Past PDE CDT Lunchtime Seminar

We investigate the long-time behaviour of solutions to a nonlocal partial differential equation on smooth Riemannian manifolds of bounded sectional curvature. The equation models self-collective behaviour with intrinsic interactions that are modeled by an interaction potential. Without the diffusion term, we consider attractive interaction potentials and establish sufficient conditions for a consensus state to form asymptotically. In addition, we quantify the approach to consensus, by deriving a convergence rate for the diameter of the solution’s support. With the diffusion term, the attractive interaction and the diffusion compete. We provide the conditions of the attractive interaction for each part to win.

The Camassa--Holm (CH) equation is a nonlocal equation that manifests supercritical behaviour in ``wave-breaking" and non-uniqueness. In this talk, I will discuss the existence of global (dissipative weak martingale) solutions to the CH equation with multiplicative, gradient type noise, derived as an inviscid limit. The goal of the talk is twofold. The stochastic CH equation will be used to illustrate aspects of a stochastic compactness and renormalisation method which is popularly used to derive well-posedness and continuous dependence results in SPDEs. I shall also discuss how a lack of temporal compactness introduces fundamental difficulties in the case of the stochastic CH equation.

This talk is based on joint works with L. Galimbert and H. Holden, both at NTNU, and with K.H. Karlsen at the University of Oslo. 

We will show asymptotic analysis for hydrodynamic system, as Navier-Stokes-Fourier system, as a useful tool in in the situation when certain parameters in the system – called characteristic numbers – vanish or become infinite. The choice of proper scaling, namely proper system of reference units, the parameters determining the behaviour of the system under consideration allow to eliminate unwanted or unimportant for particular phenomena modes of motion. The main goal of many studies devoted to asymptotic analysis of various physical systems is to derive a simplified set of equations - simpler for mathematical or numerical analysis. Such systems may be derived in a very formal way, however we will concentrate on rigorous mathematical analysis. I will concentrate on low Mach number limits with so called ill-prepared data and I will present some results which concerns passage from compressible to incompressible models of fluid flow emphasising difficulties characteristic for particular problems. In particular we will discuss Navier-Stokes-Fourier system on varying domains, a multi-scale problem for viscous heat-conducting fluids in fast rotation and the incompressible limit of compressible finitely extensible nonlinear bead-spring chain models for dilute polymeric fluids.

This talk is motivated by work on the existence theory for viscoelasticity of Kelvin-Voigt type with non-convex stored energies (joint with K. Koumatos (U. of Sussex), C. Lattanzio and S. Spirito (U. of LAquila)), which shows propagation of H1-regularity for the deformation gradient of weak solutions for semiconvex stored energies. It turns out that weak solutions with deformation gradient in H1 are in fact unique in two-space dimensions, providing a striking analogy to corresponding results in the theory of 2D Euler equations with bounded vorticity.

While weak solutions still exist for initial data in L2, oscillations on the deformation gradi- ent can now persist and propagate in time. This can be seen via a counterexample indicating that for non-monotone stress-strain relations in 1-d oscillations of the strain lead to solutions with sustained oscillations. The existence of sustained oscillations in hyperbolic-parabolic system is then studied in several examples motivated by viscoelasticity and thermoviscoelas- ticity. Sufficient conditions for persistent oscillations are developed for linear problems, and examples in some nonlinear systems of interest. In several space dimensions oscillatory exam- ples are associated with lack of rank-one convexity of the stored energy. Nonlinear examples in models with thermal effects are also developed.

The formulation of Mean Field Games (MFG) via partial differential equations typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov--Fokker--Planck equation for the density of players. However, in many cases of practical interest, the underlying optimal control problem may exhibit bang-bang controls, which typically lead to nondifferentiable Hamiltonians. In this talk we will present results on the analysis and numerical approximation of stationary second-order MFG systems for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we will propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of subdifferentials of convex functions. We present results that guarantee the existence of unique weak solutions to the stationary MFG PDI under a monotonicity condition similar to one that has been considered previously by Lasry and Lions. Moreover, we will propose a monotone finite element discretization of the weak formulation of the MFG PDI, and present results that confirm the strong H^1-norm convergence of the approximations to the value function and strong L^q-norm convergence of the approximations to the density function. The performance of the numerical method will be illustrated in experiments featuring nonsmooth solutions. This talk is based on joint work with my supervisor Iain Smears.

Irreversible processes are accompanied by an increase in the internal entropy of a continuum, and as such the entropy production function is fundamental in determining the overall state of the system. In this talk, it will be shown that the entropy production function can be utilized for a variational analysis of certain dissipative continua in two different ways. Firstly, a novel unified Lagrangian-Hamiltonian formalism is constructed giving phase space extra structure, and applied to the study of fluid flow and brittle fracture.  Secondly, a maximum entropy production principle is presented for simple bodies and its implications to the study of fluid flow discussed. 

Second-order regularity results are established for solutions to elliptic equations and systems with the principal part having a Uhlenbeck structure and square-integrable right-hand sides. Both local and global estimates are obtained. The latter apply to solutions to homogeneous Dirichlet problems under minimal regularity assumptions on the boundary of the domain. In particular, if the domain is convex, no regularity of its boundary is needed. A critical step in the approach is a sharp pointwise inequality for the involved elliptic operator. This talk is based on joint investigations with A.Kh.Balci, L.Diening, and V.Maz'ya.

Localisation is a powerful tool in proving and analysing various geometric inequalities, including isoperimertic inequality in the context of metric measure spaces. Its multi-dimensional generalisation is linked to optimal transport of vector measures and vector-valued Lipschitz maps. I shall present recent developments in this area: a partial affirmative answer to a conjecture of Klartag concerning partitions associated to Lipschitz maps on Euclidean space, and a negative answer to another conjecture of his concerning mass-balance condition for absolutely continuous vector measures. During the course of the talk I shall also discuss an intriguing notion of ghost subspaces related to the above mentioned partitions. 

In General Relativity, an impulsive gravitational wave is a localized and singular solution of the 

Einstein equations modeling the spacetime distortions created by a strongly gravitating source.
I will present a comprehensive theory allowing for ternary interactions of such impulsive gravitational waves in translation-symmetry, offering the first examples of such an interaction.  

The proof combines new techniques from harmonic analysis, Lorentzian geometry, and hyperbolic PDEs that are helpful to treat highly anisotropic low-regularity questions beyond the considered problem.  

This is joint work with Jonathan Luk.

On a smooth Riemannian manifold, the uniqueness of a geodesic given initial conditions follows from standard ODE theory. This is known to fail in the setting of RCD(K,N) spaces (metric measure spaces satisfying a synthetic notion of Ricci curvature bounded below) through an example of Cheeger-Colding. Strengthening the assumption a little, one may ask if two geodesics which agree for a definite amount of time must continue on the same trajectory. In this talk, I will show that this is true for RCD(K,N) spaces. In doing so, I will generalize a well-known result of Colding-Naber concerning the Hölder continuity of small balls along geodesics to this setting.

We derive quantitatively the weak and strong Harnack inequality for kinetic Fokker--Planck type equations with a non-local diffusion operator for the full range of the non-locality exponents in (0,1).  This implies Hölder continuity.  We give novel proofs on the boundedness of the bilinear form associated to the non-local operator and on the construction of a geometric covering accounting for the non-locality to obtain the Harnack inequalities.  Our results apply to the inhomogeneous Boltzmann equation in the non-cutoff case.

I will discuss the regularity of solutions to quasilinear systems satisfying a Legendre-Hadamard ellipticity condition. For such systems it is known that weak solutions may which fail to be C^1 in any neighbourhood, so we cannot expect a general regularity theory. However if we assume an a-priori regularity condition of the solutions we can rule out such counterexamples. Focusing on solutions to Euler-Lagrange systems, I will present an improved regularity results for solutions whose gradient satisfies a suitable BMO / VMO condition. Ideas behind the proof will be presented in the interior case, and global consequences will also be discussed.

We will consider the viscous Burgers driven by a localised one-dimensional control. The problem is considered in a bounded domain and is supplemented with the Dirichlet boundary condition. We will prove that any solution of the equation in question can be exponentially stabilised. Combining this result with an earlier result on local exact controllability we will show global exact controllability by a localised control. This is a joint work with A. Shirikyan.

We construct smooth flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We demonstrate an application of this surgery flow by improving the entropy bound on the low-entropy Schoenflies conjecture.

In this talk I will review some recent results obtained in collaboration with E. Runa and A. Kerschbaum on the one-dimensionality of the minimizers
of a family of continuous local/nonlocal interaction functionals in general dimension. Such functionals have a local term, typically the perimeter or its Modica-Mortola approximation, which penalizes interfaces, and a nonlocal term favouring oscillations which are high in frequency and in amplitude. The competition between the two terms is expected by experiments and simulations to give rise to periodic patterns at equilibrium. Functionals of this type are used  to model pattern formation, either in material science or in biology. The difficulty in proving the emergence of such structures is due to the fact that the functionals are symmetric with respect to permutation of coordinates, while in more than one space dimensions minimizers are one-dimesnional, thus losing the symmetry property of the functionals. We will present new techniques and results showing that for two classes of functionals (used to model generalized anti-ferromagnetic systems, respectively  colloidal suspensions), both in sharp interface and in diffuse interface models, minimizers are one-dimensional and periodic, in general dimension and also while imposing a nontrivial volume constraint.

The integral of mean curvature squared is a conformal invariant that measures the distance from a given immersion to the standard embedding of a round sphere. Following work of Robert Bryant who showed that all Willmore spheres in the 3-sphere are conformally minimal, Robert Kusner proposed in the early 1980s to use the Willmore energy to obtain an “optimal” sphere eversion, called the min-max sphere eversion.

We will present a method due to Tristan Rivière that permits to tackle a wide variety of min-max problems, including ones about the Willmore energy. An important step to solve Kusner’s conjecture is to determine the Morse index of branched Willmore spheres, and we show that the Morse index of conformally minimal branched Willmore spheres is equal to the index of a canonically associated matrix whose dimension is equal to the number of ends of the dual minimal surface.

Elastic strings are among the simplest one-dimensional continuum bodies and have a rich mechanical and mathematical theory dating back to the derivation of their equations of motion by Euler and Lagrange. In classical treatments, the string is either completely extensible (tensile force produces elongation) or completely inextensible (every segment has a fixed length, regardless of the motion). However, common experience is that a string can be stretched (is extensible), and after a certain amount of tensile force is applied the stretch of the string is maximized (becomes inextensible). In this talk, we discuss a model for these stretch-limited elastic strings, in what way they model elastic behavior, the well-posedness and asymptotic stability of certain simple motions, and (many) open questions.

In this talk, we discuss the feasibility of algorithms based on deep artificial neural networks (DNN) for the solution of high-dimensional PDEs, such as those arising from stochastic control and games. In the first part, we show that in certain cases, DNNs can break the curse of dimensionality in representing high-dimensional value functions of stochastic control problems. We then exploit policy iteration to reduce the associated nonlinear PDEs into a sequence of linear PDEs, which are then further approximated via a multilayer feedforward neural network ansatz. We establish that in suitable settings the numerical solutions and their derivatives converge globally, and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration. Numerical experiments on Zermelo's navigation problem and on consensus control of interacting particle systems are presented to demonstrate the effectiveness of the method. This is joint work with Kazufumi Ito, Christoph Reisinger and Wolfgang Stockinger.

I will discuss regularity properties for solutions of linear second order non-uniformly elliptic equations in divergence form. Assuming certain integrability conditions on the coefficient field, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are sharp and improve upon classical results due to Trudinger from the 1970s.

As an application of the local boundedness result, we deduce a quenched invariance principle for random walks among random degenerate conductances. If time permits I will discuss further regularity results for nonlinear non-uniformly elliptic variational problems.

We consider the homogenization of a Stokes system in a domain having many small random holes. This model mainly arises from problems of solid-fluid interaction (e.g. the flow of a viscous and incompressible fluid through a porous medium). We aim at the rigorous derivation of the homogenization limit both in the Brinkmann regime and in the one of Darcy’s law. In particular, we focus on holes that are distributed according to probability measures that allow for overlapping and clustering phenomena.

We explore one facet of an old problem: the approximation of hyperbolic conservation laws by viscous counterparts. While qualitative convergence results are well-known, quantitative rates for the inviscid limit are less common. In this talk, we consider the simplest case: a one-dimensional scalar strictly-convex conservation law started from "generic" smooth initial data. Using a matched asymptotic expansion, we quantitatively control the inviscid limit up to the time of first shock. We conclude that the inviscid limit has a universal character near the first shock. This is joint work with Sanchit Chaturvedi.

Phase transitions are present in a wide array of systems ranging from traffic to machine learning algorithms. In this talk, we will relate the concept of phase transitions to the convexity properties of the associated thermodynamic energy. Motivated by noisy stochastic gradient descent in supervised learning, we will consider the problem of understanding the thermodynamic limit of exchangeable weakly interacting diffusions (AKA propagation of chaos) from an energetic perspective. The strategy will be to exploit the 2-Wasserstein gradient flow structure associated with the thermodynamic energy in the infinite particle setting. Using this perspective, we will show how the convexity properties of the thermodynamic energy affects the homogenization limit or the stability of the log-Sobolev inequality.

A topology optimization problem for Stokes flow finds the optimal material distribution of a fluid in Stokes flow that minimizes the fluid’s power dissipation under a volume constraint. In 2003, T. Borrvall and J. Petersson [1] formulated a nonconvex optimization problem for this objective. They proved the existence of minimizers in the infinite-dimensional setting and showed that a suitably chosen finite element method will converge in a weak(-*) sense to an unspecified solution. In this talk, we will extend and refine their numerical analysis. In particular, we will show that there exist finite element functions, satisfying the necessary first-order conditions of optimality, that converge strongly to each isolated local minimizer of the problem.

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Fully nonlinear PDEs involving the eigenvalues of matrix-valued differential operators (such as the Hessian) have been the subject of intensive study over the last few decades, since the seminal work of Caffarelli, Kohn, Nirenberg and Spruck. In this talk I will discuss some recent joint work with Luc Nguyen on the regularity theory for a large class of these equations, with a particular emphasis on a special case known as the sigma_k-Yamabe equation, which arises in conformal geometry. 

[1] T. Borrvall, J. Petersson, Topology optimization of fluids in Stokes flow, International Journal for Numerical Methods in Fluids 41 (1) (2003) 77–107. doi:10.1002/fld.426.

We analyze the steady motion of a viscous incompressible fluid in a two- and three-dimensional channel containing an obstacle through the Navier-Stokes equations under different types of boundary conditions. In the 2D case we take constant non-homogeneous Dirichlet boundary data in a (virtual) square containing the obstacle, and emphasize the connection between the appearance of lift and the unique solvability of Navier-Stokes equations. In the 3D case we consider mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet. In the absence of external forcing, explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the inlet velocity. In the 3D case, this solenoidal extension is built through the Bogovskii operator and explicit bounds on its Dirichlet norm (in terms of the geometric parameters of the obstacle) are found by solving a variational problem involving the infinity-Laplacian.

The talk accounts for results obtained in collaboration with Filippo Gazzola and Ilaria Fragalà (both at Politecnico di Milano).

Solving kinetic or related models with high-dimensional random parameters has been a challenging problem. In this talk, we will discuss how to employ the bi-fidelity stochastic collocation and choose efficient low-fidelity models in order to solve a class of multi-scale kinetic equations with uncertainties, including the Boltzmann equation, linear transport and the Vlasov-Poisson equation. In addition, some error analysis for the bi-fidelity method based on these PDEs will be presented. Finally, several numerical examples are shown to validate the efficiency and accuracy of the proposed method.

I will present a nonlinear version of the open mapping principle which applies to constant-coefficient PDEs which are both homogeneous and weak* stable. An example of such a PDE is the Jacobian equation. I will discuss the consequences of such a result for the Jacobian and its relevance towards an answer to a long-standing problem due to Coifman, Lions, Meyer and Semmes. This is based on joint work with Lukas Koch and Sauli Lindberg.

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I present a general nonlinear open mapping principle suited to applications to scale-invariant PDEs in regularity regimes where the equations are stable under weak* convergence. As an application I show that, for any $p < \infty$, the set of initial data for which there are dissipative weak solutions in $L^p_t L^2_x$ is meagre in the space of solenoidal L^2 fields. This is based on joint work with A. Guerra (Oxford) and S. Lindberg (Aalto).

In the study of geometric flows it is often important to understand when a flow which converges along a sequence of times going to infinity will, in fact, converge along every such sequence of times to the same limit. While examples of finite dimensional gradient flows that asymptote to a circle of critical points show that this cannot hold in general, a positive result can be obtained in the presence of a so-called Lojasiewicz-Simon inequality. In this talk we will introduce this problem of uniqueness of asymptotic limits and discuss joint work with Melanie Rupflin and Peter M. Topping in which we examined the situation for a geometric flow that is designed to evolve a map describing a closed surface in a given target manifold into a parametrization of a minimal surface.

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The Landau equation is an important PDE in kinetic theory modelling plasma particles in a gas. It can be derived as a limiting process from the famous Boltzmann equation. From the mathematical point of view, the Landau equation can be very challenging to study; many partial results require, for example, stochastic analysis as well as a delicate combination of kinetic and parabolic theory. The major open question is uniqueness in the physically relevant Coulomb case. I will present joint work with Jose Carrillo, Matias Delgadino, and Laurent Desvillettes where we cast the Landau equation as a generalized gradient flow from the optimal transportation perspective motivated by analogous results on the Boltzmann equation. A direct outcome of this is a numerical scheme for the Landau equation in the spirit of de Giorgi and Jordan, Kinderlehrer, and Otto. An extended area of investigation is to use the powerful gradient flow techniques to resolve some of the open problems and recover known results.

I introduce a recently developed variational approach for hyperbolic PDE's. The method allows to show the existence of weak solutions to fluid-structure interactions where a visco-elastic bulk solid is interacting with an incompressible fluid governed by the unsteady Navier Stokes equations. This is a joint work with M. Kampschulte and B. Benesova.

In the talk, we study the Navier–Stokes-like problems for the flows of homogeneous incompressible fluids. We introduce a new type of boundary condition for the shear stress tensor, which includes an auxiliary stress function and the time derivative of the velocity. The auxiliary stress function serves to relate the normal stress to the slip velocity via rather general maximal monotone graph. In such way, we are able to capture the dynamic response of the fluid on the boundary. Also, the constitutive relation inside the domain is formulated implicitly. The main result is the existence analysis for these problems.

Given a curve , what is the surface  that has smallest area among all surfaces spanning ? This classical problem and its generalizations are called Plateau's problem. In this talk we consider area minimizers among the class of integral currents, or roughly speaking, orientable manifolds. Since the 1960s a lot of work has been done by De Giorgi, Almgren, et al to study the interior regularity of these minimizers. Much less is known about the boundary regularity, in the case of codimension greater than 1. I will speak about some recent progress in this direction.

In this talk, I will present different PDE models involving surface tension where it may be efficient to consider augmented versions.

Non-uniformly elliptic functionals are variational integrals like
\[
(1) \qquad \qquad W^{1,1}_{loc}(\Omega,\mathbb{R}^{N})\ni w\mapsto \int_{\Omega} \left[F(x,Dw)-f\cdot w\right] \, \textrm{d}x,
\]
characterized by quite a wild behavior of the ellipticity ratio associated to their integrand $F(x,z)$, in the sense that the quantity
$$
\sup_{\substack{x\in B \\ B\Subset \Omega \ \small{\mbox{open ball}}}}\mathcal R(z, B):=\sup_{\substack{x\in B \\ B\Subset \Omega \ \small{\mbox{open ball}}}} \frac{\mbox{highest eigenvalue of}\ \partial_{z}^{2} F(x,z)}{\mbox{lowest eigenvalue of}\  \partial_{z}^{2} F(x,z)} $$
may blow up as $|z|\to \infty$. 
We analyze the interaction between the space-depending coefficient of the integrand and the forcing term $f$ and derive optimal Lipschitz criteria for minimizers of (1). We catch the main model cases appearing in the literature, such as functionals with unbalanced power growth or with fast exponential growth such as
$$
w \mapsto \int_{\Omega} \gamma_1(x)\left[\exp(\exp(\dots \exp(\gamma_2(x)|Dw|^{p(x)})\ldots))-f\cdot w \right]\, \textrm{d}x
$$
or
$$
w\mapsto \int_{\Omega}\left[|Dw|^{p(x)}+a(x)|Dw|^{q(x)}-f\cdot w\right] \, \textrm{d}x.
$$
Finally, we find new borderline regularity results also in the uniformly elliptic case, i.e. when
$$\mathcal{R}(z,B)\sim \mbox{const}\quad \mbox{for all balls} \ \ B\Subset \Omega.$$

The talk is based on:
C. De Filippis, G. Mingione, Lipschitz bounds and non-autonomous functionals. $\textit{Preprint}$ (2020).

In this talk, we discuss Sobolev embeddings into rearrangement-invariant function spaces on (regular) domains in $\mathbb{R}^n$ endowed with measures whose decay on balls is dominated by a power $d$ of their radius, called $d$-Frostman measures. We show that these embeddings can be deduced from one-dimensional inequalities for an operator depending on $n$, $d$ and the order $m$ of the Sobolev space. We also point out an interesting feature of this theory - namely that the results take a substantially different form depending on whether the measure is decaying fast ($d\geq n-m$) or slowly ($d<n-m$). This is a
joint work with Andrea Cianchi and Lubos Pick.

I will describe new solutions to the stationary Ginzburg-Landau equation in 3 dimensions with vortex lines given by interacting helices, with degree one around each filament and total degree an arbitrary positive integer. I will also present results on the asymptotic behavior of vortices in the entire plane for a dissipative Ginzburg-Landau equation. This is work in collaboration with Manuel del Pino, Remy Rodiac, Maria Medina, Monica Musso and Juncheng Wei.

In this talk I will discuss some aspects of the potential theory, fine properties and boundary behaviour of the solutions to the Total Variation Flow. Instead of the classical Euclidean setting, we intend to work mostly in the general setting of metric measure spaces. During the past two decades, a theory of Sobolev functions and BV functions has been developed in this abstract setting.  A central motivation for developing such a theory has been the desire to unify the assumptions and methods employed in various specific spaces, such as weighted Euclidean spaces, Riemannian manifolds, Heisenberg groups, graphs, etc.

The total variation flow can be understood as a process diminishing the total variation using the gradient descent method.  This idea can be reformulated using parabolic minimizers, and it gives rise to a definition of variational solutions.  The advantages of the approach using a minimization formulation include much better convergence and stability properties.  This is a very essential advantage as the solutions naturally lie only in the space of BV functions. Our main goal is to give a necessary and sufficient condition for continuity at a given point for proper solutions to the total variation flow in metric spaces. This is joint work with Vito Buffa and Juha Kinnunen.

The second fundamental form of an embedded manifold must satisfy a set of constraint equations known as the Gauß-Codazzi equations. Since work of Chen-Slemrod-Wang, these equations are known to satisfy a particular div-curl structure: under suitable L^p bound on the second fundamental form, the curvatures are weakly continuous. In this talk we explore generalisations of this original result under weaker assumptions. We show how techniques from fluid dynamics can yield interesting insight into the weak continuity properties of isometric embeddings.

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Teichmüller harmonic map flow is a geometric flow designed to evolve combinations of maps and metrics on a surface into minimal surfaces in a Riemannian manifold. I will introduce the flow and describe known existence results, and discuss recent joint work with M. Rupflin that demonstrates how singularities can develop in the metric component in finite time.

Let $\Omega \subset \mathbb{R}^n$, $n \geq 2$, be a bounded, connected, open set with Lipschitz boundary.
Let $u$ be an eigenfunction of the Laplacian on $\Omega$ with either a Dirichlet, Neumann or Robin boundary condition.
If an eigenfunction $u$ associated with the $k$--th eigenvalue has exactly $k$ nodal domains, then we call it a Courant-sharp eigenfunction. In this case, we call the corresponding eigenvalue a Courant-sharp eigenvalue.

We first discuss some known results for the Courant-sharp Dirichlet and Neumann eigenvalues of the Laplacian on Euclidean domains.

We then discuss whether the Robin eigenvalues of the Laplacian on the square are Courant-sharp.

This is based on joint work with B. Helffer (Université de Nantes).
 

The Camassa-Holm (CH) equation is a nonlinear nonlocal dispersive equation which arises as a model for the propagation of unidirectional shallow water waves over a flat bottom. One of the most important features of the CH equation is the existence of peaked travelling waves, also called peakons. The aim of this talk is to review some asymptotic stability result for peakon solutions for CH-type equations as well as to present some new result for higher-order generalization of the CH equation.

Do classical solutions of the compressible Navier-Stokes equations converge to an entropy solution of their inviscid counterparts, the Euler equations? In this talk we present a result which answers this question affirmatively, in the one-dimensional case, for a particular class of fluids. Specifically, we consider gases that exhibit approximately polytropic behaviour in the vicinity of the vacuum, and that are isothermal for larger values of the density (which we call approximately isothermal gases). Our approach makes use of methods from the theory of compensated compactness of Tartar and Murat, and is inspired by the earlier works of Chen and Perepelitsa, Lions, Perthame and Tadmor, and Lions, Perthame and Souganidis. This is joint work with Matthew Schrecker.

We present some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient a = a(·, ·). The model case is driven by the following nonlocal double phase operator,

$$\int \frac{|u(x) − u(y)|^{p−2} (u(x) − u(y))} {|x − y|^{n+sp}} dy+ \int a(x, y) \frac{|u(x) − u(y)|^{ q−2} (u(x) − u(y))} {|x − y|^{n+tq}} dy$$

where $q ≥ p$ and $a(·, ·) = 0$. Our results do also apply for inhomogeneous equations, for very general classes of measurable kernels. By simply assuming the boundedness of the modulating coefficient, we are able to prove that the solutions are Hölder continuous, whereas similar sharp results for the classical local case do require a to be Hölder continuous. To our knowledge, this is the first (regularity) result for nonlocal double phase problems.

Formation of singularities for the relativistic Euler equations (N. Athanasiou): An archetypal phenomenon in the study of hyperbolic systems of conservation laws is the development of singularities (in particular shocks) in finite time, no matter how smooth or small the initial data are. A series of works by Lax, John et al confirmed that for some important systems, when the initial data is a smooth small perturbation of a constant state, singularity formation in finite time is equivalent to the existence of compression in the initial data. Our talk will address the question of whether this dichotomy persists for large data problems, at least for the system of the Relativistic Euler equations in (1+1) dimensions. We shall also give some interesting studies in (3+1) dimensions. This is joint work with Dr. Shengguo Zhu.

Global Well-Posedness for a Class of Stochastic McKean-Vlasov Equations in One Dimension (A. Mayorcas): We show global well-posedness for a family of parabolic McKean--Vlasov SPDEs with additive space-time white noise. The family of interactions we consider are those given by convolution with kernels that are at least integrable. We show that global well-posedness holds in both the repulsive/defocussing and attractive/focussing cases. Our strategy relies on both pathwise and probabilistic techniques which leverage the Gaussian structure of the noise and well known properties of the deterministic PDEs.

I will present recent results concerning a class of nonlinear parabolic systems of partial differential equations with small cross-diffusion (see doi.org/10.1051/m2an/2018036 and arXiv:1906.08060). Such systems can be interpreted as a perturbation of a linear problem and they have been proposed to describe the dynamics of a variety of large systems of interacting particles. I will discuss well-posedness, regularity, stability and convergence to the stationary state for (strong) solutions in an appropriate Banach space. I will also present some applications and refinements of the above-mentioned results for specific models.

In the seminar, I will talk about a parabolic toy-model for the incompressible Navier-Stokes equations, that satisfies the same energy inequality, same scaling symmetry and which is also super-critical in dimension 3. I will present some partial regularity results that this model shares with the incompressible model and other results that occur only for our model.

Federer’s characterization, which is a central result in the theory of functions of bounded variation, states that a set is of finite perimeter if and only if n−1-dimensional Hausdorff measure of the set's measure-theoretic boundary is finite. The measure-theoretic boundary consists of those points where both the set and its complement have positive upper density. I show that the characterization remains true if the measure-theoretic boundary is replaced by a smaller boundary consisting of those points where the lower densities of both the set and its complement are at least a given positive constant.

The Anderson Hamiltonian is used to model particles moving in
disordered media, it can be thought of as a Schrödiger operator with an
extremely irregular random potential. Using the recently developed theory of
"Paracontrolled Distributions" we are able to define the Anderson
Hamiltonian as a self-adjoint non-positive operator on the 2- and
3-dimensional torus and give an explicit description of its domain.
Then we use these results to solve some semi-linear PDEs whose linear part
is given by the Anderson Hamiltonian, more precisely the multiplicative
stochastic NLS and nonlinear Wave equation.
This is joint work with M. Gubinelli and B. Ugurcan.

The aim of this talk is to give an overview about the structure theory of finite dimensional RCD metric measure spaces. I will first focus on rectifiability, existence, uniqueness and constancy of the dimension of tangents up to negligible sets.
Then I will motivate why boundaries of sets of finite perimeter are natural codimension one objects to look at in this framework and present some recent structure results obtained in their study.
This is based on joint works with Luigi Ambrosio, Elia Bruè and Enrico Pasqualetto.
 

A fundamental problem in the context of Einstein's equations of general relativity is to understand precisely the dynamical evolution of small perturbations of stationary black hole solutions. It is expected that there is a discrete set of characteristic frequencies that play a dominant role at late time intervals and carry information about the nature of the black hole, much like the normal frequencies of a vibrating string. These frequencies are called quasi-normal frequencies or resonances and they are closely related to scattering resonances in the study of Schrödinger-type equations. I will discuss a new method of defining and studying resonances for linear wave equations on asymptotically flat black holes, developed from joint work with Claude Warnick.

We will discuss the similarity and difference between deterministic and stochastic NLS. Different notions (or possible formulations) of local solutions will also be discussed. We will also present a global well posedness result for stochastic mass critical NLS. Joint work with Weijun Xu (Oxford)

We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position and its orientation vector, which lies on the unit sphere. We prove that, in the low temperature regime, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into the sphere. This flow is a special case of the gradient flow to the Oseen-Frank energy functional for nematic liquid crystals and the existence of its global weak solution was first obtained by Y.M Chen, using Ginzburg-Landau approximation.  The key ingredient of our result is to show the strong compactness of the family of number density functions and the proof relies on the strong compactness of the corresponding second moment (or the Q-tensor), a spectral decomposition of the linearized operator near the limiting local equilibrium distribution, as well as the energy dissipation estimates.  This is a joint work with Wei Wang in Zhejiang university.
 

Function solutions to linear PDEs often carry rigidity properties directly associated to the equation they satsify. However, the realm of solutions covers a much larger sets of solutions. For instance, we can speak of measure solutions, as opposed to classical $C^\infty$ functions or even $L^p$ functions. It is only logical to expect that the “better” space the solution lives in, the more rigid its properties will be.

Measure solutions lie just at a comfortable half of this threshold: it is a sufficently large space which allows for a rich range of new structures; but is sufficiently rigid to preserve a meaningful geometrical pattern. For example, have you ever wondered how gradients look like in the space of measures? What about other PDE structures? In this talk I will discuss these general questions, a few examples of them, and a new theoretical approach to its understanding via PDE theory, harmonic analysis, and geometric measure theory methods.