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.

# Past Partial Differential Equations Seminar

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.

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.

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.

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.

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.

In this talk, we will discuss sequences of immersions from 2-discs into Euclidean with finite total curvature where the Willmore energy converges to zero (a minimal surface). We will show that away from finitely many concentration points of the total curvature, the surface converges strongly in $W^{2,2}$. Furthermore, we have an energy identity for the total curvature, at the concentration points after a blow-up procedure we show that there is a bubble tree consisting of complete non-compact (branched) minimal surfaces of finite total curvature which are quantised in multiples of 4\pi. We will also apply this method to the mean curvature flow, showing that sequences of surfaces that are locally converging to a self-shrinker in L^2 also develop a bubble tree of complete non-compact (branched) minimal surfaces in Euclidean space with finite total curvature quantised in multiples of 4\pi.

Linear elasticity can be rigorously derived from finite elasticity in the case of small loadings in terms of \Gamma-convergence. This was first done by Dal Maso-Negri-Percivale in the case of one-well energies with super-quadratic growth. This has been later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). I will discuss recent developments in the case when the distance between the wells is arbitrary. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions which turn out to be optimal. (This is joint work with Alicandro, Dal Maso and Lazzaroni.)

In this talk we will motivate and discuss several problems and results in harmonic analysis that involve some arithmetic or discrete structure. We will focus on pioneering work of Bourgain on discrete restriction theorems and pointwise ergodic theorems for arithmetic sets, their modern developments and future directions for the field.