In this paper, we prove that the defocusing mass-critical Hartree equation is global well-posed and scatters for any initial data $u_0\in L^2$, and prove the analogous result in the focusing case, provided the mass of $u_0$ is strictly less than that of the ground state $Q$. This result removes the radial restriction from [C. Miao, G. Xu and L. Zhao, J. Math. Pures Appl. (9), 91(2009), 49-79.], thereby proving the scattering conjecture for the Hartree equation at the scaling-critical regularity.
We prove global well-posedness for the focusing quintic nonlinear Schr\"odinger equation on the standard three-dimensional torus for initial data below the mass-corrected energy and Sobolev-norm thresholds determined by the Euclidean ground state. This gives a subthreshold focusing counterpart to the defocusing theorem...
In this paper, we study the focusing mass-critical Hartree equation \[ iu_t+\Delta u=-\bigl(|x|^{-2}*|u|^2\bigr)u \] at the ground-state mass $M(Q)$. We prove that, up to the symmetries of the equation, every global minimal-mass non-scattering solution is a solitary wave, while every finite-time minimal-mass blowup sol...
We prove that the ground state of the mass-critical Hartree equation is unique up to phase rotations and translations and nondegenerate in every dimension $d\ge3$. This makes the recent conditional $L^2$ classification of minimal-mass blow-up and non-scattering solutions unconditional; we also give a direct $H^1$ proof...
In this paper, we prove global well-posedness and scattering below the energy space for the defocusing nonlinear Schr\"odinger equation with intercritical power nonlinearities in spatial dimensions $d\geq3$, improving the regularity threshold of Vi\c{s}an and Zhang [Differ. Integr. Equations, 22(2009), 99-124].
Zu-Yu Ma, Changxing Miao, Yi-Lin Song et al.· 0 citations
We prove global well-posedness and modified scattering for the three-dimensional Hartree equation for small data in $H^{0,k}$, with $k>\frac12$ on $\mathbb R^3$ and $k=1$ on the half-space $\mathbb R^3_+$ with a real Robin boundary condition. For these data, the nonlinear and asymptotic analysis is carried out entirely...
In this short note, we prove the global well-posedness of the modified dispersion-generalized Benjamin-Ono equation with dispersion $1<\alpha<2$, $$\partial_t u + D^\alpha \partial_x u + u^2\partial_x u=0,$$ over the Sobolev space $H^s(\mathbb{R})$, for any regularity $s$ greater or equal than the sharp analytic well-p...
Mikaela Baldasso, Simão Correia, Thyago S. R. Santos· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
