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 Fabrice Planchon Mathematics , 2012, DOI: 10.1017/S1474748013000017 Abstract: We prove bilinear estimates for the Schr\"odinger equation on 3D domains, with Dirichlet boundary conditions. On non-trapping domains, they match the $\mathbb{R}^3$ case, while on bounded domains they match the generic boundary less manifold case. As an application, we obtain global well-posedness for the defocusing cubic NLS for data in $H^s_0(\Omega)$, $1  Qing Guo Mathematics , 2011, Abstract: We investigate the nonlinear Schr\"{o}dinger equation$iu_{t}+\Delta u+|u|^{p-1}u=0$with$1+\frac{4}{N}\|Q\|_{2}^{\frac{1-s_{c}}{s_{c}}}\|\nabla Q\|_{2},$then either$u(t)$~blows up in finite forward time, or$u(t)$exists globally for positive time and there exists a time sequence$t_{n}\rightarrow+\infty$such that$\|\nabla u(t_{n})\|_{2}\rightarrow+\infty.$Here$Q$is the ground state solution of$-Q+\Delta Q+|Q|^{p-1}Q=0.$A similar result holds for negative time. This extend the result of the 3D cubic Schr\"{o}dinger equation in \cite{holmer10} to the general mass-supercritical and energy-subcritical case .  Mathematics , 2014, Abstract: We consider the energy super critical nonlinear Schr\"odinger equation $$i\pa_tu+\Delta u+u|u|^{p-1}=0$$ in large dimensions$d\geq 11$with spherically symmetric data. For all$p>p(d)$large enough, in particular in the super critical regime, we construct a family of smooth finite time blow up solutions which become singular via concentration of a universal profile with the so called type II quantized blow up rates. The essential feature of these solutions is that all norms below scaling remain bounded. Our analysis fully revisits the construction of type II blow up solutions for the corresponding heat equation, which was done using maximum principle techniques following. Instead we develop a robust energy method, in continuation of the works in the energy and mass critical cases. This shades a new light on the essential role played by the solitary wave and its tail in the type II blow up mechanism, and the universality of the corresponding singularity formation in both energy critical and super critical regimes.  Mathematics , 2013, Abstract: We consider a class of defocusing energy-supercritical nonlinear Schr\"odinger equations in four space dimensions. Following a concentration-compactness approach, we show that for$1
 Mathematics , 2009, Abstract: Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as $t\to +\infty$, were constructed in previous works for the L2 critical and subcritical (NLS) and (gKdV) equations. In this paper, we extend the construction of multi-soliton solutions to the L2 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.
 Mathematics , 2015, Abstract: We consider the focusing cubic NLS in the exterior $\Omega$ of a smooth, compact, strictly convex obstacle in three dimensions. We prove that the threshold for global existence and scattering is the same as for the problem posed on Euclidean space. Specifically, we prove that if $E(u_0)M(u_0)  Mathematics , 2010, Abstract: Shearlet systems have so far been only considered as a means to analyze$L^2$-functions defined on$\R^2\$, which exhibit curvilinear singularities. However, in applications such as image processing or numerical solvers of partial differential equations the function to be analyzed or efficiently encoded is typically defined on a non-rectangular shaped bounded domain. Motivated by these applications, in this paper, we first introduce a novel model for cartoon-like images defined on a bounded domain. We then prove that compactly supported shearlet frames satisfying some weak decay and smoothness conditions, when orthogonally projected onto the bounded domain, do provide (almost) optimally sparse approximations of elements belonging to this model class.