Publications (16)0.89 Total impact
-
Article: Quintic NLS in the exterior of a strictly convex obstacle
[show abstract] [hide abstract]
ABSTRACT: We consider the defocusing energy-critical nonlinear Schr\"odinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions. For the initial-value problem with Dirichlet boundary condition we prove global well-posedness and scattering for all initial data in the energy space.08/2012; -
Article: Harmonic analysis outside a convex obstacle
[show abstract] [hide abstract]
ABSTRACT: The goal of this paper is to develop some basic harmonic analysis tools for the Dirichlet Laplacian in the exterior domain associated to a smooth convex obstacle in dimensions d\geq 3. The principle question addressed is the equivalence of Sobolev norms defined with respect to the Dirichlet and whole-space Laplacians. When equivalence holds, this reduces many important results, such as the fractional product and chain rules for the Dirichlet Laplacian, to the classical setting. Counterexamples are included to show that equivalence does not always hold. Indeed, in the case of spaces with one derivative, these examples show that our results are sharp (up to the endpoint). The choice of topics is primarily influenced by what is needed for the investigation of the energy-critical nonlinear Schr\"odinger equation in such geometries. Applications in this direction are discussed. The results in this paper play an important role in the authors' proof of large-data global well-posedness and scattering for the energy-critical NLS in three dimensional exterior domains; see arXiv:1208:4904.05/2012; -
Article: Blowup behaviour for the nonlinear Klein--Gordon equation
[show abstract] [hide abstract]
ABSTRACT: We analyze the blowup behaviour of solutions to the focusing nonlinear Klein--Gordon equation in spatial dimensions $d\geq 2$. We obtain upper bounds on the blowup rate, both globally in space and in light cones. The results are sharp in the conformal and sub-conformal cases. The argument relies on Lyapunov functionals derived from the dilation identity. We also prove that the critical Sobolev norm diverges near the blowup time.03/2012; -
Article: Global well-posedness of the Gross--Pitaevskii and cubic-quintic nonlinear Schr\"odinger equations with non-vanishing boundary conditions
[show abstract] [hide abstract]
ABSTRACT: We consider the Gross--Pitaevskii equation on $\R^4$ and the cubic-quintic nonlinear Schr\"odinger equation (NLS) on $\R^3$ with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the energy-critical NLS, we prove that they are globally well-posed in their energy spaces. In particular, we prove unconditional uniqueness in the energy spaces for these equations.12/2011; -
Article: Smooth solutions to the nonlinear wave equation can blow up on Cantor sets
[show abstract] [hide abstract]
ABSTRACT: We construct $C^\infty$ solutions to the one-dimensional nonlinear wave equation $$ u_{tt} - u_{xx} - \tfrac{2(p+2)}{p^2} |u|^p u=0 \quad \text{with} \quad p>0 $$ that blow up on any prescribed uniformly space-like $C^\infty$ hypersurface. As a corollary, we show that smooth solutions can blow up (at the first instant) on an arbitrary compact set. We also construct solutions that blow up on general space-like $C^k$ hypersurfaces, but only when $4/p$ is not an integer and $k > (3p+4)/p$.03/2011; -
Article: Global well-posedness and scattering for the defocusing quintic NLS in three dimensions
[show abstract] [hide abstract]
ABSTRACT: We revisit the proof of global well-posedness and scattering for the defocusing energy-critical NLS in three space dimensions in light of recent developments. This result was obtained previously by Colliander, Keel, Staffilani, Takaoka, and Tao.02/2011; -
Article: Scattering for the cubic Klein--Gordon equation in two space dimensions
[show abstract] [hide abstract]
ABSTRACT: We consider both the defocusing and focusing cubic nonlinear Klein--Gordon equations $$ u_{tt} - \Delta u + u \pm u^3 =0 $$ in two space dimensions for real-valued initial data $u(0)\in H^1_x$ and $u_t(0)\in L^2_x$. We show that in the defocusing case, solutions are global and have finite global $L^4_{t,x}$ spacetime bounds. In the focusing case, we characterize the dichotomy between this behaviour and blowup for initial data with energy less than that of the ground state. These results rely on analogous statements for the two-dimensional cubic nonlinear Schr\"odinger equation, which are known in the defocusing case and for spherically-symmetric initial data in the focusing case. Thus, our results are mostly unconditional. It was previously shown by Nakanishi that spacetime bounds for Klein--Gordon equations imply the same for nonlinear Schr\"odinger equations.08/2010; -
Article: Energy-Supercritical NLS: Critical [Hdot] s -Bounds Imply Scattering
[show abstract] [hide abstract]
ABSTRACT: We consider two classes of defocusing energy-supercritical nonlinear Schrödinger equations in dimensions d ≥ 5. We prove that if the solution u is a priori bounded in the critical Sobolev space, that is, , then u is global and scatters.Communications in Partial Differential Equations 06/2010; 35(6):945-987. · 0.89 Impact Factor -
Article: The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions
[show abstract] [hide abstract]
ABSTRACT: We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2}\frac4{d-2}$. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.02/2010; -
Article: The defocusing energy-supercritical nonlinear wave equation in three space dimensions
[show abstract] [hide abstract]
ABSTRACT: We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ in the energy-supercritical regime p>4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.01/2010; -
Article: On the mass-critical generalized KdV equation
[show abstract] [hide abstract]
ABSTRACT: We consider the mass-critical generalized Korteweg--de Vries equation $$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$ for real-valued functions $u(t,x)$. We prove that if the global well-posedness and scattering conjecture for this equation failed, then, conditional on a positive answer to the global well-posedness and scattering conjecture for the mass-critical nonlinear Schr\"odinger equation $(-i\partial_t + \partial_{xx})u=\pm (|u|^4u)$, there exists a minimal-mass blowup solution to the mass-critical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we can guarantee that this minimal-mass blowup solution is either a self-similar solution, a soliton-like solution, or a double high-to-low frequency cascade solution. Comment: References added/updated07/2009; -
Article: Energy-supercritical NLS: critical $\dot H^s$-bounds imply scattering
[show abstract] [hide abstract]
ABSTRACT: We consider two classes of defocusing energy-supercritical nonlinear Schr\"odinger equations in dimensions $d\geq 5$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty \dot H^{s_c}_x$, then $u$ is global and scatters.01/2009; -
Article: Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS
[show abstract] [hide abstract]
ABSTRACT: Let $d\geq 4$ and let $u$ be a global solution to the focusing mass-critical nonlinear Schr\"odinger equation $iu_t+\Delta u=-|u|^{\frac 4d}u$ with spherically symmetric $H_x^1$ initial data and mass equal to that of the ground state $Q$. We prove that if $u$ does not scatter then, up to phase rotation and scaling, $u$ is the solitary wave $e^{it}Q$. Combining this result with that of Merle \cite{merle2}, we obtain that in dimensions $d\geq 4$, the only spherically symmetric minimal-mass blowup solutions are, up to phase rotation and scaling, the pseudo-conformal ground state and the solitary wave.05/2008; -
Article: The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher
[show abstract] [hide abstract]
ABSTRACT: We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^{4/d} u$ for large spherically symmetric L^2_x(R^d) initial data in dimensions $d\geq 3$. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.09/2007; -
Article: The cubic nonlinear Schr\"odinger equation in two dimensions with radial data
[show abstract] [hide abstract]
ABSTRACT: We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry.08/2007; -
Article: Energy-critical NLS with quadratic potentials
[show abstract] [hide abstract]
ABSTRACT: We consider the defocusing $\dot H^1$-critical nonlinear Schr\"odinger equation in all dimensions ($n\geq 3$) with a quadratic potential $V(x)=\pm \tfrac12 |x|^2$. We show global well-posedness for radial initial data obeying $\nabla u_0(x), xu_0(x) \in L^2$. In view of the potential $V$, this is the natural energy space. In the repulsive case, we also prove scattering. We follow the approach pioneered by Bourgain and Tao in the case of no potential; indeed, we include a proof of their results that incorporates a couple of simplifications discovered while treating the problem with quadratic potential. Comment: Incorporates corrections to Lemma 6.511/2006;
Top co-authors
Top Journals
Institutions
-
2010
-
University of California, Los Angeles
- Department of Mathematics
Los Angeles, CA, USA
-
