arXiv subjects

Thierry Cazenave

Publications and source records attributed to Thierry Cazenave.

21 records · Page 2Linked to original sources

arXiv2012

Finite-time blowup for a complex Ginzburg-Landau equation

We prove that negative energy solutions of the complex Ginzburg-Landau equation $e^{-iθ} u_t = Δu+ |u|^α u$ blow up in finite time, where α>0 and π/2<θ<π/2. For a fixed initial value $u(0)$, we obtain estimates of the blow-up time $T_{max}^θ$ as $θ\to \pm π/2 $. It turns out that $T_{max}^θ$ stays bounded (respectively, goes to infinity) as $θ\to \pm π/2 $ in the case where the solution of the limiting nonlinear Schrödinger equation blows up in finite time (respectively, is global).

math.AP↗

arXiv2011

Scattering for the focusing energy-subcritical NLS

For the 3D focusing cubic nonlinear Schrodinger equation, Scattering of $H^1$ solutions inside the (scale invariant) potential well was established by Holmer and Roudenko~\cite{HR2} (radial case) and Duyckaerts, Holmer and Roudenko~\cite{DHR} (general case). In this paper, we extend this result to arbitrary space dimensions and focusing, mass-supercritical and energy-subcritical power nonlinearities, by adapting the method of~\cite{DHR}.

math.AP↗

arXiv2010

Continuous dependence for NLS in fractional order spaces

We consider the Cauchy problem for the nonlinear Schrödinger equation $iu_t+ Δu+ λ|u|^αu=0$ in $\R^N $, in the $H^s$-subcritical and critical cases $0<α\le 4/(N-2s)$, where $0<s<N/2$. Local existence of solutions in $H^s$ is well known. However, even though the solution is constructed by a fixed-point technique, continuous dependence in $H^s$ does not follow from the contraction mapping argument. In this paper, assuming furthermore $s<1$, we show that the solution depends continuously on the initial value in the sense that the local flow is continuous $H^s \to H^s$. If, in addition, $α\ge 1$ then the flow is Lipschitz. This completes previously known results concerning the cases $s=0,1,2$.

math.AP↗