The Analysis and Differential Equations Seminar takes place on Thursdays at 2:00, either in 4W 1.7 (Wolfson Lecture Theatre) or on Zoom.
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 Miles Wheeler and Tobias Barker.
| Date | Speaker | Title/Abstract |
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17 Feb Zoom |
Nikos Katzourakis University of Reading |
Generalised vectorial infinity-eigenvalue problems In this talk I will discuss some recent progress made in the study of nonlinear eigenvalue problems for supremal functionals. The results apply to the quasiconvex vectorial case, but they are new even for the scalar case of the well-known infinity-eigenvalue problem. |
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24 Feb 4W 1.7 |
Alexey Cheskidov University of Illinois at Chicago |
3D Navier-Stokes equations: the dynamics of a blow-up I will present an elementary introduction to the regularity and uniqueness problems for the Navier-Stokes equations. Thanks to the convex integration technique, the h-principle is starting to take shape for the uniqueness problem. It might be possible that classical scaling invariant uniqueness results (Prodi-Serrin type conditions) are the only constraints for constructing non-unique solutions of the NSE. As for the regularity problem, the situation is more delicate since there are also dynamical constraints that can potentially prevent singularity formation. |
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3 Mar 4W 1.7 | Andres Zuñiga University of O'Higgins (UOH) | A nonlocal isoperimetric problem: density perimeter We will discuss a variant of a classical geometric minimization problem, known as the “nonlocal isoperimetric problem”, which arises from studies in Nuclear Physics by Gamow in the 1930’s. By introducing a density in the perimeter functional, we obtain features that differ substantially from existing results in the framework of the classical problem without densities. In the regime of “small” non-local contribution, we completely characterize the minimizer, in the case the density is a monomial radial weight. This work is a collaboration with Stan Alama and Lia Bronsard (McMaster University) and Ihsan Topaloglu (Virginia Commonwealth University), as part of the project QUALITATIVE PROPERTIES OF WEIGHTED AND ANISOTROPIC VARIATIONAL PROBLEMS financed by ANID CHILE FONDECYT INICIACION Nº 11201259. |
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10 Mar 4W 1.7 | Bogdan Raiță Scuola Normale Superiore | On the compensated compactness method for concentration effects We study compensation phenomena for fields satisfying both a pointwise and a linear differential constraint. The compensation effect takes the form of nonlinear elliptic estimates, where constraining the values of the field to lie in a cone compensates for the lack of ellipticity of the differential operator. We give a series of new examples of this phenomenon, focusing on the case where the cone is a subset of the space of symmetric matrices and the differential operator is the divergence or the curl. One of our main findings is that the maximal gain of integrability is tied to both the differential operator and the cone, contradicting in particular a recent conjecture of G. De Philippis et al from arXiv:2106.03077. This appends the classical compensated compactness framework for oscillations with a variant designed for concentrations and also extends the recent theory of compensated integrability due to D. Serre. In particular, we find a new family of integrands that are Div-quasiconcave under convex constraints. Some of our proofs draw from a class of degenerate fully nonlinear elliptic equations, the \(k\)-hessian equations. |
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17 Mar 4W 1.7 | Julian Scheuer Cardiff University | Stability from rigidity via umbilicity The soap bubble theorem says that a closed, embedded surface of the Euclidean space with constant mean curvature must be a round sphere. Especially in real-life problems it is of importance whether and to what extent this phenomenon is stable, i.e. when a surface with almost constant mean curvature is close to a sphere. This problem has been receiving lots of attention until today, with satisfactory recent solutions due to Magnanini/Poggesi and Ciraolo/Vezzoni. The purpose of this talk is to discuss further problems of this type and to provide two approaches to their solutions. The first one is a new general approach based on stability of the so-called "Nabelpunktsatz". The second one is of variational nature and employs the theory of curvature flows. |
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24 Mar Zoom |
Uzy Smilansky Weizmann Institute of Science |
The Kronig–Penney model in a quadratic channel with \(δ\) interactions We return to an old problem – the Schödinger operator \[ H(x,y) = -\frac{\partial^2}{\partial x^2} + \frac 12 \Big(-\frac{\partial^2}{\partial y^2} + y^2\Big) + \lambda y \delta(x); \qquad x,y \in \mathbf R^2 \] a paradigm model for multi-mode quantum graphs, and review some of its surprising spectral features by addressing it using a scattering approach. One familiar with this approach we use it in order to investigate a variation on the same theme – where the delta potential is periodic along the \(x\) axis. \[ H(x,y) = -\frac{\partial^2}{\partial x^2} + \frac 12 \Big(-\frac{\partial^2}{\partial y^2} + y^2\Big) + \lambda y \sum_{n \in \mathbf Z} \delta(x-nL); \qquad x,y \in \mathbf R^2 \] The Floquet spectrum and its unconventional dependence on the spacing parameter \(L\) will be displayed and discussed qualitatively using a semi-classical model. |
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31 Mar Zoom | Mariya Ptashnyk Heriot Watt University | From individual-based models to continuum descriptions: Modelling and analysis of interactions between different populations First we will show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations is given by a free-boundary problem for the cell densities. Then, in addition to interactions, we will consider the microscopic movement of cells and derive a fractional cross-diffusion system as the many-particle limit of a multi-species system of moderately interacting particles. |
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7 Apr 4W 1.7 | Qian Wang University of Oxford | Rough solutions of the \(3\)-D compressible Euler equations I will talk about my work on the compressible Euler equations. We prove the local-in-time existence the solution of the compressible Euler equations in \(3\)-D, for the Cauchy data of the velocity, density and vorticity \((v,\varrho, \omega) \in H^s\times H^s\times H^{s'}\), \(2 < s' < s\). The result extends the sharp result of Smith–Tataru and Wang, established in the irrotational case, i.e. \(\omega=0\), which is known to be optimal for \(s > 2\). At the opposite extreme, in the incompressible case, i.e. with a constant density, the result is known to hold for \(\omega\in H^s\), \(s>3/2\) and fails for \(s\le 3/2\), see the work of Bourgain–Li. It is thus natural to conjecture that the optimal result should be \((v,\varrho, \omega) \in H^s\times H^s\times H^{s'}\), \(s>2, \, s'>\frac{3}{2}\). We view our work as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of sound wave from the transported part, and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustic spacetime. |
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14 Apr 4W 1.7 2:00 | First of two talks Helena J. Nussenzveig Lopes Federal University of Rio de Janeiro | Energy balance for 2D incompressible fluid flow I will discuss some recent results on energy balance for weak solutions of the 2D incompressible Euler equations with and without external forcing. This is based on joint works with Alexey Cheskidov, Milton Lopes Filho and Roman Shvydkoy. |
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14 Apr 4W 1.7 3:00 | Second of two talks Milton Lopes Federal University of Rio de Janeiro | Small obstacle limit for the inviscid Euler-alpha system We consider a family of solutions of the 2D Euler-alpha equations with no-slip boundary conditions, in the region \(\{ \varepsilon < |x| \}\). We prove that this family converges to a solution of a modified Euler system in the full plane when epsilon approaches zero. |
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Canceled |
Antonio Fernández ICMAT, Madrid |
Canceled |
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Canceled |
Arianna Giunti Imperial College London |
Canceled |
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5 May 4W 1.7 |
Douglas S. Seth NTNU |
Three-dimensional steady waves with vorticity: on the surface and internally The first rigorous existence result for the three-dimensional water wave problem was not published until 1981 and is due to Reeder and Shinbrot. In this talk we begin by highlighting a few of the difficulties that arise in the three-dimensional problem that are not present in two dimensions. In particular if we omit the common assumption that the solutions are irrotational. The rest of the talk will be dedicated to two existence results for the three-dimensional problem where the vorticity is nonzero. In both we assume that the velocity field is Beltrami. In the first result we consider the classical water wave problem with surface waves over a body of water. In the second result we also consider internal waves between different fluids. This talk is based on joint work with Erik Wahlén (Lund University) and Evgeniy Lokharu (Lund University). |
| Date | Speaker | Title/Abstract |
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10:30 30 Sep 4W 1.7 |
Steven Flynn University of Bath |
Talk at 10:30 Unraveling X-ray transforms on Heisenberg groups The classical X-ray Transform maps a function on Euclidean space to a function on the space of lines on this Euclidean space by integrating the function over the given line. Inverting the X-ray transform has wide-ranging applications, including to medical imaging and seismology. Much work has been done to understand this inverse problem in Euclidean space, Euclidean domains, and more generally, for symmetric spaces and Riemannian manifolds with boundary where the lines become geodesics. We formulate a sub-Riemannian version of the X-ray transform on the simplest sub-Riemannnian manifold, the Heisenberg group. Here serious geometric obstructions to classical inverse problems, such as existence of conjugate points, appear generically. With tools adapted to the geometry, such as an operator-valued Fourier Slice Theorem, we prove that an integrable function on the Heisenberg group is indeed determined by its line integrals over sub-Riemannian (as well as over its compatible Riemannian and Lorentzian) geodesics. We also pose an abundance of accessible follow-up questions, standard in the inverse problems community, concerning the sub-Riemannian case, and report progress answering some of them. |
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7 Oct Zoom |
Amol Sasane London School of Economics |
On the existence of spatially tempered null solutions to linear constant coefficient PDEs Given a linear, constant coefficient partial differential equation in \(\mathbb R^{d+1}\), where one independent variable plays the role of ‘time’, a distributional solution is called a null solution if its past is zero. Motivated by physical considerations, we consider distributional solutions that are tempered in the spatial directions alone (and do not impose any restriction in the time direction). Considering such spatially tempered distributional solutions, we give an algebraic-geometric characterization, in terms of the polynomial describing the PDE at hand, for the null solution space to be trivial (that is, consisting only of the zero distribution). |
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14 Oct 4W 1.7 |
Leonardo Tolomeo University of Bonn |
Quasi-invariance and growth of Sobolev norms for Hamiltonian PDEs
In this talk, we consider a Hamiltonian PDE with a suitable gaussian initial data, with the goal of studying the growth in time of the solution to this equation. The study of this kind of problems was started by Bourgain in ’96, who considered the Schrödinger equation with cubic nonlinearity posed on the 2-dimensional torus (2d-NLS). In his work, Bourgain exploited the formal invariance of the Gibbs measure in order to construct solutions to 2d-NLS in negative Sobolev regularity. These solutions satisfy a logarithmic bound on the growth of their norm. A crucial observation is that Bourgain’s argument relies only on quasi-invariance, i.e. the existence of a density for the law of the solution with respect to the initial gaussian measure, together with some assumptions on the density. In 2015, N. Tzvetkov developed a strategy to prove quasi-invariance that depends only on the structure of the transport equation associated with the flow. This approach was expanded in 2018 by Planchon, Tzvetkov and Visciglia, who managed to incorporate some information about the solution that come from the deterministic study of the PDE. This allows to obtain quasi-invariance as a consequence of deterministic global well posedness in a plethora of situations. In this talk, we suggest a further improvement to the previous techniques, that relies on finer space-time properties of the density of the transported measure. As an application, we obtain quasi-invariance and polynomial growth of solutions for the fourth-order Schrödinger equation with initial data in negative Sobolev regularity. This is a joint work with J. Forlano (UCLA). |
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21 Oct |
No seminar | |
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28 Oct |
Tobias Barker University of Bath |
A quantitative approach to the Navier–Stokes equations Recently, Terence Tao used a new quantitative approach to infer that certain ‘slightly supercritical’ quantities for the Navier–Stokes equations must become unbounded near a potential blow-up time. In this talk I’ll discuss a new strategy for proving quantitative bounds for the Navier–Stokes equations, as well as applications to behaviours of potentially singular solutions. This talk is based upon joint work with Christophe Prange (CNRS, Cergy Paris Université). |
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4 Nov 4W 1.7 |
Michele Dolce Imperial College London |
Long-time behaviour in the 2D Euler-Boussinesq equations near a stably stratified Couette flow Fluids in the ocean are often inhomogeneous, incompressible and, in relevant physical regimes, can be described by the 2D Euler–Boussinesq system. Equilibrium states are then commonly observed to be stably stratified, namely the density increases with depth. We are interested in considering the case when also a background shear flow is present. In the talk, I will describe quantitative results for small perturbations around a stably stratified Couette flow. We show that the density variation and velocity decay in \(L^2\) with a rate \(O(t^{-1/2})\), namely they undergo inviscid damping. On the other hand, the vorticity and density gradient grow as \(O(t^{1/2})\), a phenomenon that we call shear-buoyancy instability. This is first precisely quantified at the linear level. For the nonlinear problem, the result holds on the optimal time-scale on which a perturbative regime can be considered. Namely, given an initial perturbation of size \(O(ε)\), thanks to the linear instability, it becomes of size \(O(1)\) on a time-scale of order \(O(ε^{-2})\). This is joint work with J. Bedrossian, R. Bianchini and M. Coti Zelati. |
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11 Nov 4W 1.7 |
Lisa Kreusser University of Bath |
Rigorous Continuum Limit for the Discrete Network Formation Problem Motivated by recent physics papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. We study the existence of solutions to this model, propose an adaptation so that a macroscopic system can be obtained as its formal continuum limit, and show the global existence of weak solutions of the macroscopic gradient flow. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. This is joint work with J. Haskovec and P. Markowich. |
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18 Nov Zoom |
Costante Bellettini University College London |
Existence of hypersurfaces with prescribed-mean-curvature Let \(N\) be a compact Riemannian manifold of dimension 3 or higher, and \(g\) a Lipschitz non-negative (or non-positive) function on \(N\). We prove that there exists a closed hypersurface \(M\) whose mean curvature attains the values prescribed by \(g\) (joint work with Neshan Wickramasekera, Cambridge). Except possibly for a small singular set (of codimension 7 or higher), the hypersurface \(M\) is \(C^2\) immersed and two-sided (it admits a global unit normal); the scalar mean curvature at \(x\) is \(g(x)\) with respect to a global choice of unit normal. More precisely, the immersion is a quasi-embedding, namely the only non-embedded points are caused by tangential self-intersections: around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially (as in the case of two spherical caps touching at a point). A special case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when \(g\) is a constant, in which the above result gives a CMC (constant-mean-curvature) hypersurface for any prescribed value of the mean curvature. The construction of \(M\) is carried out largely by means of PDE principles: (i) a minmax for an Allen–Cahn (or Modica-Mortola) energy, involving a parameter that, when sent to 0, leads to an interface from which the desired PMC hypersurface is extracted; (ii) quasi-linear elliptic PDE and geometric-measure-theory arguments, to obtain regularity conclusions for said interface; (iii) parabolic semi-linear PDE (together with specific features of the Allen-Cahn framework), to tackle cancellation phenomena that can happen when sending to 0 the Allen-Cahn parameter. |
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25 Nov Zoom |
Jan Kristensen University of Oxford |
Oscillation and concentration in sequences of PDE constrained maps It is well-known that for exponents \(p>1\), any \(L^p\)-weakly converging sequence of PDE constrained maps admits, up to a subsequence, a decomposition into sequences of PDE constrained maps where one converges in measure (no oscillation) and the other is \(p\)-equi-integrable (no concentration). For \(p=1\) the relevant corresponding result concerns weakly* convergent sequences of PDE constrained measures and is false: the oscillation and concentration cannot be separated while simultaneously satisfying the PDE constraint. In this talk we explain how the concentration regardless of the failure of a decomposition result retains its PDE character. The presented results are parts of joint works with Andre Guerra (IAS/ETH) and Bogdan Raita (Pisa). |
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2 Dec Zoom |
Evgeniy Lokharu Linköping University |
Fine properties of steady water waves In this talk we will discuss some recent results on two dimensional steady water waves. We will explain how the Benjamin and Lighthill conjecture can be significantly refined and will prove a new bound for the amplitude of an arbitrary Stokes wave in terms of the non-dimensional Bernoulli constant. Our result, in particular, implies the inequality \(a \leq C c^2 / g\), where \(a\) is the amplitude, \(c\) is the speed of the wave, and \(g\) is the gravitational constant. This fact is valid for arbitrary Stokes waves irrespectively of the amplitude with an absolute constant \(C\). Another observation is that any extreme Stokes wave over a sufficiently deep stream has necessarily a small amplitude, provided the non-dimensional mass flux is much smaller than the depth. |
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9 Dec Zoom |
Filip Rindler University of Warwick |
Elasto-plasticity driven by dislocation movement The modelling of large-strain elasto-plasticity poses many challenges and, despite its great practical importance in industry, no fully satisfactory theory has emerged so far. On the other hand, it has been known for a long time that the principal mechanism behind macroscopic plastic deformation in crystalline materials (such as metals) is the movement of dislocations, that is, 1-dimensional topological defects in the crystal lattice. In this talk, I will present some recent progress towards the challenge of coupling elasto-plasticity with the movement of systems of (discrete) dislocations. In particular, I will explain how a new geometric language, built on 2-dimensional "slip trajectories" in space-time (based on the theory of integral currents), yields a natural mathematical framework for dislocation evolutions. This yields the first existence result for solutions to the evolutionary system describing a crystal undergoing large-strain elasto-plastic deformations where the plastic part of the deformation is driven directly by the movement of dislocations. This is joint work with T. Hudson (Warwick). |
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16 Dec 4W 1.7 |
Jan Burczak Universität Leipzig |
From ketchup to concentration-driven convex integration It is much easier to make hair gel or shaving foam flow after applying some force to it. Such fluid is called non-Newtonian: it changes its viscosity under applied force. This behaviour is abundant in nature: ketchup, ice, concrete, molten lava, blood, certain polymers, porridge, not forgetting the eponymous ketchup are all non-Newtonian. A simple model of such fluid (a power-law model) is known to be well-posed in the 'subcritical' regime and to have energy solutions above its 'compactness threshold'. A recent result obtained with S. Modena and L. Székelyhidi shows that also a dual picture holds. Namely, the power-law model is ill posed below the 'compactness threshold' and it has many (very) weak solutions in the 'supercritical regime'. The last result is of consequence to the classical Navier–Stokes equations. |
This seminar series was previously called PDE seminar and the programme since the academic year 2010/11 can be found here.