Nonlinear Wave Equations

2-6 September, 2013

Instituto Superior Técnico

Lisboa, Portugal

Nonlinear Wave Equations

2-6 September, 2013

Instituto Superior Técnico

Lisboa, Portugal

Time/Day | Monday | Tuesday | Wednesday | Thursday | Friday |
---|---|---|---|---|---|

10:00-11:00 | P. Raphaël | E. Aubry | N. Tzvetkov | J.Drumond Silva | |

11:30-12:30 | C. Rigault | E. Delay | T. Boulenger | F. Planchon | |

Lunch Break | |||||

14:00-15:00 | R. Carles | M. Lemou | P. Girão | N. Godet | |

15:30-16:30 | O. Ivanovici | J. Smulevici | V. Banica | O. Pocovnicu |

**Erwann Aubry (Univ. Nice)**- Title:
- Abstract:

**Valeria Banica (Univ. Évry Val d'Essonne)**- Title:
- Dispersion for the Schrödinger equation on the line with multiple Dirac delta potentials
- Abstract:
- In this talk we consider the time dependent one-dimensional Schrödinger equation with multiple Dirac delta potentials. We prove that the classical dispersion property holds. The result is obtained in a more general setting of a Laplace operator on a tree with
*δ*-coupling conditions at the vertices. The proof relies on a careful analysis of the properties of the resolvent of the associated Hamiltonian. This is a joint work with Liviu Ignat.

**Thomas Boulenger (Univ. Basel)**- Title:
- Stability to metric and nonlinear perturbations of NLS pseudo-conformal blow up solutions
- Abstract:
- The present work aims at investigating the effects of a perturbation of the euclidean metric around blow up point on the 2-dimensional
*L*^{2}-critical focusing NLS :*i∂*+ Δ_{t}u_{g}*u*+*k*(*x*) |*u*|^{2}*u*= 0. We prove a stability result for the pseudo-conformal blow up regime, with respect to some geometrical perturbations, embodied by the metric*g*, and some nonlinear perturbations, embodied by the potential*k*. We use modulation techniques to treat perturbatively the additional terms, in the very same spirit as in a previous paper from Raphaël and Szeftel (http://arxiv.org/abs/1001.1627, 2010). We will present a quick overview of the adapted method to derive existence and uniqueness.

**Rémi Carles (Univ. Montpellier 2)**- Title:
- Scattering for NLS with partial harmonic confinement
- Abstract:
- We consider the nonlinear Schrodinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates. We infer the existence of wave operators. Asymptotic completeness stems from suitable Morawetz estimates, which in turn follow from the latest approach, applied to a marginal of the position density. This is a joint work with Paolo Antonelli and Jorge Drumond Silva.

**Erwann Delay (Univ. Avignon)**- Title:
- Localized gluing of initial data set for the vacuum Einstein equations, applications and generalizations
- Abstract:
- A method originally due to Corvino and Schoen allows to glue locally two initial data sets for the vacuum Einstein equations. We will explain this technique in the time symmetric case for simplicity, it correspond to glue two Riemannian metrics of zero scalar curvature. We will also see the multiple applications and generalizations which have been done until now.

**Pedro Girão (I.S.T., Lisbon)**- Title:
- Bifurcation curves of a diffusive logistic equation with harvesting orthogonal to the first eigenfunction
- Abstract:
- We study the global bifurcation curves of a diffusive logistic equation, when harvesting is orthogonal to the first eigenfunction of the Laplacian, for values of the linear growth up to
*λ*_{2}+*δ*, examining in detail their behavior as the linear growth rate crosses the first two eigenvalues. We observe some new behavior with regard to earlier works concerning this equation. Namely, the bifurcation curves suffer a transformation at*λ*_{1}, they are compact above*λ*_{1}, there are precisely two families of degenerate solutions with Morse index equal to zero, and the whole set of solutions below*λ*_{2}is not a two dimensional manifold. This is joint work with Mayte Pérez-Llanos.

**Nicolas Godet (Univ. Nice)**- Title:
- Abstract:

**Oana Ivanovici (Univ. Nice)**- Title:
- Strichartz estimates for the wave equation inside strictly convex domains
- Abstract:
- We will review recent results on dispersive estimates for the wave equation in the Friedlander model case of a strictly convex domain. In particular, we extend previous work on the dispersive estimate to average over singularities and get optimal Strichartz estimates. This is joint work with G. Lebeau and F. Planchon.

**Mohammed Lemou (Univ. Rennes 1)**- Title:
- Extended Hardy-Littlewood inequalities for rearrangements and some quantitative stability results
- Abstract:
- The main goal here is to prove a new extended Hardy-Littlewood type inequality for generalized rearrangement of functions. This inequality yields quantitative stability results for some dynamical systems that essentially have the important property to preserve the rearrangement and the Hamiltonian. In particular we derive a quantitative stability result for a large class of steady state solutions to the gravitational Vlasov-Poisson systems; more precisely we show that this type of inequality implies a quantitative control of the
*L*^{1}norm of the perturbation by the Hamiltonian (the energy functional). We then review the main arguments in the proof of the non linear stability of these steady state solutions that we have recently obtained, and show how the compactness arguments on which this proof was based can be replaced by a quantitative control directly derived from this extended Hardy-Littlewood inequality. Some other known stability results are also covered by this inequality: e.g. the relativistic Vlasov-Poisson and 2D-Euler systems.

**Fabrice Planchon (Univ. Nice)**- Title:
- Geometric uniqueness for the Cauchy problem in General Relativity
- Abstract:
- Solutions to the Einstein equations are, by default, unique in a reduced form, where a gauge has been (conveniently) chosen. Changing gauge involves losing regularity. We will review how to overcome this difficulty by using the geometric structure of the equations (this is a series of works by Igor Rodnianski-FP, David Parlongue and the 3 of us).

**Oana Pocovnicu (Princeton Univ.)**- Title:
- Scattering for a cubic-quintic nonlinear Schrödinger equation on R
^{3} - Abstract:
- In this talk, we consider the cubic-quintic nonlinear Schrödinger equation (NLS) on R
^{3}whose nonlinearity is given by the sum of a defocusing quintic power and a focusing cubic power. Notice that the quintic nonlinearity is energy-critical on R^{3}, while the cubic one is energy-subcritical. We consider the issue of scattering for the cubic-quintic NLS. - Previously, Tao, Visan, and Zhang proved global well-posedness in
*H*^{1}and scattering for solutions with small mass. The main result in this talk is scattering for solutions that are situated below a branch of (rescaled) solitons in the mass-energy plane. Our result shows that while all solutions exist globally in time, the region of scattering is bounded by (rescaled) solitons, namely the situation is very different from the energy-critical defocusing/focusing quintic NLS. Our proof combines variational arguments and concentration-compactness and rigidity arguments. This is joint work with Rowan Killip (UCLA), Tadahiro Oh (University of Edinburgh), and Monica Visan (UCLA).

**Pierre Raphaël (Univ. Nice)**- Title:
- On the degenerate Patlak-Keller-Segel model
- Abstract:
- I will consider the classical Patlak-Keller-Segel model of chemotactic aggregation of bacteria in biology. This critical nonlinear diffusion equation has attracted a considerable attention for the past ten years in particular in connection with the possibility of singularity formation for this type of model. In this talk, I will first reformulate the problem within a larger class of both dispersive and parabolic blow up problems. I will then revisit the seminal construction of Herrero, Velasquez (1996) and prove the existence and stability of a blow up mechanism, together with a complete description of the associated blow up bubble.

**Cyril Rigault (Univ. Rennes 1)**- Title:
- The Vlasov-Manev system: stability of ground states and self-similar blow-up solutions
- Abstract:
- The gravitational Vlasov-Manev system is a kinetic model which has a similar Vlasov structure as the classical Vlasov-Poisson system, but is coupled to a potential in -1/r - 1/r^2 (Manev potential) instead of the usual gravitational potential in -1/r, and in particular the potential field does not satisfy a Poisson equation but a fractional- Laplacian equation. This modification of the potential is a relativistic correction, which can be seen as a alternative to the classical relativistic model. I will first talk about the orbital stability of the ground states type solutions which are constructed as minimizers of the Hamiltonian, following the classical strategy: compactness of the minimizing sequences and the rigidity of the flow. However, in driving this analysis, there are two mathematical obstacles: the first one is related to the possible blow-up of solutions to the VM system, which we overcome by imposing a sub-critical condition on the constraints of the variational problem. The second difficulty (and the most important) is related to the nature of the Euler-Lagrange equations (fractional-Laplacian equations) to which classical results for the Poisson equation do not extend. We overcome this difficulty by proving the uniqueness of the minimizer under equimeasurabilty constraints, using only the regularity of the potential and not the fractional- Laplacian Euler-Lagrange equations itself. In the second part of my talk, I will give the strategy of the proof of the existence of exact self-similar blow-up solutions to the Vlasov-Manev equation in the "pur case", with initial data arbitrarily close to ground states. This construction is based on a suitable variational problem with equimeasurability constraint. This is joint work with M. Lemou and F. Méhats.

**Jorge Drumond Silva (I.S.T., Lisbon)**- Title:
- Nonlinear Schrödinger equation with time dependent potential: large time properties
- Abstract:
- We present recent results on large time behaviour of solutions to the nonlinear Schrödinger equation with time dependent external potential and defocusing nonlinearity. The potential is assumed to grow at most quadratically in space, uniformly for all time, for which a typical example is a (possibly anisotropic) harmonic potential with bounded coefficients in time.
- We start by presenting a global in time well posedness result without further assumptions on the potential, with a general exponential growth control of its first order derivatives and momenta. As a consequence, these yield a double exponential growth rate of the higher Sobolev norms and momenta. We also show that if the potential is harmonic and isotropic, with coefficients decaying sufficiently fast in time, then there is scatteting, the Sobolev norms remain bounded and momenta grow polinomially in time.
- This is joint work with Rémi Carles.

**Jacques Smulevici (Univ. Paris-Sud, Orsay)**- Title:
- Symmetric solutions to the Einstein equations with weak regularity
- Abstract:
- After an introduction to the Einstein equations, I will present joint work with Philippe G. LeFloch concerning the study of so-called T2-symmetric solutions to these equations, under weak regularity assumptions. Our main results are local and global existence in the so-called areal gauge as well as a description of the asymptotics of the solutions in the case of T3-Gowdy solutions.

**Nikolay Tzetkov (Univ. Cergy-Pontoise)**- Title:
- Multi-solitons for the water-waves system
- Abstract:
- We will present the construction of multi-solitons solutions (that is to say solutions that are time asymptotics to a sum of decoupling solitary waves) for the full water waves system with surface tension. This is a generalization of similar results obtained for strongly simplified model equations such as the KdV equation. The construction is quite different compared with the one solitons obtained in the classical work by Amick-Kirchgassner. The talk is based on a joint work with Mei Ming and Frederic Rousset.