Past Partial Differential Equations Seminar

16 October 2017
16:00
Abstract

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.

  • Partial Differential Equations Seminar
12 June 2017
16:30
Abstract

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.

  • Partial Differential Equations Seminar
12 June 2017
15:30
Abstract

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 $\widehat \rho<0$. 

  • Partial Differential Equations Seminar
5 June 2017
16:00
Heiko Gimperlein
Abstract

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)
 

  • Partial Differential Equations Seminar
29 May 2017
16:00
Barbara Zwicknagl
Abstract
I will report some recent analytical results on microstructures in low-hysteresis shape memory alloys. The modelling assumption is that the width of the thermal hysteresis is closely related to the minimal energy that is necessary to build a martensitic nucleus in an austenitic matrix. This energy barrier is typically modeled by (singularly perturbed) nonconvex elasticity functionals. In this talk, I will discuss recent results on the resulting variational problems, including stress-free inclusions and microstructures in the case of almost compatible phases. This talk is partly based on joint works with S. Conti, J. Diermeier, M. Klar, and D. Melching.
  • Partial Differential Equations Seminar
22 May 2017
16:00
Tim Healey
Abstract

We study a model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations. We also numerically compute a special class of such solutions, namely those possessing icosahedral symmetry. We overcome several difficulties along the way. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly under-determined mid-plane deformation. We then use well known group-theoretic selection techniques combined with global bifurcation methods to obtain our results.

  • Partial Differential Equations Seminar
15 May 2017
16:00
Abstract

Various concepts of weak solution have been suggested for the fundamental equations of fluid dynamics over the last few decades. However, such weak solutions may be non-unique, or at least their uniqueness is unknown. Nevertheless, a conditional notion of uniqueness, the so-called weak-strong uniqueness, can be established in various situations. We present some recent results, both positive and negative, on weak-strong uniqueness in the realm of incompressible and compressible fluid dynamics. Applications to the convergence of numerical schemes will be indicated.

  • Partial Differential Equations Seminar
8 May 2017
16:00
Matthias Winter
Abstract

Results on the existence and stability of clustered spike patterns for biological reaction‐diffusion systems with two small diffusivities will be presented. In particular we consider a consumer chain model and the Gierer‐Meinhardt activator-inhibitor system with a precursor gradient. A clustered spike pattern consists of multiple spikes which all approach the same limiting point as the diffusivities tend to zero. We will present results on the asymptotic behaviour of the spikes including their shapes, positions and amplitudes. We will also compute the asymptotic behaviour of the eigenvalues of the system linearised around a clustered spike pattern. These systems and their solutions play an important role in biological modelling to account for the bridging of lengthscales, e.g. between genetic, nuclear, intra‐cellular, cellular and tissue levels, or for the time-hierarchy of biological processes, e.g. a large‐scale structure, which appears first, induces patterns on smaller scales. This is joint work with Juncheng Wei.
 

  • Partial Differential Equations Seminar
1 May 2017
16:00
Patrick Farrell
Abstract

Computing the solutions $u$ of an equation $f(u, \lambda) = 0$ as the parameter $\lambda$ is varied is a central task in applied mathematics and engineering. In this talk I will present a new algorithm, deflated continuation, for this task.

Deflated continuation has three main advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only aimed to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is extremely simple: it only requires a minor modification to any existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available.

Among other problems, we will apply this to a famous singularly perturbed ODE, Carrier's problem. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the singular perturbation parameter tends to zero. The analysis yields a novel and complete taxonomy of the solutions to the problem, and demonstrates that a claim of Bender & Orszag (1999) is incorrect. We will also use the algorithm to calculate distinct local minimisers of a topology optimisation problem via the combination of deflated continuation and a semismooth Newton method.

  • Partial Differential Equations Seminar

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