Partial differential equations and modelling contributions

We construct a smooth branch of travelling wave solutions for the 2 dimensional Gross-Pitaevskii equations for small speed. These travelling waves exhibit two vortices far away from each other. We also compute the leading order term of the derivatives with respect to the speed. We construct these solutions by an implicit function type argument. In collaboration with David Chiron

We consider the non linear wave equation (NLW) on the d-dimensional torus

$$u_{tt} - \Delta u + \mu + f(u) =0\quad x \in \mathbb T^d$$

where $f=\partial_u F$ is analytic on a neighborhood of the origin and which is at least of order 2 at the origin. Let $u(t)$ be a solution corresponding to a small initial datum $u(0)\in H^s(\mathbb T^d)$. We prove that we control $[u(t)]_s$ that mix the $H^s$ norm of the $\varepsilon^{-\beta(r)}$ lower Fourier modes of the solution $u$ and the energy norm of the remaining higher modes during long times of order $\varepsilon^{-r}$.

Our general strategy applies to any Hamiltonian PDEs whose linear frequencies satisfy only a first Melnikov condition. In particular it also applies to the Hamiltonian Boussinesq $abcd$ system and the Whitham-Boussinesq system in water waves theory. Joint work with Joackim Bernier and Erwan Faou.

The inviscid Burgers, Euler and Saint-Venant equations are nonlinear hyperbolic PDEs modeling fluid flows and surface water waves propagating in shallow water. These equations, prominent in physics, are the subject of numerous mathematical and numerical investigations. It is well-known that these equations develop shocks in finite time, even for regular initial conditions. These shocks are problematic, in particular, for numerical simulations. Therefore, several techniques have been proposed to regularized these equations. Adding viscosity or/and dispersion into the equations can avoid the formation of shocks. Here, we study a regularization of barotropic Euler equations, which conserves the energy, and generalize the conservative regularization of the Saint-Venant equations proposed by Clamond and Dutykh in 2017.

It is by now well known that the use of Carleman estimates allows to establish the controllability to trajectories of nonlinear parabolic equations. However, by this approach, it is not clear how to decide whether a given function is indeed reachable. That issue has obtained very recently almost sharp results in the linear case. In this talk, we investigate the set of reachable states for a nonlinear heat equation in dimension one. The nonlinear part is assumed to be an analytic function of the spatial variable $x$, the unknown $y$, and its derivative $y_x$.
By investigating carefully a nonlinear Cauchy problem in x in some space of Gevrey functions, and the relationship between the jet of space derivatives and the jet of time derivatives, we derive an exact controllability result for small initial and final data that can be extended as analytic functions on some ball of the complex plane. This is a joint work with Camille Laurent (Sorbonne Université). It time allows, works in progress about the reachable states for KdV and for ZK will be outlined.

In this talk, I will present a result on the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Next, I will illustrate this result with two examples: a nonlinear Schrödinger equation for a nucleon and a Schrödinger equation with a double power non-linearity. This talk is based on joint works with Mathieu Lewin.