# A conference in honour of Helge Holden

## Non-linear PDEs, mathematical physics, and stochastic analysis

### Titles and abstracts

#### Sergio Albeverio

##### A unified approach to infinite dimensional integrals (of probabilistic and oscillatory type)

**Abstract:**
We present a recent approach to infinite dimensional integration, which covers both probabilistic (Wiener type ) and oscillatory (Feynman type) integrals. This is work developed in collaboration with S. Mazzucchi (Trento). Some recent applications to quantum mechanical problems and to higher order heat-type equations are also briefly discussed.

#### Alberto Bressan

##### Optimal transport distances for nonlinear wave equations

**Abstract:**
Various nonlinear wave equations, including the Camassa–Holm
and a class of second order variational wave equation, admit unique
global conservative solutions, for arbitrary initial data in \( H^1(R) \).
However, dependence on the initial data is not continuous in the
\( H^1 \) norm.
In this talk we shall explain how to construct a Riemann-type distance
which renders Lipschitz continuous the semigroup of solutions.
This metric is determined by an optimal transportation problem.

By a well known change of variables, solutions can be represented in terms of a semilinear system. Using Thom's transversality theorem one can show that, for generic smooth initial data, the solution remains piecewise smooth. Moreover, a detailed description of all generic singularities can be given.

Using the density of piecewise smooth solutions, Lipschitz dependence w.r.t. the Riemann-type distance is first proved within a class of piecewise smooth solutions, then extended by continuity to the set of all \( H^1 \) solutions.

#### Gui-Qiang G. Chen

##### Weak Compactness/Rigidity Problems in Nonlinear Partial Differential Equations

**Abstract:**
In this lecture, we will discuss a program in the analysis of weak compactness/rigidity problems in nonlinear partial differential equations through several fundamental examples in mechanics and geometry.
The examples especially include the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations and the global stochastic entropy solutions to the barotropic Euler equations driven by random force terms, the sonic-subsonic limit of solutions of the full Euler equations for multidimensional steady compressible fluids, as well as the global weak continuity/rigidity of the Gauss–Codazzi–Ricci system and corresponding isometric embeddings of Riemannian manifolds by developing some intrinsic compensated compactness theorems on the manifolds.

#### Snorre Christiansen

##### Uniformly stable finite elements for small viscosity problems

**Abstract:**
For convection diffusion equations on bounded domains, boundary layers are expected at the outflow boundary when the viscosity becomes small. Standard Galerkin methods then exhibit spurious oscillations extending far beyond the boundary layer. For an exponentially fitted Petrov-Galerkin method on the other hand, we can prove optimal stability estimates, up to a logarithm of the viscosity.

#### Giuseppe Maria Coclite

##### A model for the morphogens evolution in a growing tissue

**Abstract:**
Recently Averbukh, Ben-Zvi, Mishra, and Barkai proposed a model for the regulation of growth and patterning in developing tissues by diffusing morphogens.
The model consists of a free boundary nonhomogeneous Neumann problem with two equations, a diffusion one and a transport one.
We show that solutions of the underlying coupled system of nonlinear PDEs exist, are unique and are stable in a suitable sense. The key tool in the analysis is the transformation of the underlying system to a porous medium equation. These results were obtained in collaboration with M. M. Coclite (University of Bari) and S. Mishra (ETH-Zurich).

#### Rinaldo M. Colombo

##### Roads and Kinks from a Hyperbolic Point of View

**Abstract:**
This talk will overview recent results about the modeling of junctions/networks in roads/pipes and about the analysis of the resulting equations. Consequences of these results reach game theory as well as the dynamics of fluids in curved pipes.

#### Adrian Constantin

##### The Hamiltonian structure of nonlinear equatorial wave-current interactions

**Abstract:**
We present a Hamiltonian formulation for the nonlinear equations of motion of equatorial wave-current interactions in a two-layer fluid with a flat bed and a free surface, for flows with piecewise constant vorticity. By proving, without recourse to approximations, that the flow can be viewed as an irrotational perturbation of a mean flow (representing the underlying current with flow-reversal), we can associate to it perturbed velocity potential functions. This permits us to obtain a computationally efficient expression of the Hamiltonian functional, in terms of the deformations of the free surface and of the free interface, the traces of the perturbed velocity potential functions on them, and the Dirichlet- Neumann operators for the two layers. The talk will be based on joint work with R. Ivanov and C.-I. Martin.

#### Constantine Dafermos

##### Redistribution of Damping in Hyperbolic Conservation Laws

The lecture will survey a program for constructing BV solutions to the Cauchy problem for hyperbolic systems of balance laws with partially dissipative source.

#### Victor Enolski

##### Analytic formulae for charge two monopole fields

**Abstract:**
The ADHMN-construction (Atiyah-Drinfeld-Hitchin-Manin-Nahm) of the Higgs and
gauge fields for a non-abelian monopole leads to solving of a
Weyl equation, that is a linear ODE with “potentials”
given by the so called Nahm data. Even in the case of
charge two, where the Nahm data are expressible in terms of
elliptic functions, the analytical expressions
for the monopole fields are still unknown. We overcome the problem using as we call it
*Lesser known Nahm Ansatz* comparatively to the well known Nahm Ansatz which reduces
ADHM instanton construction to static, ADHMN monopole case. We report complete analytic
description in \(\mathbb{R}^3\) of charge two monopole fields
in terms of four solutions of a quartic (Atiyah–Ward constraint) and four transcendents,
given as incomplete second kind elliptic integrals depending in these solutions.

#### Maria J. Esteban

##### Nonlinear flows and optimality for functional inequalities

**Abstract:**
In this talk I will present some recent results about the qualitative properties of extremal functions for functional inequalities. I will show that linear and nonlinear flows are a very good tool to prove this kind of result in an optimal way.
The main subject of my talk will be the symmetry of extremals for Caffarelli–Kohn–Nirenberg inequalities on the Eucledian espace. But I will also explain how this method can be used to derive similar inequalities and results on Riemaniann manifolds.

#### Pavel Exner

##### Schrödinger operators with singular interactions on hypersurfaces

**Abstract:**
The subject of this talk are Schrödinger operators with an attractive singular ‘potential’ supported by a hypersurface of a lower dimensionality. Some of them can be formally written as \(-\Delta-\alpha \delta(x-\Gamma)\) with \(\alpha>0\), where \(\Gamma\) is a manifold in~\(\mathbb{R}^d\) of codimension one, but we consider also more singular cases. In this talk we discuss relations between the spectra of these operators and the geometry of \(\Gamma\) using, in particular, inequalities between operators corresponding to different ‘potentials’.

#### Ulrik Skre Fjordholm

##### Well-posedness of statistical solutions of the uncertain Cauchy problem for conservation laws

**Abstract:**
We consider the scalar conservation law
\[
\partial_t u+\nabla_x\cdot f(u)=0, \qquad u(x,0)=u_0(x).
\]
In practically all real-world applications, the initial data \(u_0\) will not be known exactly due to measurement error, low measurement resolution, inherent physical uncertainties, etc. We argue that the correct way of representing uncertain data is via one of two equivalent representations:

- As a family \((\mu_t)_{t\geq0}\) of probability measures on a function space (\(L^1(\mathbb{R}^d)\) in our case). This concept of solutions was originally developed by Foias (1970) in the study of turbulence for the Navier–Stokes equation.
- As a family \((\nu_t=(\nu^1_t,\nu^2_t,\dots))_{t\geq0}\) of Young measures giving statistics and correlations of the solution at different spatial points. The "one-point statistics" \(\nu^1_t\) will be a measure-valued solution in the sense of DiPerna (1985).

We derive a hierarchy of evolutionary PDE in terms of either \(\mu_t\) or \(\nu_t\), obtaining so-called *statistical solutions*. We prove existence, uniqueness and stability for scalar conservation laws under a Kruzkov-type entropy condition.

This is joint work with Siddhartha Mishra (ETH Zürich) and Samuel Lanthaler (EPFL).

#### Hermano Frid

##### Normal Traces of \(L^2\)-Divergence Measure Fields, Strong Trace Property and Application to a Nonlinear Degenerate Parabolic Problem

**Abstract:**
We first recall a recent result on the normal trace for \(L^2\)-divergence measure fields. Then, as an application of this result, we present a new strong trace theorem.
We then show an application of this new strong trace property to an initial-boundary value problem for a nonlinear degenerate anisotropic parabolic equation.

#### Katrin Grunert

##### Blow-up for the two-component Camassa-Holm system

**Abstract:**
One property, that attracted considerable attention in the context of the Camassa–Holm (CH) equation is that even smooth initial data can lead to classical solutions which only exist locally due to wave breaking. In the case of the two-component Camassa–Holm (2CH) system, a generalization of the CH equation, there is not only a huge class of solutions that enjoys wave breaking within finite time, but there is also a regularising effect which prevents many solutions from blow up and makes it possible to approximate every weak conservative solution of the CH equation by smooth solutions of the 2CH system. Hence the aim of this talk is twofold. On the one hand we will study this regularising effect in some detail and on the other hand we will focus on how to predict if a solution enjoys wave breaking in the nearby future or not. In addition, we will present some initial data which yield solutions with accumulating breaking times. Joined work with H. Holden and X. Raynaud.

#### Poul G. Hjorth

##### The Numbers Lead a Dance

**Abstract:**
I will discuss some examples from the interface between
mathematics and syntax based poetry. In particular I will examine a certain
13th century lyrical style, the generalization of which gives rise
to some number theoretical questions. In the search for answers we find
ourselves in 20th century mathematics of iterative maps and chaotic dynamics. Joint with A.R.Champneys, U.of Bristol, UK.

#### Henrik Kalisch

##### Singular Solutions for Systems of Conservation Laws

**(no abstract)**

#### Werner Kirsch

##### On the semicircle law for random matrices

**Abstract:**
I will review some recent results on Wigner's semicircle law with emphasis on band random matrices and matrices with ependent entries.

#### Knut-Andreas Lie

##### Challenges and developments in simulation of hydrocarbon recovery and CO2 storage

**Abstract:**
Computer simulation of multiphase flow in porous rock formations has been conducted since the early 1950s. In this talk, I will review some of the mathematical characteristics of the underlying mathematical models and try to explain some of the numerical challenges that still remain. I will also outline a few ideas that have been used lately to make simulation technology more accurate, efficient, and versatile.

#### Peter Lindqvist

##### From Local to Global Estimates via BMO

**Abstract:**
Functions of Bounded Mean Oscillation were introduced by F. John and L. Nirenberg in 1961. They are useful auxiliary tools for the study of elliptic and parabolic partial equations; for example the celebrated Moser iteration depends on this concept. I will focus on a less known property of BMO functions: in a wide class of domains the local and global BMO norms are comparable. In this way some interior estimates for the (super)solutions of PDE's can be verified up to the boundary.

The talk starts almost from first principles!

#### Eugenia Malinnikova

##### Complex Jacobi matrices and sharp uniqueness results for discrete evolutions

**Abstract:**
We consider the (semi-)discrete heat and Schroninger equations, or more generally the equation \(u_t=\alpha(\Delta_du+Vu)\), where \(u:\mathbb{Z}^d\times\mathbb{R}\to\mathbb{C}\), \(\Delta_d\) is the discrete Laplace operator ,and \(\alpha\) is a fixed complex number. Assuming that the potential \(V:\mathbb{Z}^d\to\mathbb{C}\) is a bounded function, we apply the Shohat–Favard theorem for the corresponding Jacobi matrix, to show that a solution \(u(t,x)\) to such equation can not decay fast at two distinct times. The result is sharp. This is a joint work with Yu. Lyubarskii

#### Siddhartha Mishra

##### Systems of conservation laws in the Geosciences

Traditionally, many developments, particularly in the design of efficient numerical methods, for systems of conservation laws, were motivated by problems in aerodynamics and astrodynamics. In the last decade or so, the situation has been gradually changing and a large number of interesting examples and issues have been motivated by the geosciences. We consider two such models from the geosciences and the issues they raise. The first model concerns simulations of tsunamis, generated by rockslides or earthquakes and the quantification of uncertainty in the wave heights due to uncertainties in the model parameters. The second model arises in the numerical simulation of stratocumulus clouds, which are essential in the prediction of global climate dynamics. We consider the vexing issue of interaction between efficient numerical methods and turbulence models such as large eddy simulations for this problem. For both sets of problems, we present novel solutions based on state of the art numerical methods.

#### Xavier Raynaud

##### Variational approach of the Camassa-Holm system

**Abstract:**
The Camassa–Holm equation,
\[\begin{aligned}
m_t + (um)_x + u_xm &= 0,\\
m& = u - u_{xx},
\end{aligned}\]
is a geodesic equation for the \(H^1\) norm. We start by describing briefly how the Camassa–Holm
system,
\[\begin{aligned}
m_t + (um)_x + u_xm + \varepsilon \rho\rho_x&= 0,\\
m& = u - u_{xx},
\end{aligned}\]
can be derived from a variational principle, using a Lagrangian functional composed of the kinetic
energy (we will discuss the relevance of this name), \(E_{\text{kin}}=\frac12\int_{\mathbb
R}(u^2+u_x^2)\,dx\), and the potential energy, \(E_{\text{opt}} =\frac{\varepsilon}2\int_{\mathbb
R}(\rho - 1)^2\,dx\). The solution of the Camassa–Holm equation may blow-up. It happens when a
family of characteristics converge to a unique point and in this case, the energy density
\(u^2+u_x^2\,dx\) converges to a measure \(\mu\) with a singular part \(\mu_s\). This particular
blow-up configuration enables the prolongation of the solutions in a conservative way. The elastic
force, which is introduced through the potential energy, acts as a repulsive force and prevents the
characteristics to merge. The solutions remains regular and we show that the singular part \(\mu_s\)
corresponds to the limit of the potential energy density, that is
\[\varepsilon(\rho^\varepsilon - 1)^2 \,dx\overset{\ast}{\rightharpoonup}\mu_s,\]
as \(\varepsilon\) tends to zero.

#### Wen Shen

##### On some models for reservoir simulation

**Abstract:**
We present two recents works in reservoir simulation, one on two-phase flow with polymer flooding, and the other on gas flooding with three components. Both models share a common feature such that, in a suitably defined Langragian coordinates, the equation of thermo-dynamics is decoupled from that of hydro-dynamics. The solutions of the resulted triangle system is closely related to a scalar conservation with discontinuous flux where nonlinear resonance is present. Under general assumptions, we construct global solutions for Riemann problems and front tracking approximation. For the polymer flooding model, a suitable wave strength functional is introduced to control the total wave strength, which leads tot he convergence of the front tracking approximations.

#### Barry Simon

##### Large Deviations and a Conjecture of Lukic

**Abstract:**
I will begin by discussing the role of sum rules in the spectral theory of orthogonal polynomials and the conjectures of Simon and Lukic for higher order sum rules in OPUC. Then I will discuss the new approach of Gamboa, Nagel and Roualt obtaining sum rules form the theory of large deviations. Finally, I will describe some recent joint work with Breuer and Zeitouni on using sum rules to partially prove a new case of Lukic's conjecture.

#### Eitan Tadmor

##### Hydrodynamic flocking: alignment with non-local means and critical thresholds

**Abstract:**
Starting with agent-based descriptions, different models of large crowds flocking dynamics lead to hydrodynamic descriptions which involve alignment around non-local means. The global behavior of the resulting flocking hydrodynamics is dictated by the balance between nonlinear convection and these alignment forcing. In particular, there is a sharp dichotomy between the existences of global smooth solutions vs. finite-time blowup of strong solutions. This dichotomy is quantified in terms of critical thresholds (CT) in the space of initial configurations.

We demonstrate this CT phenomena with several n-dimensional prototype models. These include prolonged life-span of sub-critical 2D shallow-water solutions, 3D restricted Euler and Euler–Poisson equations, and the hydrodynamic descriptions of flocking hydrodynamics.

#### Gerald Teschl

##### Jacobi Polynomials, Bernstein-type Inequalities and Dispersion Estimates for the Discrete Laguerre Operator

**Abstract:**
This talk is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the Jacobi operator associated with generalized Laguerre polynomials. This operator features prominently in the recent study of nonlinear waves in (2 + 1)-dimensional noncommutative scalar field theory. We use known and new uniform estimates for Jacobi polynomials to establish some new dispersive estimates.

#### Ragnar Winther

##### The bubble transform: A new tool for analysis of finite element methods

**Abstract:**
We will discuss the construction of a linear operator,
referred to as the bubble transform, which maps scalar
functions defined on an \(n\)-dimensional domain \(\Omega\) into a collection of
functions with local support. In fact, for a given simplicial
triangulation \(\mathcal T\) of \(\Omega\), the associated bubble transform
\({\mathcal B}_{\mathcal T}\) produces a decomposition of functions on \(\Omega\) into a sum
of functions with support on the corresponding macroelements. The
transform is bounded in both \(L^2\) and the Sobolev space \(H^1\), it is
local,
and it preserves the
corresponding continuous piecewise polynomial spaces of all degrees. As a
consequence, this transform is a useful tool for analysing finite element methods.
Important applications are the construction of local
projection operators which are uniformly bounded in the polynomial degree, and the
study of condition numbers of local bases and frames.

#### Bernt Øksendal

##### Optimal insider control of stochastic partial differential equations

**Abstract:**
We study the problem of optimal inside control of an SPDE (a stochastic evolution equation) driven by a Brownian motion and a Poisson random measure. Our optimal control problem is new in two ways:

- The controller has access to inside information, i.e. access to information about a future state of the system,
- The integro-differential operator of the SPDE might depend on the control.

In the first part of the paper, we formulate a sufficient and a necessary maximum principle for this type of control problem, in two cases:

- When the control is allowed to depend both on time \(t\) and on the space variable \(x\).
- When the control is not allowed to depend on \(x\).

In the second part of the paper, we apply the results above to the problem of optimal control of an SDE system when the inside controller has only noisy observations of the state of the system. Using results from nonlinear filtering, we transform this noisy observation SDE inside control problem into a \emph{full observation} SPDE insider control problem.

The results are illustrated by explicit examples.