Nonlinear waves and interface problems

A workshop to take place June 26–28, 2012, at the Centre for Mathematical Sciences, Lund University.

## Programme

 Tuesday 26/6 Wednesday 27/6 Thursday 28/6 Opening 09.00–09.15 09.15–10.00 A. Himonas G. Zhang E. Varvaruca 10.00–10.30 10.30–11.15 M. Groves A. Castro V. Kozlov 11.15–12.00 A. Aleman T. Alazard S. Walsh 12.00–13.30 13.30–14.15 J.-M. Vanden-Broeck H. Kalisch N. G. Kuznetsov 14.15–15.00 E. I. Parau M. Haragus B. Buffoni 15.00–15.30 15.30–16.15 E. Lenzmann M. Ehrnström 16.15–17.00 G. Schneider E. Wahlén 17.00–18.00 18.00–

### Tuesday

#### 09.00-09.15

pdf Opening and information

#### 09.15-10.00

##### Alex Himonas, University of Notre Dame

pdf The Cauchy problem for the CH and DP equations

We shall consider the initial value problem for the Camassa-Holm (CH) and Degasperis-Procesi (DP) equations and discuss their well-posedness properties in Sobolev spaces $H^s$. For $s>3/2$ these equations are well-posed and their data-to-solution map is continuous but not uniformly continuous. When $s<3/2$, both CH and DP are ill-posed in Sobolev spaces $H^s$, and this will be the main focus of our presentation. This talk is based on work with Carlos Kenig, Gerard Misiolek, Curtis Holliman and Katelyn Grayshan.

Coffee break

#### 10.30-11.15

##### Mark Groves, Universität des Saarlandes

pdf On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type

We consider a class of pseudodifferential evolution equations of the form $u_t + ( n(u) + Lu )_x = 0$, in which L is a linear smoothing operator and n is at least quadratic near the origin; this class includes in particular the Whitham equation. A family of solitary-wave solutions is found using a constrained minimisation principle and concentration-compactness methods for noncoercive functionals. The solitary waves are approximated by (scalings of) the corresponding solutions to partial differential equations arising as weakly nonlinear approximations; in the case of the Whitham equation the approximation is the Korteweg-deVries equation. We also demonstrate that the family of solitary-wave solutions is conditionally energetically stable.

#### 11.15-12.00

##### Alexandru Aleman, Lund University

pdf Harmonic maps and ideal flows

Using harmonic maps we provide an approach towards obtaining explicit solutions to the incompressible two-dimensional Euler equations. More precisely, the problem of finding all solutions which in Lagrangian coordinates (describing a particle path) present a labeling by harmonic functions is reduced to solving an explicit nonlinear differential system in $\mathbb{C}^n$ with $n=3$ or $n=4$. While the general solution is not available in explicit form, the structural properties of the system permit us to identify several classes of explicit solutions.

Joint work with A. Constantin.

Lunch

#### 13.30-14.15

##### Jean-Marc Vanden-Broeck, University College London

pdf Nonlinear hydroelastic waves

Waves propagating under an ice sheet are considered. The ice sheet is modelled as an elastic membrane. The mathematical formulation leads to a set of nonlinear equations similar to the classical equations for gravity-capillary water waves. The main difference is that the curvature term in the dynamic boundary condition is replaced by a higher order derivative. Waves generated by a moving load, periodic waves and solitary waves are presented. Both two and three-dimensional configurations are considered. It is shown that there are hydroelastic waves which do not have an equivalent in the gravity-capillary problem.

#### 14.15-15.00

##### Emilian I. Parau, University of East Anglia

pdf Numerical computations of solitary waves in a two-layer ﬂuid

A conﬁguration consisting of two superposed ﬂuids is considered. We compute numerically waves which propagate on the interface between two ﬂuids, with a lighter ﬂuid lying above a heavier one. The lower ﬂuid is bounded below by a rigid bottom and the upper ﬂuid layer is bounded above by a free surface. The effect of gravity and surface tension is considered. The two dimensional fully nonlinear problem is formulated as an integro-diﬀerential equations system by using the Cauchy integral formula in each layer and the dynamic boundary conditions.

Both in-phase and out-of-phase solitary waves are presented. Two types of solitary waves are found: solitary waves which bifurcate from the uniform ﬂow and decay monotonically, as KdV-like solutions and solitary waves bifurcating from the minimum of the phase velocity and have damped oscillations, as NLS-type solutions.

Forced waves generated by moving pressures will also be presented. Three-dimensional waves will be discussed.

Coffee break

#### 15.30-16.15

##### Enno Lenzmann, Universität Basel

Uniqueness of nonlinear ground states for the fractional Laplacian

In this talk, we will discuss uniqueness and non-degeneracy of ground state solutions to nonlinear problems involving the fractional Laplacian (e.g., solitary waves for the generalized Benjamin-Ono equation). The proof will depend on two main ingredients: 1.) sharp nodal bounds for eigenfunctions of fractional Schrödinger operators , and 2.) a nonlinear flow argument. Time permitting, we will also discuss some applications and recent uniqueness results in higher space dimensions.

This is joint work with Rupert Frank (Princeton).

#### 16.15-17.00

##### Guido Schneider, Universität Stuttgart

pdf Validity and non-validity of the NLS approximation

We consider a number of examples where the NLS equation can be derived as an amplitude equation. We discuss the question: When makes the NLS approximation correct predictions and when not. Especially we consider the NLS approximation for the water wave problem.

### Wednesday

#### 09.15-10.00

##### Guanghui Zhang, Heinrich-Heine-Universität Düsseldorf

pdf Regularity of two dimensional steady capillary gravity water waves

We consider the two-dimensional steady capillary water waves with vorticity. In the case of zero surface tension, it is well known that the free surface of a wave of maximal amplitude is not smooth at a free surface point of maximal height, but forms a sharp crest with an angle of 120 degrees. When the surface tension is not zero, physical intuition suggests that the corner singularities should disappear. In this talk we prove that for suitable weak solutions, the free surfaces are smooth. On a technical level, solutions of our problem are closely related to critical points of the Mumford-Shah functional, so that our main task is to exclude cusps pointing into the water phase. This is a joint work with Georg Weiss.

Coffee break

#### 10.30-11.15

##### Ángel Castro, École normale supérieure, Paris

pdf Splash singularty for the Water Waves problem

We exhibit smooth initial data for the 2D water wave equation for which we prove that smoothness of the interface breaks down in fi nite time into a splash singularity (self intersecting curve in one point). Moreover, we show a stability result together with numerical evidence that there exist solutions of the 2D water wave equation that start from a graph, turn over and collapse in a splash singularity in fi nite time.

#### 11.15-12.00

##### Thomas Alazard, École normale supérieure, Paris

pdf On the Cauchy problem for the water-waves equations

The water-waves problem consists in describing the motion, under the influence of gravity, of a fluid occupying a domain delimited below by a fixed bottom and above by a free surface. We consider the Cauchy theory for low regularity solutions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of $C_{3/2}$ class and consequently have unbounded curvature. Furthermore, no regularity assumption is assumed on the bottom. We also take benefit from an elementary observation to solve a question raised by Boussinesq on the water-wave equations in a canal.

Lunch

#### 13.30-14.15

##### Henrik Kalisch, University of Bergen

pdf Mechanical Balance Laws in the Boussinesq Scaling: Theory and Applications

The focus will be on a family of Boussinesq systems used for modeling of small-amplitude long waves in open channel flow. It will be shown how the reconstruction of the velocity field from the principal dependent variables of these equations yields information about properties of the associated fluid flow. We will discuss a few applications where this analysis can be used advantageously.

#### 14.15-15.00

##### Mariana Haragus, Université de Franche-Comté

pdf Existence of Defects in Swift-Hohenberg Equations

We show the existence of grain boundaries and dislocations in the classical Swift-Hohenberg equation and in an anisotropic Swift-Hohenberg equation, respectively. We find these defects as traveling waves connecting roll patterns with different wavenumbers. The analysis relies upon a spatial dynamics formulation of the bifurcation problem, a local center-manifold reduction, and normal form theory. We also discuss possible extensions and limitations of our approach.

Joint work with Arnd Scheel, University of Minnesota.

#### Guided tour

Starting in front of Lund cathedral we will be guided through the historical centre of the town.

#### Dinner

At Kulturen restaurant, located in connection to the open-air museum Kulturen, slightly to the north of Lund cathedral.

### Thursday

#### 09.15-10.00

##### Eugen Varvaruca, University of Reading

pdf Variational formulation for steady gravity waves with constant vorticity

A new variational formulation will be given for the problem of steady periodic gravity waves with constant vorticity in water of finite depth, generalizing that of Babenko for irrotational waves of infinite depth. I will also present some new existence results for waves of large amplitude, obtained by means of global bifurcation theory applied in the setting of the new formulation. This is joint work with Adrian Constantin (King's College London) and Walter Strauss (Brown University).

Coffee break

#### 10.30-11.15

pdf Stokes waves on vortical flows with counter-currents

The nonlinear problem of steady waves with vorticity is considered. The problem describes two-dimensional, rotational motion of an inviscid, incompressible and heavy fluid, say, water in a horizontal open channel of uniform rectangular cross-section. The aim is to investigate the bifurcation mechanism resulting in the formation of Stokes waves on the horizontal free surface of a parallel shear flow in which counter-currents may be present. These flows were studied in the joint paper with N. Kuznetsov [see QJMAM, 64 (2011)], where equations for their depths were derived. Here, the explicit conditions are presented that guarantee the existence of Stokes waves on a parallel shear flow. The general theorem is illustrated by several examples. This is a joint work with N. Kuznetsov, Russian Academy of Sciences, St Petersburg.

#### 11.15-12.00

##### Samuel Walsh, Courant Institute, New York University

pdf Steady water waves with compactly supported vorticity

In this talk, we shall discuss some recent results on the existence of two-dimensional, traveling, water waves with the special property that the vorticity is a Dirac measure (a point vortex), or supported in a compact set (a vortical patch). Such waves arise naturally, for instance, if we think of a classical, irrotational traveling wave with some interesting but localized dynamics below the surface (caused by the presence of a fish, say). This is joint work with C. Zeng and J. Shatah.

Lunch

#### 13.30-14.15

##### Nikolay G. Kuznetsov, Russian Academy of Sciences, St. Petersburg

pdf On integral properties of steady gravity waves on water of finite depth

In the mid-1970s, Longuet-Higgins initiated studies of integral properties of steady gravity waves on water of uniform depth (only solitary and Stokes waves were known at that time), and used these properties in his numerical computations of wave characteristics. An approach described here allows us to derive in a unified manner some integral property for all steady waves on water of finite depth. (Notice that periodic waves with more than one crest per period have been discovered by now, whereas Longuet-Higgins considered only solitary waves and Stokes waves on deep water.) A modified form of Bernoulli's equation is applied; this equation has many other applications, in particular, it provides necessary conditions for the existence of non-trivial waves. The main part of material is based on a joint work with Vladimir Kozlov, Linköping University, Sweden.

#### 14.15-15.00

##### Boris Buffoni, École Polytechnique Fédérale de Lausanne

pdf Travelling periodic surface waves: minimisation method in a given rearrangement class of vorticity and stability.

With G. R. Burton, we modify the approach of G. R. Burton and J. F. Toland to show the existence of periodic surface water waves with vorticity in order that it becomes suited to a stability analysis.

Travelling periodic waves are obtained by a direct minimisation under constraints of a functional that corresponds to the total energy. The constraints are on the circulation along the upper free boundary, the total horizontal impulse (the velocity becoming a Lagrange multiplier) and the rearrangement class of the vorticity.

By stability, we mean conditional energetic stability of the set of minimizers as a whole, the perturbations being spatially periodic of given period.

Coffee break

#### 15.30-16.15

##### Mats Ehrnström, Norwegian University of Science and Technology

In the setting of the two-dimensional Euler equations with a free surface, a flat bed, and otherwise periodic boundary conditions, we discuss the presence (and further possibility) of exact travelling gravity water waves consisting, on the linear level, of more than one Fourier mode. Such wave packets have surface profiles with several crests in each minimal period and have earlier been constructed, in the periodic and irrotational setting, in work by Toland and Jones; recent work together with J. Escher, G. Villari and E. Wahlén extend their presence to rotational currents. Using a Lyapunov–Schmidt reduction for the problem, kernels of the linearised problem with more than one dimension are constructed using vorticity distributions which allow for interior stagnation and so-called critical layers—fluid regions consisting solely of closed streamlines. We focus here on the analysis of the kernels and the pits and falls encountered when passing from the linear to the nonlinear level.

#### 16.15-17.00

##### Erik Wahlén, Lund University

pdf A dimension-breaking phenomenon for steady water waves with weak surface tension

Iooss and Kirchgässner proved that the two-dimensional water wave problem with weak surface tension admits two families of solitary wave solutions of envelope form. The solutions are to leading order described by the nonlinear Schrödinger equation. In this talk I will discuss how these waves generate families of three-dimensional periodically modulated solitary waves in a dimension-breaking bifurcation. The new solutions decay in the direction of propagation and are periodic in the transverse direction. They are related to the Davey-Stewartson equation. The proof is based on a reversible Hamiltonian spatial-dynamics formulation and an infinite-dimensional version of the Lyapunov centre theorem.

This is joint work with M. Groves and S.-M. Sun.