Analysis Seminar 2022/23

The Analysis and Differential Equations Seminar takes place on Thursdays at 2:15 in 4W 1.7 (Wolfson Lecture Theatre).

This is the seminar for the Analysis Group in the Department of Mathematical Sciences at the University of Bath. If you have any queries, or if you would like to be on our e-mail list, please contact the organisers Tobias Barker, Steven Flynn, Søren Mikkelsen and Miles Wheeler.

Spring 2023

Date	Speaker	Title/Abstract
9 Feb 4W 4.5 2:30–3:30		Coffee and biscuits in the social space (4W 4.5) Free coffee pods and biscuits! Bring your own mug, or grab one from the fourth floor kitchen down the hall.
16 Feb	Ed Gallagher University of Bath Marco Badran University of Bath	Weighted \(\infty\)-Willmore Spheres Consider the following problem: on the topological two-sphere \(S^2\), minimise a weighted version of the \(L^\infty\) norm of the mean curvature among all immersions \(f \colon S^2 \rightarrow \mathbb{R}^3\) with prescribed surface area. For this problem, we show under a low-energy assumption that not only minimisers but also a wider class of "pseudo-minimal" immersions are intimately connected to a system of PDEs, obtained as the limit as \(p \rightarrow \infty\) of the Euler-Lagrange equations for the approximating \(L^p\) problems. This system governs the geometry of our solutions, as its nodal set determines the mean curvature of the immersion. In particular, the mean curvature takes on at most three values: it vanishes on the nodal set, and elsewhere has constant magnitude with sign matching the sign of the PDEs' solution. Intriguingly, when attempting to solve the problem on surfaces other than the sphere we run into both energetic and analytical issues which prevent us from generalising our results further, although I am hopeful that some of these can be resolved by the time I finish my PhD! This is joint work with Roger Moser. Concentrating solutions for the magnetic Ginzburg–Landau equations Given a closed manifold \(N\) and a compact, non-degenerate minimal submanifold \(M\) of codimension 2 we prove the existence of a collection of solutions \(U_\varepsilon\) to the magnetic Ginzburg–Landau equations such that \(U_\varepsilon\) concentrates around \(M\) as \(\varepsilon\to 0\). Moreover, we give a precise asymptotic expansion of the profile of this concentration and we show that the method we use can be extended to the non-compact case.
23 Feb	Clotilde Fermanian Kammerer Université Paris Est-Créteil	Some avatars of the correspondence principle in semi-classical analysis The correspondence principle, as stated by Niels Bohr in 1923, is at the root of the traditional results in semi-classical analysis. It offers a natural insight into the world of semi-classical pseudodifferential operators, Egorov Theorem, Wigner measures, etc… The aim of this talk will be to present this general setting and explain how recent results of semi-classical analysis express in that framework.
2 Mar	Melanie Rupflin University of Oxford	Quantitative estimates for almost harmonic maps For geometric variational problems one often only has weak, rather than strong, compactness results and hence has to deal with the problem that sequences of (almost) critical points \(u_j\) can converge to a limiting object with different topology. A major challenge posed by such singular behaviour is that the seminal results of Simon on Lojasiewicz inequalities, which are one of the most powerful tools in the analysis of the energy spectrum of analytic energies and the corresponding gradient flows, are not applicable. In this talk we present a method that allows us to prove Lojasiewicz inequalities for almost harmonic maps that form simple singularities and explain how these results allow us to draw new conclusions about the energy spectrum of harmonic maps and the convergence of harmonic map flow for low energy maps from surfaces into analytic manifolds.
9 Mar	Eugene Shargorodsky King's College London	Variations on Liouville's theorem A classical theorem of Liouville states that a function that is analytic and bounded on the entire complex plane is in fact constant. The same conclusion is true for a function that is harmonic and bounded on all of a Euclidean space. I will discuss generalisations of Liouville’s theorem to generators of Levy processes due to D. Berger, F. Kuehn, and R.L. Schilling, and will describe a related joint work in progress with D. Berger, R.L. Schilling, and T. Sharia.
16 Mar	Giacomo Ageno University of Bath Lorenzo Quarisa University of Warwick	Infinite time blow-up for the 3D energy critical heat equation in bounded domains A typical positive solution to the Dirichlet problem for the energy-critical heat equation decays exponentially fast or blows up in finite time. Other behaviours have been known since 1984, but until 2020 they have been precisely detected only in the radial case. In this talk, I will present the first examples of 3-dimensional non-radial global unbounded solutions, precisely describing asymptotic behaviour and stability. The dimension plays a crucial role, making the heart of the problem nonlocal, and revealing an interesting connection with the Brezis–Nirenberg number. This is joint work with Manuel del Pino. The adjoint Rayleigh and Orr–Sommerfeld equations The Rayleigh and Orr–Sommerfeld equations are ODEs which arise from the linearised incompressible Euler and Navier–Stokes equations around a shear flow. In this upcoming work joint with Jose Rodrigo, we consider the adjoints of the Rayleigh and Orr–Sommerfeld equations with respect to the complex \(L^2\) product. In the viscous case, we fix a family of viscosity-dependent Navier boundary conditions, which in the limit corresponds to the no-slip condition. We establish existence and asymptotic properties of their eigenvalues, eigenmodes and Green functions away from critical layers. An application of the adjoint operators is that they allow us to deduce properties of the kernels and images of the original operators in correspondence with their eigenvalues.
23 Mar	Paolo Bonicatto University of Warwick	Transport of currents and geometric Rademacher-type theorems In the classical theory, given a vector field \(b\) on \(\mathbb R^d\), one usually studies the transport/continuity equation drifted by \(b\) looking for solutions in the class of functions (with certain integrability) or at most in the class of measures. In this seminar I will talk about recent efforts, motivated by the modelling of defects in crystals, aimed at extending the previous theory to the case when the unknown is instead a family of k-currents in \(\mathbb R^d\), i.e. generalised \(k\)-dimensional surfaces. The resulting equation involves the Lie derivative \(L_b\) of currents in direction \(b\) and reads \(\partial_t T_t + L_b T_t = 0\). In the first part of the talk I will briefly introduce this equation, with a special attention to its space-time formulation. I will then shift the focus to some rectifiability questions and Rademacher-type results: given a Lipschitz path of integral currents, I will discuss the existence of a “geometric derivative”, namely a vector field advecting the currents. Based on joint works with G. Del Nin (MPI, Leipzig) and F. Rindler (Warwick).
30 Mar 4W 4.5 2:30–3:30		Coffee and biscuits in the social space (4W 4.5) Free coffee pods and biscuits! Bring your own mug, or grab one from the fourth floor kitchen down the hall.
6 Apr	Eliot Pacherie NYU Abu Dhabi	Nonlinear enhanced dissipation in viscous Burgers type equations We consider the viscous Burgers equation on the real line for a particular class of initial data with infinite mass. We show that the solutions decay at a better rate than the solutions of the heat equation for the same initial datas: a nonlinear transport term improves the dissipation. We compute the asymptotic profile, which presents a discontinuity, similar to boundary layer problems. This is a collaboration with Tej-Eddine Ghoul and Nader Masmoudi.
20 Apr	Søren Mikkelsen University of Bath	Schrödinger evolution in a low-density random potential – Convergence to solutions of the linear Boltzmann equation It is a fundamental problem in mathematical physics to derive macroscopic transport equation from the underlying microscopic transport equations. In this talk, we will consider problems of this kind. To be precise we will consider solutions to a time-dependent Schrödinger equation for a potential localised at the points of a Poisson point process. For these solutions we will present a result stating that the phase-space distribution converges in the annealed Boltzmann-Grad limit to a semiclassical Wigner measure which solves the linear Boltzmann equation.
27 Apr 6W 1.1	Matthias Kurzke University of Nottingham	This seminar will take place in 6W 1.1 Tetrahedral Frame Fields via Constrained Third-Order Symmetric Tensors In this talk I will present some results on Ginzburg–Landau approximations of frame fields. Frame fields with octahedral order (cross fields) have been used for mesh generation, while tetrahedral symmetry occurs in some liquid crystals. Compared to their 2D analog (triangular or “Mercedes-Benz” frames), tetrahedral frames have a more interesting topology with non-abelian fundamental group. Tetrahedral frame fields have a natural isometric embedding into third-order symmetric tensors that allows a Ginzburg–Landau type approximation, and I will show some of the topological singularities than can be observed. This is joint work with Dmitry Golovaty, Alberto Montero and Daniel Spirn.
4 May	Jonathan Bevan University of Surrey	Mean Hadamard inequalities This talk is about integral functionals of the form \[I_n(\varphi)=\int_Q \|\nabla \varphi\|^n+ f(x)\det \nabla \varphi \;\mathrm dx\] which can appear in the Calculus of Variations as 'excess functionals'. This makes it important to know for which \(f \in L^{\infty}(Q)\) it holds that \(I_n(\varphi) \geq 0\) for all \(\varphi \in W_0^{1,n}(Q;\mathbb{R}^n)\). We prove that there are piecewise constant \(f\) such that \(I_n\geq 0\) holds and, moreover, that this is strictly stronger than any inequality obtained by using only the pointwise Hadamard inequality \(n^{\frac{n}{2}}\|\det A\|\leq \|A\|^n\) for \(n \times n\) matrices \(A\). When \(f\) takes just two values we find that \(I_n\geq 0\) holds if and only if the variation of \(f\) in \(Q\) is at most \(2n^{\frac{n}{2}}\), a fact that is connected to work of A. Mielke and P. Sprenger (in J. Elasticity, 1998) on quasiconvexity at the boundary. For more general (but still piecewise constant) \(f\), we show that \(I_n \geq 0\) is decided by both the geometry of the jump sets and the values taken by \(f\). This is joint work with Martin Kruzik and Jan Valdman.
11 May 2:15	Walter Strauss Brown University	Continua of steadily rotating stars I will present a survey of some recent mathematical work on rotating stars that is joint with Yilun Wu and also partly with Juhi Jang. The rotating star is modeled as a compressible fluid subject to gravity. Under certain conditions there exists a large family of solutions on which the supports of the stars become unbounded. The stars have a fixed mass and they rotate around a fixed axis at a speed that varies along the family. I will also mention more elaborate models, including one that permits the entropy to be variable.
~~11 May~~ ~~2:15~~	Mark Wilkinson Nottingham Trent University	Cancelled

Autumn 2022

Date	Speaker	Title/Abstract
6 Oct 4W 4.5 2:30–3:30		Coffee and biscuits in the social space (4W 4.5) Free coffee pods and biscuits! Bring your own mug, or grab one from the fourth floor kitchen down the hall.
13 Oct 4W 1.7	Matthew Schrecker UCL	Gravitational Collapse of Self-Similar Stars The Euler-Poisson equations give the classical model of a self-gravitating star under Newtonian gravity. It is widely expected that, in certain regimes, initially smooth initial data may give rise to blow-up solutions, corresponding to the collapse of a star under its own gravity. In this talk, I will present recent work with Yan Guo, Mahir Hadzic and Juhi Jang that demonstrates the existence of smooth, radially symmetric, self-similar blow-up solutions for this problem. At the heart of the analysis is the presence of a sonic point, a singularity in the self-similar model that poses serious analytical challenges in the search for a smooth solution.
20 Oct 4W 1.7	Michael Shearer NC State University	The Kawahara Equation The Kawahara equation is a 5th order dispersive equation, a higher order version of the KdV equation. Traveling waves satisfy a fourth-order ordinary differential equation in which the traveling wave speed \(c\) and a constant of integration \(A\) are parameters. A further integration yields the Hamiltonian, an invariant of all solutions. Periodic solutions are computed with an iterative spectral method, resulting in a family of periodic solutions depending on the three constants \(c\), \(A\) and wave number \(k\). We derive jump conditions between periodic solutions with different wave numbers but equal speeds and Hamiltonian. The jump conditions are necessary conditions for the existence of traveling waves that asymptote to the periodic orbits at infinity. Bifurcation theory and parameter continuation are then used to compute multiple solution branches for the jump conditions. From these, we construct heteroclinic orbits from the intersection of stable and unstable manifolds of compatible periodic solutions. Each branch terminates at an equilibrium-to-periodic solution in which the equilibrium is the background for a solitary wave that connects to the associated periodic solution. The jump conditions are closely related to Whitham shocks, discontinuous solutions of the Whitham modulation equations, suggesting the existence of wave structures more complex than the traveling waves presented here.
27 Oct 4W 1.7	Charles Collot CY Cergy Paris Université	Stability of the Kolmogorov-Zakharov spectrum of weak turbulence Weakly turbulent waves can be described by the kinetic wave equation. The first part of the talk will present results on the derivation of this effective equation from a microscopic system described by the nonlinear Schrodinger equation. This kinetic wave equation has isotropic steady states called Kolmogorov–Zakharov spectra, that represent cascades of mass and energy between from small spatial scales toward large ones and conversely. The second part of the talk will discuss the stability of the spectrum associated to the indirect mass cascade. This is based on joint works with I. Ampatzoglou, H. Dietert, and P. Germain.
3 Nov 4W 1.7	André de Laire Université de Lille	On traveling waves for the Gross–Pitaevskii equations In this talk, we will discuss some properties of traveling waves solutions for some variants of the classical Gross–Pitaevskii equation in the whole space, in order to include new physical models in Bose–Einstein condensates and nonlinear optics. We are interested in the existence of finite energy localized traveling waves solutions with nonvanishing conditions at infinity, i.e. dark solitons. After a review of the state of the art in the classical case, we will show some results for a family of Gross–Pitaevskii equations with nonlocal interactions in the potential energy, obtained by variational techniques. Then, we will discuss the existence and behavior of the dark solitons for the Gross–Pitaevskii equation is a strip, according to its width. This is joint work with Philippe Gravejat, Salvador Lopez-Martinez, and Didier Smets.
10 Nov 4W 1.7	Mahir Hadžić University College London	Oscillation vs. relaxation in galactic dynamics We give an overview of recent developments in the stability study of nontrivial equilibria of the gravitational Vlasov–Poisson system. We discuss the existence of linear oscillating modes and the validity of the gravitational Landau damping in this context. Time permitting, we will also discuss the Einstein–Vlasov equilibria proving that steady galaxies with very dense cores lead to existence of growing mode instabilities in contrast to the Newtonian case. Our results highlight the importance of the Hamiltonian structure of these systems. The talk is based on published and ongoing works with Z. Lin, G. Rein, M. Schrecker, and C. Straub.
17 Nov 4W 1.7	Hyunju Kwon ETH Zürich	A strong Onsager conjecture on the Euler equations Smooth (spatially periodic) solutions to the incompressible 3D Euler equations have kinetic energy conservation in every local region, while turbulent flows exhibit anomalous dissipation of energy. Toward verification of the anomalous dissipation, the Onsager theorem, the threshold Hölder regularity of the total kinetic energy conservation is 1/3, has been proved by Isett. In this talk, I'll discuss a strong Onsager conjecture, which combines the Onsager theorem with the local energy inequality.
24 Nov 4W 1.7	Mark Wilkinson Nottingham Trent University	Cancelled
1 Dec 4W 1.7	María Medina de la Torre Universidad Autónoma de Madrid	From sign-changing solutions of the Yamabe equation to critical competitive systems In this talk we will analyze the existence and the structure of different sign-changing solutions to the Yamabe equation in the whole space and we will use them to find solutions to critical competitive systems in dimension 4.
8 Dec 4W 1.7	Lashi Bandara Brunel University London	Boundary value problems for first-order elliptic operators with compact and noncompact boundary The index theorem for compact manifolds with boundary, established by Atiyah-Patodi-Singer in the mid-70s, is considered one of the most significant mathematical achievements of the 20th century. An important and curious fact is that local boundary conditions are topologically obstructed for index formulae and non-local boundary conditions lie at the heart of this theorem. Consequently, this has inspired the study of boundary value problems for first-order elliptic differential operators by many different schools, with a class of induced operators adapted to the boundary taking centre stage in formulating and understanding non-local boundary conditions. That being said, much of this analysis has been confined to the situation when adapted boundary operators can be chosen self-adjoint. Dirac-type operators are the quintessential example. Nevertheless, natural geometric operators such as the Rarita-Schwinger operator on 3/2-spiniors, arising from physics in the study of the so-called Delta baryon, falls outside of this class. Analytically, this requires analysis beyond self-adjoint operators. In recent work with Bär, the compact boundary case is handled for general first-order elliptic operators, using spectral theory to choose adapted boundary operators to be invertible bi-sectorial. The Fourier circle methods present in the self-adjoint analysis are replaced by the bounded holomorphic functional calculus, coupled with pseudo-differential operator theory and semi-group techniques. This allows for a full understanding of the maximal domain of the interior operator as a bounded surjection to a space on the boundary of mixed Sobolev regularity, constructed from spectral projectors associated to the adapted boundary operator. Regularity and Fredholm extensions are also studied. For the noncompact case, a preliminary trace theorem as well as regularity theory are handed by resorting to the case with compact boundary. This necessitates deforming the coefficients of the interior operator in a compact neighbourhood. Therefore, even for Dirac-type operators, allowing for fully general symbols in the compact boundary case is paramount. Under slightly stronger geometric assumptions near the noncompact boundary (automatic for the compact case) and when the interior operator admits a self-adjoint adapted boundary operator, an upgraded trace theorem mirroring the compact setting is obtained. Importantly, there is no spectral assumptions other than self-adjointness on the adapted boundary operator. This, in particular, means that the spectrum of this operator can be the entire real line. Again, the primarily tool that is used in the analysis is the bounded holomorphic functional calculus.
15 Dec 4W 1.7	Camilla Nobili University of Surrey	The role of boundary conditions in scaling laws for the Rayleigh–Bénard convection problem Rayleigh–Bénard convection is the buoyancy-driven flow of a fluid heated from below and cooled from above and is a paradigm for nonlinear dynamics with important applications in meteorology, oceanography and engineering. We are interested in obtaining quantitative bounds on the Nusselt number, the vertical heat transport enhancement factor. The Nusselt number, besides being an interesting quantity for engineering applications, is the natural quantity to measure the intensity and effectiveness of the motion. For this reason, we are interested in proving (upper) bounds which catch the relation between the Nusselt number and the (nondimensional control parameter) Rayleigh number, in turbulent regimes. Despite great scientific developments in this field in the last 30 years, it is still not clear what role the boundary conditions play in the scaling laws for the Nusselt number. In this talk we address this problem, establishing rigorous bounds for the Rayleigh–Bénard convection problem with Navier-slip boundary conditions for the velocity. We employ the background field method and deal with a careful PDE analysis, due to the production of vorticity at the walls. In conclusion, we relate this result to other bounds derived for no-slip and stress-free boundary conditions and discuss open problems. This seminar is based on a joint work with T. Drivas and H. Nguyen and on an ongoing project with F. Bleitner.

Previous Analysis Seminars

This seminar series was previously called PDE seminar and the programme since the academic year 2010/11 can be found here.