Section 4 Seminar, Spring 2025

In this talk, I will present my recent work on boundary-preserving numerical schemes for scalar stochastic partial differential equations (SPDEs) whose solutions remain in some bounded subset of the real line. Models with solutions in bounded or half-bounded domains show up in the study of heat (temperature in degrees Kelvin should be positive) and of concentrations (concentrations should be in [0,1]). Classical numerical schemes are not guaranteed to remain in such domains and can hence be interpreted as non-physical. I will present a novel boundary-preserving numerical scheme to weakly, in the stochastic sense, approximate the solutions of such semilinear scalar SPDEs whose solutions remain in a bounded domain.

In this talk I will introduce a class of continuity equations with noisy perturbations on transport form. I will briefly introduce the theory of rough paths and show how this perspective can give meaning to the noisy part of the equation. In the linear setting, we shall see how to obtain well-posedness of the continuity equations under DiPerna-Lions regularity conditions on the drift. If time permits, we shall also see how these results can be extended to a non-linear drift and how this leads to so-called Yudovich theory of the Euler equation with rough perturbations when $d=2$.

I will show that there exist stochastic flows of diffeomorphisms generated by SDEs with drift in $L^q_t C^{0, \alpha}_x$ for any $q \in [2, \infty)$ and $\alpha \in (0, 1)$. This result is achieved using a Zvonkin-type transformation for the SDE. I will give an outline of the argument, focusing on the PDE aspect which deals with well-posedness and optimal regularity for a class of parabolic PDEs related to the transformation. In the end, applications of such stochastic flows will be discussed.

We develop the Riemann–Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schrödinger (NLS) equation. The method is based on the simultaneous spectral analysis of both equations of the Lax pair and the subsequent analysis of the so-called Global Relation coupling, in the spectral terms, the appropriate spectral transforms of the initial data and all boundary values. Building on this method, a finite-band solution to the NLS equation can be given in terms of the solution of an associated RH problem, the jump conditions for which are characterized by specifying the endpoints of the arcs defining the contour of the RH problem and the constants (so-called phases) involved in the jump matrices. Particularly, we solve the problem of retrieving the phases given the solution of the NLS equation evaluated at a fixed time.

The Landau-Lifschitz-Navier-Stokes equations are stochastic partial differential equations, which introduce a stochastic forcing term to describe the macroscopic fluctuations away from the deterministic Navier-Stokes equations. In dimension $d=3$, we consider the large deviations in a scaling regime where the noise intensity and correlation length go to zero simultaneously, with a coupled rate. The large deviations from the Navier-Stokes coincide with those found by Quastel and Yau for a lattice gas, linking this mathematical model to the form of the noise predicted on physical grounds by Landau-Lifshitz. The question of how closely the upper and lower bounds match is circumscribed by problems in the analysis of the deterministic, forced Navier-Stokes, including uniqueness, the equality case of the energy inequality, and the loss of regularity higher than that of the Leray class.

Derived 60 years ago as a model for internal waves, the Benjamin–Ono equation (BO) is the scalar, one-dimensional evolution equation given by $u_t​=(|D|u−u^2)_x$​. Here, $|D|$ is the nonlocal Fourier multiplier on $\mathbb{R}$ that multiplies $\hat{u}(\xi)$ by $|\xi|$. Although the equation's Lax-pair structure was discovered 40 years ago, it did not attract much attention until recently, when Gérard and Kappeler (2021) published a series of papers unveiling much of BO's integrable structure. Since then, many other authors have contributed to further the understanding of BO which, by now, is better understood than the KdV equation (the prototypical example of a completely integrable system).

The integrable structure of BO yields numerous useful identities and properties; most notably is the "explicit formula" for solutions. Armed with these, one can accurately answer many questions for BO that are typically very difficult for other equations. This includes questions on the long-term behavior of solutions, global well-posedness in low-regularity spaces, and the zero-dispersion limit.

In this talk, we will explore the notion of "complete integrability", what one can expect from such structure, and its consequences for BO. I will also, if there is time, discuss my current work on the zero-dispersion limit of BO for periodic data; in contrast to Lax and Levermore’s work on KdV, the situation is much simpler for BO, and allows for an explicit expression of the limit.

Solutions to the Hunter—Saxton (HS) equation $$ u_t(t, x) + uu_x(t, x) = \frac{1}{4} \left(\int_{-\infty}^x u_x^2(t, z){\rm d}z - \int_x^{\infty}u_x^2(t, z){\rm d}z\right)\!, $$ might develop singularities in finite time in the sense that the spatial derivative blows up pointwise, while the solution itself remains bounded and continuous --- a phenomenon known as wave breaking. Even classical solutions may therefore have a finite lifespan. Moreover, wave breaking induces energy concentrations on sets of zero measure. Hence, to continue solutions beyond breakdown, one has to choose how to manipulate this concentrated energy, leading to a continuum of possible weak solutions.

In this talk we discuss a numerical method which removes an $\alpha$-fraction of the concentrated energy at each wave breaking occurrence and thereby gives rise to $\alpha$-dissipative solutions, for $\alpha \in [0, 1]$. In short, the method is based on applying a novel projection operator to the initial data and subsequently evolving forward in time with a generalized method of characteristics. The projection operator is tailored to preserve some of the important underlying structure inherent to solutions of the HS equation and takes advantage of the fact that if the initial data is piecewise linear, then so is the corresponding $\alpha$-dissipative solution.

A brief outline of the convergence proof is given, along with a short discussion about a condition under which we can deduce a robust convergence rate.

This talk is based on joint work with Katrin Grunert (NTNU), Susanne Solem (NMBU), and Anders Nordli (UiT).

Whittle—Matérn Gaussian random fields are popular in spatial statistics. They can be interpreted as solutions to specific stochastic PDEs extending the original concept also to manifolds such as hypersurfaces. In this talk, I will discuss the simulation of such and more general random fields on hypersurfaces using surface finite element approximations. We close with an outlook towards time-dependent problems and the use of the methods for the generation of static and time-evolving stochastic manifolds.

In this talk I will discuss broadly the concept of iterated integral signatures from rough paths theory. I will survey some recent ideas on universal approximation with signatures and applications to stochastic analysis of functionals arising from (partial) differential equations and/or financial mathematics. If time permits, I will also discuss the ideas behind the recent extension of the path-signature to 2D surfaces (images), and some open challenges in this direction. The talk is aimed at an audience with a general mathematical background, and will focus on ideas and concepts. Some of the results will be based on various joint works with K. Ebrahimi-Fard, J. Diehl, S. Tindel, F. Benth, and F. Straum.

In this talk we consider approximations of scalar conservation laws by adding nonlocal diffusive operators. In particular, we consider solutions associated to fractional Laplacian and fractional Rosenau perturbations and show that, for any \(t > 0\), the mutual \(L^1\) distance of their profiles is negligible as compared to their common distance to the underlying inviscid entropy solution.

We provide explicit examples showing that our rates are optimal in the supercritical and critical cases, in one space dimension and for strictly convex fluxes. For subcritical equations, our rates are not optimal but they remain explicit.

Those results were obtained in collaboration with N. Alibaud, M. Dalery, and C. Donadello.