$ onto $\mathcal{H}^1(\mathbb{R}^2)$.

I present the proof of my recent result that $J \colon W^{1,n}(\mathbb{R}^n,\mathbb{R}^n) \to \mathcal{H}^1(\mathbb{R}^n)$ is non-surjective for every $n \ge 2$. The surjectivity question is still open when the domain of definition of $J$ is the inhomogeneous Sobolev space $\dot{W}^{1,n}(\mathbb{R}^n,\mathbb{R}^n)$. I also shortly discuss my work on T. Iwaniec's conjecture from 1997 which states that for every $n \ge 2$ and $p \in [1,\infty[$ the operator $J \colon W^{1,np}(\mathbb{R}^n,\mathbb{R}^n) \to \mathcal{H}^p(\mathbb{R}^n)$ has a continuous right inverse.

The contact line problem in interfacial fluid mechanics concerns the triple-junction between a fluid, a solid, and a vapor phase. Although the equilibrium configurations of contact lines have been well-understood since the work of Young, Laplace, and Gauss, the understanding of contact line dynamics remains incomplete and is a source of work in experimentation, modeling, and mathematical analysis. In this talk we consider a 2D model of contact point (the 2D analog of a contact line) dynamics for an incompressible, viscous, Stokes fluid evolving in an open-top vessel in a gravitational field. The model allows for fully dynamic contact angles and points. We show that small perturbations of the equilibrium configuration give rise to global-in-time solutions that decay to equilibrium exponentially fast. This is joint with with Yan Guo.

Reinhard Farwig and Chenyin Qian

Consider the autonomous quasi-geostrophic equation with fractional dissipation in $\mathbb{R}^2$

\begin{equation} \label{a}

\theta_t+u\cdot\nabla\theta+(-\Delta)^{\alpha}\theta=f(x,\theta)

\end{equation}

in the subcritical case $1/2<\alpha\leq1$, with initial condition $\theta(x, 0)= \theta^{0}$ and given external force $f(x,\theta)$. Here the real scalar function $\theta$ is the so-called potential temperature, and the incompressible velocity field $u=(u_1,u_2)=(-\mathcal {R}_2\theta,\mathcal {R}_1\theta)$ is determined from $\theta$ via Riesz operators. Our aim is to prove the existence of the compact global attractor $\mathcal{A}$ in the Bessel potential space $H^s(\mathbb{R}^2)$ when $s>2(1-\alpha)$.

The construction of the attractor is based on the existence of an absorbing set in $L^2(\mathbb{R}^2)$ and $H^s(\mathbb{R}^2)$ where $s>2(1-\alpha)$. A second major step is usually based on compact Sobolev embeddings which unfortunately do not hold for unbounded domains. To circumvent this problem we exploit compact Sobolev embeddings on balls $B_R \subset \mathbb{R}^2$ and uniform smallness estimates of solutions on $\mathbb{R}^2 \setminus B_R$. In the literature the latter estimates are obtained by a damping term $\lambda\theta$, $\lambda<0$, as part of the right hand side $f$ to guarantee exponential decay estimates. In our approach we exploit a much weaker nonlocal damping term of convolution type $\rho*\theta$ where $\widehat \rho<0$.

In everyday language, this talk studies the question about the optimal shape and location of a thermometer of a given volume to reconstruct the temperature distribution in an entire room. For random initial conditions, this problem was considered by Privat, Trelat and Zuazua (ARMA, 2015), and for short times we remove both the randomness and geometric assumptions in their article. Analytically, we obtain quantitative estimates for the well-posedness of an inverse problem, in which one determines the solution in the whole domain from its restriction to a subset of given volume. Using a new decomposition of $L^2(\Rd)$ into heat packets from microlocal analysis, we conclude that there exists a unique optimal such subset, that it is semi-analytic and can be approximated numerically by solving a sequence of finite-dimensional optimization problems. (joint with Alden Waters)