## Periodic Reporting for period 3 - ANADEL (Analysis of Geometrical Effects on Dispersive Equations)

Reporting period: 2021-02-01 to 2022-07-31

o Objective 1 : Wave and / or Schrödinger equations in exterior domains (i.e. exterior of a convex - not necessary strict, a (weakly) non-trapping domain, etc) ; in this case we want to understand: sharp dispersive / Strichartz estimates; diffractive effects ;

o Objective 2 : Wave and / or Schrödinger equations in bounded domains (inside a convex - not necessary strict, or inside a general bounded domain (with points on the boundary where the curvature changes its sign) ; in this case we want to understand : sharp dispersive / Strichartz estimates (in semi-classical time for Schrödinger, spectral projector bounds; concentration of eignemodes, etc) ;

o Objective 3 : Exponential sums estimates : so far, number theoretic methods have been a key tool for developing PDEs on compact manifolds. Any (small) progress on the above questions would make progress on analytic number theory from a PDE perspective...

Besides, some of the techniques we develop (on Objectives 1 and 2) may yield advances for dispersive equations on domains without boundary.

Advances on the first two objectives have been achieved during the first reporting period.

The PI had spent an important amount of time on Objective 3 also, and will continue to do so (as any small progress on this issues would represent a huge achievement).

[6], STRICHARTZ ESTIMATES FOR THE WAVE EQUATION ON A 2D MODEL CONVEX DOMAIN, by O.Ivanovici G. Lebeau and F.Planchon

We prove sharper Strichartz estimates than expected from the optimal dispersion bounds. This follows from taking full advantage of the space-time localization of caustics. Several

improvements on the parametrix construction of our previous work are obtained along the way and are of independent interest. This is related to Objective 2.

[7]. NEW COUNTEREXAMPLES TO STRICHARTZ ESTIMATES FOR THE WAVE EQUATION ON A 2D MODEL CONVEX DOMAIN, by O.Ivanovici G. Lebeau and F.Planchon

We prove that the range of Strichartz estimates on a model 2D convex domain may be further restricted compared to the known counterexamples from previous works by O.Ivanovici. Our new family of counterexamples is built on the parametrix construction from [6]. Interestingly enough, it is sharp in at least some regions of phase space. This is related to Objective 2.

[8]. DISPERSION FOR THE WAVE EQUATION INSIDE STRICTLY CONVEX DOMAINS II: THE GENERAL CASE, by O.Ivanovici R.Lascar G. Lebeau and F.Planchon

We consider the wave equation on a strictly convex domain of dimension d >1 with smooth boundary and with Dirichlet boundary conditions. We construct a sharp local in time parametrix and then proceed to obtain dispersion estimates: our fixed time decay rate for the Green function exhibits a 1/4 loss with respect to the boundary less case. Moreover, we precisely

describe where and when these losses occur and relate them to swallowtail type singularities in the wave front set, proving that the resulting decay is optimal.

[9]. DISPERSIVE ESTIMATES FOR THE SEMI-CLASSICAL SCHRÖDINGER EQUATION IN A STRICTLY CONVEX DOMAIN, by O.Ivanovici

We consider a model case for a strictly convex domain of dimension d >1 with smooth boundary and we describe dispersion for the semi-classical Schrödinger equation with Dirichlet boundary condition. More specifically, we obtain the optimal fixed time decay rate for the linear semi-classical flow : a loss of 1/4 occurs with respect to the boundary less case due to repeated swallowtail type singularities.

[10]. DISPERSIVE ESTIMATES INSIDE THE FRIEDLANDER DOMAIN FOR THE KLEIN-GORDON EQUATION AND THE WAVE EQUATION IN LARGE TIME, by O.Ivanovici

We prove dispersive estimates for the wave equation in large time and for the Klein-Gordon equation inside a convex domain. This follows from taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at a given, large time. This is related to Objective 2.

[5].Transport of gaussian measures by the flow of the nonlinear Schrödinger equation, by F. Planchon, N. Tzvetkov, N. Visciglia

We prove a new smoothing type property for solutions of the 1d quintic Schrödinger equation.

[1]. A non-linear Egorov theorem and Poincaré-Birkhoff normal forms for quasi-linear pdes on the circle, by Roberto Feola, Felice Iandoli

We consider an abstract class of quasi-linear para-differential equations on the circle. For each equation in the class we prove the existence of a change of coordinates which conjugates the equation to a diagonal and constant coefficient para-differential equation.

[2]. Long time existence for fully nonlinear NLS with small Cauchy data on the circle, by Roberto Feola, Felice Iandoli

We prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrödinger equations on the one dimensional torus.

[3].Local well-posedness for the quasi-linear Hamiltonian Schrödinger equation on tori, by Roberto Feola, Felice Iandoli

We prove a local in time well-posedness result for quasi-linear Hamiltonian Schrödinger equations on the d-dimensional torus for any d. For any initial condition in the Sobolev space H^s, with s large, we prove the existence and unicity of classical solutions of the Cauchy problem associated to the equation.

[4].Long time solutions for quasi-linear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori, by Roberto Feola, Benoit Grébert and Felice Iandoli

We consider quasi-linear, Hamiltonian perturbations of the cubic Schrödinger on the d dimensional torus. We prove that the lifespan of solutions is strictly larger than the local existence time.

Works in progress

[i] DISPERSION FOR THE WAVE AND THE SCHR7fODINGER EQUATIONS OUTSIDE A BALL AND COUNTEREXAMPLES, by O.Ivanovici and G.Lebeau

We consider the linear wave equation and the linear Schrodinger equation outside a ball in R^d. In dimension d = 3 we show that the linear wave flow and the linear Schrodinger

floow satisfy the dispersive estimates as in R3. For d >3 we show that there exists points where the dispersive estimates fail for both wave and Schr7fodinger equations.

[ii].DISPERSIVE ESTIMATES FOR THE WAVE EQUATION OUTSIDE OF A CYLINDER, by O.Ivanovici and F.Iandoli

We prove sharp dispersive for the wave equation with Dirichlet boundary conditions outside a cylinder . This is related to Objective 1.

[iii]. Dispersive estimates for the wave equation outside two balls in R^3, by O.Ivanovici and D.Lafontaine

We investigate dispersion for the wave equation with Dirichlet boundary conditions outside two balls in R^3; this is one of the simplest examples of (weak-) trapping geometry. Objective 1

Parametrix construction for the wave equation in a general convex domain

sharp dispersion estimates for the wave equation on a model convex domain

dispersion estimates for Schrodinger equation on convex domains

long time behavior of nonlinear Schrodinger equations on compact manifolds

Expected results:

sharp propagation of singularities inside convex domains

dispersive estimates on general domains (no convexity assumption)

exponential sum estimates