Past 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
24 April 2017
16:00
Epifanio Virga
Abstract

In the mathematical theory of liquid crystals, a hedgehog is a universal equilibrium solution for Frank's elastic free-energy functional. It is characterized by a radial defect for the nematic director, reminiscent of the way spines are arranged in the spiny mammal. For certain choices of Frank's elastic constants, the free energy stored in a ball subject to radial boundary conditions for the director is minimized by a hedgehog with its defect in the centre of the ball. For other choices of Frank's constants, it is known that a radial hedgehog cannot be a minimizer for this variational problem. We shall gather evidence supporting the conjecture that a "twisted" hedgehog takes the place of a radial hedgehog as an energy minimizer (and we shall not fail to say in which sense it is "twisted"). We shall also show that a twisted hedgehog often accompanies, unseen, a radial hedgehog, as its virtual double, ready to beat its energy as a certain elastic anisotropy is reached.

  • Partial Differential Equations Seminar
6 March 2017
16:00
Peter Topping
Abstract


A familiar technique in PDE theory is to use mollification to adjust a function controlled in some weak norm into a smooth function with corresponding control on its $C^k$ norm. It would be extremely useful to be able to do the same sort of regularisation for Riemannian metrics, and one might hope to use Ricci flow to do this. However, attempting to do so throws up some fundamental problems concerning the well-posedness of Ricci flow. I will explain some recent developments that allow us to use Ricci flow in this way in certain important cases. In particular, the Ricci flow will now allow us to adjust a `noncollapsed’ 3-manifold with a lower bound on its Ricci curvature through a family of such manifolds, without disturbing the Riemannian distance function too much, and so that we instantly obtain uniform bounds on the full curvature tensor and all its derivatives. These ideas lead to the resolution of some long-standing open problems in geometry.

No previous knowledge of Ricci flow will be assumed, and differential geometry prerequisites will be kept to a minimum.

Joint work with Miles Simon.
 

  • Partial Differential Equations Seminar
20 February 2017
16:00
Piotr B. Mucha
Abstract

I will talk about connections between the compressible and incompressible Navier-Stokes systems. In case of the compressible model, as the bulk (volume) viscosity is very high, the divergence of the velocity becomes small, in the limit it is zero and we arrive at the case of incompressible system. An important role here is played by the inhomogeneous version of the classical Navier-Stokes equations. I plan to discuss analytical obstacle appearing within the analysis. The considerations are done in the framework of regular solutions in Besov and Sobolev spaces. The results which will be discussed are joint with Raphael Danchin from Paris.

  • Partial Differential Equations Seminar

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