Autumn 2019

Time/place: Tuesday 03rd of December, 14:15–15:00, room 734
Speaker: Hung Le (NTNU) Title: On the existence and instability of solitary waves with a finite dipole
Abstract: In this talk, we consider the existence and stability properties of two-dimensional solitary waves traversing an infinitely deep body of water. We assume that above the water is air and that the waves are acted upon by gravity with surface tension effects on the air-water interface. In particular, we study the case where there is a finite dipole in the bulk of the fluid, that is, the vorticity is a sum of two weighted \(\delta\)-functions. Using an implicit function theorem argument, we construct a family of solitary waves solutions for this system that is exhaustive in a neighborhood of 0. Our main result is that this family is conditionally orbitally unstable. This is proved using a modification of the Grillakis—Shatah–Strauss method recently introduced by Varholm, Wahlén, and Walsh.

Time/place: Tuesday 03rd of December, 14:15–15:00, room 734

Speaker: Hung Le (NTNU)
Title: On the existence and instability of solitary waves with a finite dipole

Abstract: In this talk, we consider the existence and stability properties of two-dimensional solitary waves traversing an infinitely deep body of water. We assume that above the water is air and that the waves are acted upon by gravity with surface tension effects on the air-water interface. In particular, we study the case where there is a finite dipole in the bulk of the fluid, that is, the vorticity is a sum of two weighted \(\delta\)-functions. Using an implicit function theorem argument, we construct a family of solitary waves solutions for this system that is exhaustive in a neighborhood of 0. Our main result is that this family is conditionally orbitally unstable. This is proved using a modification of the Grillakis—Shatah–Strauss method recently introduced by Varholm, Wahlén, and Walsh.

Time/place: Thursday 28th of November, 10:15–11:00, room 656
Speaker: Balázs Kovács (Tübingen) Title: A convergent algorithm for mean curvature flow with and without forcing
Abstract: We will sketch a proof of convergence for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The proposed and studied numerical method combines evolving surface finite elements, whose nodes determine the discrete surface like in Dziuk's algorithm proposed in 1990, and linearly implicit backward difference formulae for time integration. The proposed method differs from Dziuk's approach in that it discretizes Huisken's evolution equations (from [Huisken (1984)]) for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space. This numerical method admits a convergence analysis, which combines stability estimates and consistency estimates to yield optimal-order \(H^1\)-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix–vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. We will also present various numerical experiments to illustrate and complement the theoretical results. Furthermore, we will give an outlook towards forced mean curvature flow, that is for problems coupling mean curvature flow with a surface PDE. The talk is based on joint work with B. Li (Hong Kong) and Ch. Lubich (Tübingen).

Time/place: Thursday 28th of November, 10:15–11:00, room 656

Speaker: Balázs Kovács (Tübingen)
Title: A convergent algorithm for mean curvature flow with and without forcing

Abstract: We will sketch a proof of convergence for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The proposed and studied numerical method combines evolving surface finite elements, whose nodes determine the discrete surface like in Dziuk's algorithm proposed in 1990, and linearly implicit backward difference formulae for time integration. The proposed method differs from Dziuk's approach in that it discretizes Huisken's evolution equations (from [Huisken (1984)]) for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space. This numerical method admits a convergence analysis, which combines stability estimates and consistency estimates to yield optimal-order \(H^1\)-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix–vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. We will also present various numerical experiments to illustrate and complement the theoretical results. Furthermore, we will give an outlook towards forced mean curvature flow, that is for problems coupling mean curvature flow with a surface PDE.
The talk is based on joint work with B. Li (Hong Kong) and Ch. Lubich (Tübingen).

Time/place: Tuesday 8th of October, 14:15–15:00, room 734
Speaker: Gunnar Taraldsen (NTNU) Title: The spectrum of a random operator is a random set
Abstract: The theory of random sets is demonstrated to prove useful for the theory of random operators. A random operator is here defined by requiring the graph to be a random set. It is proved that the spectrum and the set of eigenvalues of random operators are random sets. These results seem to be a novelty even in the case of random bounded operators. The main technical tools are given by the measurable selection theorem, the measurable projection theorem, and a characterisation of the spectrum by approximate eigenvalues of the operator and the adjoint operator. A discussion of some of the existing definitions of the concept of a random operator is included at the end of the paper.

Time/place: Tuesday 8th of October, 14:15–15:00, room 734

Speaker: Gunnar Taraldsen (NTNU)
Title: The spectrum of a random operator is a random set

Abstract: The theory of random sets is demonstrated to prove useful for the theory of random operators. A random operator is here defined by requiring the graph to be a random set. It is proved that the spectrum and the set of eigenvalues of random operators are random sets. These results seem to be a novelty even in the case of random bounded operators. The main technical tools are given by the measurable selection theorem, the measurable projection theorem, and a characterisation of the spectrum by approximate eigenvalues of the operator and the adjoint operator. A discussion of some of the existing definitions of the concept of a random operator is included at the end of the paper.

Time/place: Tuesday 1st of October, 14:15–15:00, room 734
Speaker: Mark Groves (Saarland University) Title: Solitary-wave solutions to the full dispersion Kadomtsev-Petviashvili equation
Abstract: The KP-I equation \[u_t + m(D) u_x - 2uu_x = 0,\] where \(m(D)\) is the Fourier multiplier operator with multiplier \[m(k)=1+\frac{k_2^2}{2k_1^2} + \frac{1}{2}(\beta-{\textstyle\frac{1}{3}})k_1^2,\] arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number \(\beta>\frac{1}{3}\)). This equation admits – as an explicit solution – a "fully localised" or "lump" solitary wave which decays to zero in all spatial directions. Recently there has been interest in the full dispersion KP-I equation \[u_t + \tilde{m}(D) u_x - 2uu_x = 0\] obtained by retaining the exact dispersion relation from the water-wave problem, that is, replacing \(m\) by \[\tilde{m}(k)=\left((1+\beta\|k\|^2)\frac{\tanh \|k\|}{\|k\|}\right)^{1/2}\left(1+\frac{k_2^2}{k_1^2}\right).\] In this talk I show that the full dispersion KP-I equation also has a fully localised solitary-wave solution. The existence theory is variational and perturbative in nature. This project is joint work with Mats Ehrnström (NTNU, Norway).

Time/place: Tuesday 1st of October, 14:15–15:00, room 734

Speaker: Mark Groves (Saarland University)
Title: Solitary-wave solutions to the full dispersion Kadomtsev-Petviashvili equation

Abstract: The KP-I equation \[u_t + m(D) u_x - 2uu_x = 0,\] where \(m(D)\) is the Fourier multiplier operator with multiplier \[m(k)=1+\frac{k_2^2}{2k_1^2} + \frac{1}{2}(\beta-{\textstyle\frac{1}{3}})k_1^2,\] arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number \(\beta>\frac{1}{3}\)). This equation admits – as an explicit solution – a "fully localised" or "lump" solitary wave which decays to zero in all spatial directions.
Recently there has been interest in the full dispersion KP-I equation \[u_t + \tilde{m}(D) u_x - 2uu_x = 0\] obtained by retaining the exact dispersion relation from the water-wave problem, that is, replacing \(m\) by \[\tilde{m}(k)=\left((1+\beta|k|^2)\frac{\tanh |k|}{|k|}\right)^{1/2}\left(1+\frac{k_2^2}{k_1^2}\right).\] In this talk I show that the full dispersion KP-I equation also has a fully localised solitary-wave solution. The existence theory is variational and perturbative in nature.
This project is joint work with Mats Ehrnström (NTNU, Norway).

Time/place: Thursday 26th of September, 14:15–15:00, room F3 (gamle fysikk)
Speaker: David Ambrose (Drexel University, Philadelphia) Title: Existence Theory for a Mean Field Games Model of Household Wealth
Abstract: We study a nonlinear system of partial differential equations arising in macroeconomics which utilizes a mean field approximation. This equation together with the corresponding data, subject to two moment constraints, is a model for debt and wealth across a large number of similar households, and was introduced in a recent paper of Achdou, Burea, Lasry, Lions, and Moll. We introduce a relaxation of their problem, generalizing one of the moment constraints; any solution of the original model is a solution of this relaxed problem. We prove existence and uniqueness of strong solutions to the relaxed problem, under the assumption that the time horizon is small. Since these solutions are unique and since solutions of the original problem are also solutions of the relaxed problem, we conclude that if the original problem does have solutions, then such solutions must be the solutions we prove to exist. Furthermore, for some initial data and for sufficiently small time horizons, we are able to show that solutions of the relaxed problem are not solutions of the original problem. In this way we demonstrate nonexistence of solutions for the original problem in certain cases.

Time/place: Thursday 26th of September, 14:15–15:00, room F3 (gamle fysikk)

Speaker: David Ambrose (Drexel University, Philadelphia)
Title: Existence Theory for a Mean Field Games Model of Household Wealth

Abstract: We study a nonlinear system of partial differential equations arising in macroeconomics which utilizes a mean field approximation. This equation together with the corresponding data, subject to two moment constraints, is a model for debt and wealth across a large number of similar households, and was introduced in a recent paper of Achdou, Burea, Lasry, Lions, and Moll. We introduce a relaxation of their problem, generalizing one of the moment constraints; any solution of the original model is a solution of this relaxed problem. We prove existence and uniqueness of strong solutions to the relaxed problem, under the assumption that the time horizon is small. Since these solutions are unique and since solutions of the original problem are also solutions of the relaxed problem, we conclude that if the original problem does have solutions, then such solutions must be the solutions we prove to exist. Furthermore, for some initial data and for sufficiently small time horizons, we are able to show that solutions of the relaxed problem are not solutions of the original problem. In this way we demonstrate nonexistence of solutions for the original problem in certain cases.

Time/place: Thursday 26th of September, 15:15–16:00, room F3 (gamle fysikk)
Speaker: Miles Wheeler (University of Bath) Title: New exact solutions to the steady 2D Euler equations
Abstract: We present a large class of explicit "hybrid" equilibria for the 2D Euler equations, consisting of point vortices embedded in a smooth sea of "Stuart-type" vorticity. Mathematically, these are singular solutions of the elliptic Liouville equation satisfying some additional constraints at each singularity. This is joint work with Vikas Krishnamurthy, Darren Crowdy, and Adrian Constantin.

Time/place: Thursday 26th of September, 15:15–16:00, room F3 (gamle fysikk)

Speaker: Miles Wheeler (University of Bath)
Title: New exact solutions to the steady 2D Euler equations

Abstract: We present a large class of explicit "hybrid" equilibria for the 2D Euler equations, consisting of point vortices embedded in a smooth sea of "Stuart-type" vorticity. Mathematically, these are singular solutions of the elliptic Liouville equation satisfying some additional constraints at each singularity.
This is joint work with Vikas Krishnamurthy, Darren Crowdy, and Adrian Constantin.