# Master projects for Mats Ehrnstrom

I am a mathematician specialising in partial differential equations using analytic methods , but can also offer projects in numerical applications of the same, harmonic analyis , and data-driven PDEs , as well as collaborative projects with other researchers. My main strength is in analysis as applied to nonlocal and frequency models, and I have a research group in water waves and dispersive equations. Most of my students have gone on to pursue a PhD degree. If your basic interest is in industry, there are probably better supervisors, but if your are interested in analysis, programming or numerics, modelling or data-driven methods, chances are you could find a suitable project. Browse some of the earlier master projects here.

If you are interested in writing a master's thesis, don't hesitate to send an Email to mats [dot] ehrnstrom [at] ntnu [dot] no. Master students get meetings one hour weekly, and each student is given a secondary contact in the group, that can help with smaller details and discussions outside the ordinary supervision. Students are expected to take responsibility for the advancement of the project, and to be fairly independent.

## Water waves and dispersive equations

Surface water waves travelling in one direction may in general be described via a class of equations of the form

$u_t + L u_x + n(u)_x = 0,$

where $u(t,x)$ describes the horizontal velocity, $L$ is a linear and symmetric differential or pseudo-differential operator like $\partial_x^2$, and $n(u)$ is a nonlinear term like $u^2$. By changing the operator $L$, one can describe different forms of dispersion — the speed at which small wave components of different Fourier frequencies spread out. The nonlinearity $n$, on the other hand, accounts for concentration, wave-breaking and singularity formation in the wave. This gives a rich picture of different behaviours, see the image below.

Travelling waves are special solutions of the form $u(t,x) = \varphi(x - ct)$, where $c$ is the speed of the wave; they are propagating with a fixed shape and speed, therefore also called steady solutions, as they appear such in a reference frame moving with the same speed as the wave. Steady waves are prototypical of water waves, as it is expected that over long times time-dependent solutions will split up into different packages of nearly steady waves. The study of steady waves include the existence of solitary and periodic solutions , spatial symmetries, and regularity of highest waves. The study of time-dependent solutions, on the other hand, include well-posedness, stability, wave-breaking and long-term behaviour.

In water waves I will have the following themes to start from, each of them adjustable.

Solitary solutions via optimisation

Solitary solutions via optimisation

Equations as those mentioned above have a Hamiltonian energy of the form $\int ( u L u + N(u) ) dx,$ where $N$ is an anti-derivative of $n$, and $\int u^2 dx$ is a further conserved quantity for solutions. By minimising the Hamiltonian in an open ball in a Sobolev space, under the constraint that the second conserved quantity, which is the square of the $L^2$-norm, is fixed, one finds critical points that are in fact solutions to the equation. In variants to this, one fixes a different part of the energy, and instead maximises an energy. In both cases, the resulting waves have their main mass localised in space, so-called solitary waves.

Prerequisites: Basic knowledge of at least two of L^p-spaces, Sobolev spaces, functional analysis.

• Arnesen: Existence of solitary-wave solutions to nonlocal equations. Discrete Contin. Dyn. Syst. 36 (2016), 3483–3510.
• Hildrum: Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity. Nonlinearity 33 (2020), 1594–1624.
• Arnesen, Ehrnstrom, Stefanov: A maximisation technique for solitary waves: the case of the nonlocally dispersive Whitham equation. arXiv:2303.14036

Periodic solutions via bifurcation theory

Periodic solutions via bifurcation theory

Steady periodic waves in water-wave equations are most commonly constructed by perturbing the linear problem. This is local bifurcation: starting from a parameterised family of trivial solutions, such as running water with speed as the parameter, one can find certain values of the parameter where the implicit function theorem breaks down, and a different curve of solutions crosses the trivial one, yielding small-amplitude sinusoidal solutions. Continuing such a curve to larger solutions is called global bifurcation. In this project one analyzes small and larger periodic solution of water wave equations, using functional analysis and elliptic estimates.

Prerequisites: Fourier series; Functional analysis, alternatively god understanding of Linear methods; ODEs or PDEs.

• Ehrnstrom, Kalisch. Global bifurcation for the Whitham equation. Math. Model. Nat. Phenom. 8 (2013), 13–30.
• Aasen, Varholm. Traveling gravity water waves with critical layers. J. Math. Fluid Mech. 20 (2018), 161–187.
• Hildrum, Xue. Periodic Hölder waves in a class of negative-order dispersive equations. J. Differential Equations 343 (2023), 752–789.

Highest waves

Highest waves

The study of highest waves is about the existence, regularity and precise properties of solutions to water wave equations attaining a largest possible height. Not all equations possess such solutions: in water waves, it is primarily the gravitational force that enforces such a maximal height, and speed, of surface waves. The mathematical objective is mainly a regularity study, using ad hoc and very precise estimates to determine the singularity arising at the top of the wave, the possible angle or cusple, and further properties such as convexity between consequtive crests.

Prerequisites: Fourier series; Functional analysis, alternatively god understanding of Linear methods; handiness in analytic estimates. May be combined with, or reshaped as, a numerical project.

• Ehrnstrom, Wahlén: On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation. Ann. Inst. H. Poincaré C Anal. Non Linéaire 36 (2019), 1603–1637.
• Ehrnstrom, Mæhlen, Varholm: On the precise cusped behaviour of extreme solutions to Whitham-type equations. arXiv:2302.08856. In print in Ann. Inst. H. Poincaré C Anal. Non Linéaire.
• Ørke: Highest waves for fractional Korteweg–De Vries and Degasperis–Procesi equations. arXiv:2201.13159. In print in Arkiv för MAtematik.

Wave-breaking

Wave-breaking

Dispersive PDEs model surfaces and media that are such that a wave’s velocity depends on its length. Water surfaces are one such medium, and there are numerous dispersive PDEs modelling water waves. A characteristic feature of many water wave models is that a wave can break (as water waves often do near the shore). Mathematically, we say that a function $u$, depending on time $t$ and space $x$, breaks at time $T$ if $u_x$ becomes unbounded as t approaches T while u remains bounded.

Weakly dispersive PDEs take the form $\partial_t u + \partial_x u^2 = K*u$, where $K$ is a convolution kernel in space, which is odd, $K(x) = K(-x)$, and absolutely integrable. It is expected that all such equations feature wave breaking, but it has only been proved for the case where $K$ is monotone for positive $x$. This project generalizes the result to all $K$ by employing new methods. It is also possible to extend the result to a periodic setting for more general dispersive PDEs.

This project is in co-supervision with Ola Mæhlen, UiO.

Prerequisites: The mathematics rely on the method of characteristic, fixed-point iteration, and operator splitting.

• Mæhlen, Xue: One-sided Hölder regularity of global weak solutions of negative order dispersive equations. J. Differential Equations364(2023), 412–455.

Three-dimensional water waves

Three-dimensional water waves

Three-dimensional water waves are governed by the Euler equations with a two-dimensional free upper surface. Only few solutions are known, especially in the presence of an underlying current. In this project, one starts from such known solutions, and investigates the trajectories traversed by the fluid particles. The project is built in steps, where the student can choose to focus more on numerical programming, or more on theoretical solutions. The main question asked is whether one can find patterns in the particles paths, which are known in a simpler two-dimensional setting. An open problem is whether trajectories will cross each other, something which is not possible in two dimensions.

This project is in joint supervision with Douglas Svensson Seth, NTNU.

Prerequisites: Basic knowledge of ordinary differential equations/dynamical systems, Fourier series and programming is necessary. Knowledge of partial differential equations and functional analysis is beneficial. The project can be more mathematically or programming oriented, depending on the student's preferences.

• Lokharu, Seth, Wahlén: An existence theory for small-amplitude doubly periodic water waves with vorticity. Arch. Ration. Mech. Anal. 238 (2020), 607–637.
• Ehrnstrom, Villari: Linear water waves with vorticity: rotational features and particle paths. J. Differential Equations 244 (2008), 1888–1909.

## Modelling, applied problems, and data-driven partial differential equations

Dynamic wave-field analysis

Dynamic wave-field analysis

When water waves propagate over areas with varying currents or depth, the wave characteristics change. The aim of this project is to combine Fourier- and wavelet analysis with object tracking and feature extraction, and develop a method to estimate these changes from video observations of waves.

This project is in co-supervision with Adrian Kirkeby, Simula Research Laboratory.

Prerequisites: Knowledge of one or more of the following topics: partial differential equations, wave propagation, Fourier analysis, programming (Python, Julia, or Matlab). The project can be more mathematically or programming oriented, depending on the students preferences.

Statistical models for differential equations

Statistical models for differential equations

In co-supervision with Ingelin Steinsland.