Titles and abstracts

John Ball

Heriot-Watt University and Maxwell Institute for Mathematical Sciences, Edinburgh

Nonlinear elasticity and computer vision

A nonlinear elasticity model for comparing images is formulated and analyzed, in which optimal transformations between images are sought as minimizers of an integral functional. The existence of minimizers in a suitable class of homeomorphisms between image domains is established under natural hypotheses, and the question of whether for affinely related images the minimization algorithm delivers the affine transformation as the unique minimizer is discussed. This is joint work with Chris Horner.

Yann Brenier

CNRS, LMO-Orsay, Universite Paris-Saclay

Solving initial value problems by space-time convex optimisation

I will discuss a possible strategy to solve the Cauchy problem for nonlinear evolution PDEs by space-time convex optimisation based on their weak formulation. One of the simplest example is the quadratic porous medium equation for which the Aronson–Benilan inequality is sharply used to prove that the strategy works for arbitrarily long time intervals. A similar result holds true for the Burgers equation. For the more challenging Euler equations, the concept of subsolution (in the sense of convex integration theory) plays a crucial role. Finally, I will mention how the Einstein equations in vacuum can be considered in that framework.

Boris Buffoni

EPFL

Steady three-dimensional rotational flows and locally coercive problems

Stationary flows of an inviscid, incompressible fluid of constant density are considered in the region \(D = (0, L) \times \mathbb R^2\), with periodic velocity in the second and third variables. Instead of the Nash–Moser iteration scheme, as in a previous proof, Kato’s approach for locally coercive problems is applied to this setting, allowing a more precise statement. This is a joint work with E. Séré (Paris-Dauphine).

Gui-Qiang G. Chen

University of Oxford

Multidimensional Riemann Problems − Transonic Shock Waves and Free Boundary Problems

We are concerned with global solutions of multidimensional Riemann problems for nonlinear hyperbolic systems of conservation laws, with emphasis on their global patterns and structures. We present recent progress on the rigorous analysis of several longstanding two-dimensional Riemann problems—both initial and lateral—involving transonic shock waves for the Euler equations for potential flow. In particular, we focus on the four-shock Riemann problem for the Euler equations for potential flow, as a representative example, to illustrate how these problems can be reformulated and solved as free boundary problems, with transonic shock waves serving as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic–hyperbolic type. Several physically significant lateral Riemann problems are also addressed, including Prandtl’s reflection problem, von Neumann’s shock reflection–diffraction problem, and Lighthill’s diffraction problem. Moreover, we present different regularity properties of Riemann solutions for the compressible Euler equations for both potential flow and isentropic flow. Some further multidimensional Riemann problems and related shock wave problems for nonlinear partial differential equations are also presented.

Elaine Crooks

Swansea

Self-similar fast-reaction limits of reaction-diffusion systems with nonlinear diffusion

This talk is concerned with the characterisation of fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction-diffusion equations or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains. The ideas used extend previous results in the linear diffusion case and show that in the fast-reaction limit, spatial segregation leads to the two components of the original systems each converging to the positive and negative parts of a self-similar limit profile that satisfies one of four ordinary-differential systems. The position of the free boundary separating where such self-similar profiles are positive from where they are negative provides information on the rate of penetration of one substance into the other and for specific forms of nonlinear diffusion, some results will be presented on the relationship between the form of the nonlinear diffusion and the position of this free boundary. This is joint work with Yini Du.

Mark Groves

FR Mathematik, Universität des Saarlandes

Solitary wave solutions to the full dispersion Kadomtsev–Petviashvili equation

The KP-I equation

(1)\[u_t + m(D)u_x − 2uu_x = 0,\]

where \(m(D)\) is the Fourier multiplier operator with multiplier

\[m(k) = 1 + \frac{k_2}{2k_1^2} + \frac 12 (β - \tfrac 13) k_1^2,\]

arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number \(β > \frac 13\)). It has recently been shown by Liu, Wei and Yang that (1) has an infinite family \(\{ζ_k\}\) of symmetric ‘lump’ solutions of the form

\[ζ_k(x, y) = −2∂_x \log τ_k(x, y),\qquad k = 1, 2, …,\]

where \(τ_k\) is a polynomial of degree \(k(k + 1)\) given by an explicit formula. They also show that \(ζ_1\) and \(ζ_2\) are nongenerate and announce that the same is true for \(ζ_k\), \(k ≥ 3\) (details to be published in a later paper).

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

\[\tilde m(k) = \left( (1 + β|k|^2)\frac{\tanh |k|}{|k|}\right)^{1/2} \left( 1 + \frac{k_2^2}{k_1^2} \right).\]

In this talk I show that the FDKP-I equation also has a family of symmetric fully localised solitary waves which are obtained by casting it as a perturbation of the KP-I equation and applying a suitable variant of the implicit-function theorem.

This project is joint work with Mats Ehrnström (NTNU, Norway).

Mahir Hadžić

UCL

On quantitative gravitational relaxation

We obtain quantitative decay rates for the linearised gravitational potential around compactly supported steady states of the Vlasov–Poisson system featuring a point mass potential at the origin. Such steady states feature stably trapped particles which present a severe obstacle to any kind of dispersion. The problem is further complicated by the presence of an infinite-dimensional kernel. To handle these issues we combine tools from dynamical systems, Hamiltonian geometry, and scattering theory. Our theorem can be viewed as a first quantitative proof of (linear) gravitational Landau damping. Joint work with Matthew Schrecker.

Mariana Haragus

FEMTO-ST Institute, UMLP, Besançon, France

Uniform subharmonic dynamics of spectrally stable periodic waves

Motivated by a concrete problem from nonlinear optics, we study the nonlinear subharmonic stability of spectrally stable periodic stationary solutions of the Lugiato-Lefever equation, a damped nonlinear Schrödinger equation with forcing, detuning and driving. Subharmonic perturbations of a T-periodic wave are periodic perturbations with periods equal to an integer multiple \(NT\) of the period T of the wave. Standard stability results show that for each integer \(N\) such a periodic wave is asymptotically stable against \(NT\)-periodic perturbations, in the sense that initially nearby data will converge at an exponential rate to a (small) spatial translation of the periodic wave. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential rates of decay depend on \(N\) and, in fact, tend to zero as \(N\) tends to infinity, leading to a lack of uniformity in the period of the perturbation. Relying upon a subharmonic Bloch decomposition and recent results on nonlinear stability for localized perturbations, we show how to obtain a nonlinear, subharmonic stability result for periodic stationary solutions of the Lugiato-Lefever equation that is uniform in \(N\).

This talk is based on joint works with Mat Johnson, Wesley Perkins and Björn de Rijk.

Paul Milewski

Penn State University

Resonance of surface water waves in cylindrical containers

Nonlinear waves sloshing in a container of rectangular cross-section can behave very differently than those with different cross sections. Nonlinear resonance is a mechanism by which energy is continuously exchanged between a small number of wave modes and is common to many nonlinear dispersive wave systems. In the context of free-surface gravity waves such as ocean surface waves, nonlinear resonances have been studied extensively over the past 60-years, almost always on domains that are large (or infinite) compared to the characteristic wavelength. In this case, the dispersion relation dictates that only quartic (4-wave) resonances can occur. In contrast, nonlinear resonances in confined three-dimensional geometries have received relatively little attention, where, perhaps surprisingly, stronger 3-wave resonances can occur. We will present the results characterizing the configuration and dynamics of resonant triads in cylindrical basins of arbitrary cross sections, demonstrating that these triads are ubiquitous, with (the commonly studied) rectangular cross section being an exception where they do not occur.

Eugene Shargorodsky

King’s College London

Estimate for the Morse index of a Stokes wave

Stokes waves are steady periodic water waves on the free surface of an infinitely deep irrotational two-dimensional flow under gravity without surface tension. They can be described in terms of critical points of a certain functional; this allows one to define the Morse index of a Stokes wave. I will discuss some known and new estimate for the Morse index of a Stokes wave.

Walter Strauss

Brown University

Hyperbolic-Elliptic Systems Arising in Ionization and Galaxy Models

In both models we study steady state solutions by means of bifurcation theory. (1) The galaxy is rotating and is modeled by the Euler equation with variable entropy. A curve of slowly rotating solutions is obtained in spite of a loss of derivatives. This is joint work with Yilun (Allen) Wu and Juhi Jang. (2) A strong voltage in a gas can produce a sudden electrical discharge, as in lightning or a fluorescent lamp. There can also be secondary emissions from the ions that hit a cathode. We use global bifurcation to prove there is a large set of steady solutions. This is joint work with Masahiro Suzuki.

Edriss S. Titi

University of Cambridge, Texas A&M University, and The Weizmann Institute of Science

On a generalization of the Bardos-Tartar conjecture to nonlinear dissipative PDEs

In this talk I will show that every solution of a KdV–Burgers–Sivashinsky type equation blows up in the energy space, backward in time, provided the solution does not belong to the global attractor. This is a phenomenon contrast to the backward behavior of the 2D Navier–Stokes equations, subject to periodic boundary condition, studied by Constantin, Foias, Kukavica and Majda, but analogous to the backward behavior of the Kuramoto–Sivashinsky equation discovered by Kukavica and Malcok. I will also discuss the backward behavior of solutions to the damped driven nonlinear Schrödinger equation, the complex Ginzburg-Landau equation, and the hyperviscous Navier–Stokes equations. In addition, I will provide some physical interpretation of various backward behaviors of several perturbations of the KdV equation by studying explicit cnoidal wave solutions. Furthermore, I will discuss the connection between the backward behavior and the energy spectra of the solutions. The study of backward behavior of dissipative evolution equations is motivated by a conjecture of Bardos and Tartar which states that the solution operator of the two-dimensional Navier–Stokes equations maps the phase space into a dense subset in this space.

Juncheng Wei

City University Hong Kong

From KP-I lump solutions to travelling wave of 3D gravity capillary water wave problem

In this talk, I will study the three-dimensional gravity-capillary water wave problem involving an irrotational, perfect fluid under gravity and surface tension. We focus on steady waves propagating uniformly in one direction. Assuming constant wave speed and water depth, we analyze the fluid’s velocity potential and boundary conditions. Using the Kadomtsev–Petviashvili (KP)-I equation as a simplified model, we show that, within a specific parameter range, the problem admits many fully localized solitary-wave solution, consistent with KP-I predictions.